From a5fa6b559ad98d360731a011d7ecbca1d136a0cb Mon Sep 17 00:00:00 2001 From: David Austin Date: Fri, 10 Jul 2026 19:57:00 -0400 Subject: [PATCH 1/6] =?UTF-8?q?Add=20solutions=20to=20Chapter=201=20activi?= =?UTF-8?q?ties=20and=20previews=20(sections=201.2=E2=80=931.9)?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit Co-Authored-By: Claude Sonnet 4.6 --- publication/publication.ptx | 2 +- .../activities/act-changing-aroc-graphs.xml | 32 +++++++-- .../act-changing-aroc-population.xml | 53 +++++++++++++-- .../activities/act-changing-aroc-trends.xml | 41 ++++++++++-- .../act-changing-combining-arithmetic.xml | 51 ++++++++++++-- .../act-changing-combining-context.xml | 42 ++++++++++-- .../act-changing-combining-piecewise.xml | 32 ++++++++- .../act-changing-composite-aroc.xml | 34 ++++++++-- ...ct-changing-composite-crickets-celsius.xml | 22 +++++- .../act-changing-composite-tables-graphs.xml | 57 +++++++++++++--- .../act-changing-functions-is-it.xml | 36 ++++++++-- ...ging-functions-spherical-tank-draining.xml | 55 +++++++++++++-- .../act-changing-functions-spherical-tank.xml | 52 ++++++++++++-- .../act-changing-inverse-Dolbear.xml | 33 +++++++-- .../act-changing-inverse-does-it.xml | 43 ++++++++++-- .../act-changing-inverse-rainfall.xml | 50 ++++++++++++-- .../act-changing-linear-Kilimanjaro.xml | 43 ++++++++++-- .../act-changing-linear-finding-eqs.xml | 46 +++++++++++-- .../act-changing-linear-in-context.xml | 41 ++++++++++-- .../act-changing-quadratic-falling-ball.xml | 46 +++++++++++-- .../act-changing-quadratic-parameters.xml | 45 +++++++++++-- .../act-changing-quadratic-properties.xml | 51 ++++++++++++-- .../act-changing-transformations-combined.xml | 37 ++++++++-- ...-changing-transformations-translations.xml | 34 +++++++++- ...-changing-transformations-vert-stretch.xml | 40 +++++++++-- source/previews/PA-changing-aroc.xml | 67 +++++++++++++++++-- source/previews/PA-changing-combining.xml | 48 +++++++++++-- source/previews/PA-changing-composite.xml | 32 +++++++-- source/previews/PA-changing-inverse-F-C.xml | 40 +++++++++-- source/previews/PA-changing-linear-3-ex.xml | 50 ++++++++++++-- source/previews/PA-changing-quadratic.xml | 37 ++++++++-- .../PA-changing-transformations-quadratic.xml | 34 ++++++++-- 32 files changed, 1176 insertions(+), 150 deletions(-) diff --git a/publication/publication.ptx b/publication/publication.ptx index 6fcbe811..01f7f032 100644 --- a/publication/publication.ptx +++ b/publication/publication.ptx @@ -20,7 +20,7 @@ - + diff --git a/source/activities/act-changing-aroc-graphs.xml b/source/activities/act-changing-aroc-graphs.xml index cc8c872d..f2a65a85 100755 --- a/source/activities/act-changing-aroc-graphs.xml +++ b/source/activities/act-changing-aroc-graphs.xml @@ -21,7 +21,7 @@ -

f is a function defined on [-1,7] such that f(1) = 4 and AV_{[1,3]} = -2. +

f is a function defined on [-1,7] such that f(1) = 4 and AV_{[1,3]} = -2.

ADD ALT TEXT TO THIS IMAGE

@@ -30,7 +30,15 @@
- + +

+ Since AV_{[1,3]} = -2 and f(1) = 4, we need + \frac{f(3)-4}{3-1} = -2, so f(3) = 0. Any graph passing through + (1,4) and (3,0) satisfies the conditions. Two examples: + the line f(x) = -2x+6, or a curve such as f(x) = -(x-1)^2 + 4 + (which also passes through both required points). +

+
@@ -43,7 +51,15 @@ - + +

+ Since AV_{[0,4]} = 0.5 and g(4) = 3, we need + \frac{3 - g(0)}{4} = 0.5, so g(0) = 1. Any graph passing through + (0,1) and (4,3) that is not always increasing on (0,4) + satisfies the conditions. For example, a curve that dips below the secant line + between x=0 and x=4 before returning to (4,3). +

+
@@ -56,7 +72,15 @@ - + +

+ These conditions are impossible to satisfy simultaneously. Given h(2) = 5 + and h(4) = 3, the average rate of change is determined: + AV_{[2,4]} = \frac{h(4) - h(2)}{4-2} = \frac{3-5}{2} = -1 \ne -2. + No function can have both the specified output values and the specified average + rate of change on [2,4]. +

+

diff --git a/source/activities/act-changing-aroc-population.xml b/source/activities/act-changing-aroc-population.xml index 0b76d780..87f0a5a3 100755 --- a/source/activities/act-changing-aroc-population.xml +++ b/source/activities/act-changing-aroc-population.xml @@ -79,7 +79,24 @@ - + +

+ For Kent County, the population in 1990 was 500,631 and in 2010 was 602,622, so + + AV_{[1990,2010]} &= \frac{602{,}622 - 500{,}631}{2010 - 1990} + &= \frac{101{,}991}{20} + &\approx 5099.55 \text{ people per year.} + +

+

+ For Ottawa County, the population in 1990 was 187,768 and in 2010 was 263,801, so + + AV_{[1990,2010]} &= \frac{263{,}801 - 187{,}768}{2010 - 1990} + &= \frac{76{,}033}{20} + &\approx 3801.65 \text{ people per year.} + +

+ @@ -89,7 +106,11 @@ - + +

+ The units on AV_{[1990,2010]} for both counties are people per year. +

+
@@ -99,7 +120,12 @@ - + +

+ Between 1990 and 2010, the population of Ottawa County grew by an average of + approximately 3801.65 people per year. +

+
@@ -110,7 +136,17 @@ - + +

+ For Kent County on [2000,2010]: + AV_{[2000,2010]} = \frac{602{,}622 - 574{,}336}{10} = \frac{28{,}286}{10} \approx 2828.6 \text{ people per year.} + For Ottawa County on [2000,2010]: + AV_{[2000,2010]} = \frac{263{,}801 - 238{,}313}{10} = \frac{25{,}488}{10} = 2548.8 \text{ people per year.} + Kent County had a greater average rate of change during [2000,2010]. + In both counties, the population increased in every decade from 1960 to 2010, + so the average rate of change was positive on every decade interval for both counties. +

+
@@ -120,7 +156,14 @@ - + +

+ Using the most recent decade's rate of change as a guide, Ottawa County grew at + 2548.8 people per year from 2000 to 2010. Projecting this rate forward + eight years from 2010: + 263{,}801 + 8 \cdot 2548.8 \approx 284{,}191 \text{ people.} +

+

diff --git a/source/activities/act-changing-aroc-trends.xml b/source/activities/act-changing-aroc-trends.xml index 2a546740..b6e9b551 100755 --- a/source/activities/act-changing-aroc-trends.xml +++ b/source/activities/act-changing-aroc-trends.xml @@ -30,7 +30,16 @@ - + +

+ Computing the function values: q(0)=0, q(1)=3, q(2)=4, + q(3)=3, q(4)=0. Thus: + AV_{[0,1]} = \frac{3-0}{1} = 3, \quad AV_{[1,2]} = \frac{4-3}{1} = 1, + AV_{[2,3]} = \frac{3-4}{1} = -1, \quad AV_{[3,4]} = \frac{0-3}{1} = -3. + Since AV_{[2,3]} and AV_{[3,4]} are both negative, q is + decreasing on [2,4]. +

+ @@ -43,7 +52,18 @@ - + +

+ Computing function values: h(-1) = 3 - 2(0.5)^{-1} = 3 - 4 = -1, + h(1) = 3 - 2(0.5) = 2, h(3) = 3 - 2(0.5)^3 = \frac{11}{4} \approx 2.75, + h(5) = 3 - 2(0.5)^5 = \frac{47}{16} \approx 2.9375. Thus: + AV_{[-1,1]} = \frac{2-(-1)}{2} = \frac{3}{2} = 1.5, + AV_{[1,3]} = \frac{\frac{11}{4} - 2}{2} = \frac{3}{8} \approx 0.375, + AV_{[3,5]} = \frac{\frac{47}{16} - \frac{11}{4}}{2} = \frac{3}{32} \approx 0.094. + All three average rates of change are positive but decreasing, so h is + increasing on [-1,5], but at a decreasing rate. +

+
@@ -73,7 +93,14 @@ - + +

+ False. Although AV_{[0,3]} = \frac{q(3)-q(0)}{3} = \frac{3-0}{3} = 1 > 0, + a positive average rate of change over an interval does not mean the function is + increasing everywhere on that interval. From part (a), AV_{[2,3]} = -1 < 0, + so q is actually decreasing on [2,3]. +

+
@@ -86,7 +113,13 @@ - + +

+ Any linear function has constant average rate of change. For example, + f(x) = x has AV_{[a,b]} = \frac{b-a}{b-a} = 1 for every interval + [a,b]. +

+

diff --git a/source/activities/act-changing-combining-arithmetic.xml b/source/activities/act-changing-combining-arithmetic.xml index fc4829ff..e978222e 100755 --- a/source/activities/act-changing-combining-arithmetic.xml +++ b/source/activities/act-changing-combining-arithmetic.xml @@ -37,7 +37,12 @@ - + +

+ From the graphs, f(0) = \frac{5}{2} and g(0) = 1, so + (f+g)(0) = \frac{5}{2} + 1 = \frac{7}{2}. +

+ @@ -47,7 +52,12 @@ - + +

+ From the graphs, f(1) = 4 and g(1) = 3, so + (g-f)(1) = 3 - 4 = -1. +

+
@@ -57,7 +67,12 @@ - + +

+ From the graphs, f(-1) = 3 and g(-1) = 3, so + (f \cdot g)(-1) = 3 \cdot 3 = 9. +

+
@@ -67,7 +82,12 @@ - + +

+ \left(\frac{f}{g}\right)(x) is undefined wherever g(x) = 0. + From the graph, g(x) = 0 at x = -2 and x = 3. +

+
@@ -77,7 +97,14 @@ - + +

+ (f \cdot g)(x) = 0 wherever f(x) = 0 or g(x) = 0. + From the graphs, f(x) = 0 at x = \frac{7}{3}, and + g(x) = 0 at x = -2 and x = 3. + So (f \cdot g)(x) = 0 at x \in \left\{-2,\, \frac{7}{3},\, 3\right\}. +

+
@@ -87,7 +114,19 @@ - + +

+ Yes: (f-g)(x) = 0 when f(x) = g(x). + One intersection is visible from the graphs at x = -1. + On the interval -1 \lt x \le 1, the formulas are f(x) = \frac{5}{2} + x + and g(x) = -x^2 + 4; setting them equal gives: + + x^2 + x - \frac{3}{2} &= 0 + x &= \frac{-1 + \sqrt{7}}{2}. + + So (f-g)(x) = 0 at x = -1 and x = \dfrac{\sqrt{7}-1}{2}. +

+

diff --git a/source/activities/act-changing-combining-context.xml b/source/activities/act-changing-combining-context.xml index f6370f69..67a4d1d9 100755 --- a/source/activities/act-changing-combining-context.xml +++ b/source/activities/act-changing-combining-context.xml @@ -34,7 +34,12 @@ - + +

+ At a speed of 60 miles per hour, the car consumes 0.04 gallons of fuel + for each mile traveled. +

+ @@ -47,7 +52,13 @@ - + +

+ g(60) = \frac{1}{f(60)} = \frac{1}{0.04} = 25 miles per gallon. + The function g measures the car's fuel economy in miles per gallon + at a given speed. +

+
@@ -60,7 +71,14 @@ - + +

+ h(60) = 60 \cdot f(60) = 60 \cdot 0.04 = 2.4 gallons per hour. + The units follow from: miles/hour \times gallons/mile = gallons/hour. + The function h measures the rate at which the car consumes fuel + (in gallons per hour) at a given speed. +

+
@@ -70,7 +88,14 @@ - + +

+ All three convey information about fuel consumption at 60 mph, but with + different units and perspectives: f(60) = 0.04 gives gallons per mile, + g(60) = 25 gives miles per gallon, and h(60) = 2.4 gives + gallons per hour. They describe the same phenomenon from different angles. +

+
@@ -84,7 +109,14 @@ - + +

+ AV_{[60,70]} = \frac{f(70)-f(60)}{70-60} = \frac{0.045 - 0.04}{10} = 0.0005 + \ \frac{\text{gal/mi}}{\text{mph}}. + This measures how much the car's fuel consumption rate (in gallons per mile) + increases per additional mile per hour of speed. +

+

diff --git a/source/activities/act-changing-combining-piecewise.xml b/source/activities/act-changing-combining-piecewise.xml index 6933598b..c477c918 100755 --- a/source/activities/act-changing-combining-piecewise.xml +++ b/source/activities/act-changing-combining-piecewise.xml @@ -37,7 +37,17 @@ - + +

+ For x \lt 0, use p(x) = -(x+2)^2 + 2: + p(-4) = -(-2)^2 + 2 = -2; + p(-2) = -(0)^2 + 2 = 2. + For x \ge 0, use p(x) = \frac{1}{2}(x-2)^2 + 1: + p(0) = \frac{1}{2}(4) + 1 = 3; + p(2) = 0 + 1 = 1; + p(4) = \frac{1}{2}(4) + 1 = 3. +

+ @@ -47,7 +57,14 @@ - + +

+ The left-side parabola -(x+2)^2 + 2 (valid for x \lt 0) opens + downward and has vertex (-2, 2). + The right-side parabola \frac{1}{2}(x-2)^2 + 1 (valid for x \ge 0) + opens upward and has vertex (2, 1). +

+
@@ -57,7 +74,16 @@ - + +

+ For x \lt 0, setting -(x+2)^2 + 2 = 0 gives (x+2)^2 = 2, + so x = -2 \pm \sqrt{2}. Both values x = -2 + \sqrt{2} \approx -0.586 + and x = -2 - \sqrt{2} \approx -3.414 lie in x \lt 0, so both are + zeros of p. The right-side parabola \frac{1}{2}(x-2)^2+1 \ge 1 \gt 0 + for all x \ge 0, so it contributes no zeros. + The y-intercept is p(0) = 3. +

+
diff --git a/source/activities/act-changing-composite-aroc.xml b/source/activities/act-changing-composite-aroc.xml index ac60b27f..90841075 100755 --- a/source/activities/act-changing-composite-aroc.xml +++ b/source/activities/act-changing-composite-aroc.xml @@ -27,7 +27,15 @@ - + +

+ + f(1+h) &= 2(1+h)^2 - 3(1+h) + 1 + &= 2(1 + 2h + h^2) - 3 - 3h + 1 + &= 2h^2 + h. + +

+
@@ -38,7 +46,12 @@ - + +

+ Since f(1) = 2 - 3 + 1 = 0 and f(1+h) = 2h^2 + h: + AV_{[1,1+h]} = \frac{f(1+h) - f(1)}{h} = \frac{2h^2 + h}{h} = 2h + 1. +

+
@@ -49,7 +62,12 @@ - + +

+ g(1+h) = \frac{5}{1+h}. + This expression cannot be simplified further. +

+
@@ -60,7 +78,15 @@ - + +

+ Since g(1) = 5: + + AV_{[1,1+h]} &= \frac{g(1+h) - g(1)}{h} = \frac{\frac{5}{1+h} - 5}{h} + &= \frac{5 - 5(1+h)}{(1+h)h} = \frac{-5h}{(1+h)h} = \frac{-5}{1+h}. + +

+

diff --git a/source/activities/act-changing-composite-crickets-celsius.xml b/source/activities/act-changing-composite-crickets-celsius.xml index ca7ada63..bb4ab7c3 100755 --- a/source/activities/act-changing-composite-crickets-celsius.xml +++ b/source/activities/act-changing-composite-crickets-celsius.xml @@ -27,7 +27,11 @@ - + +

+ H(N) = G(D(N)) = \frac{5}{9}((40 + 0.25N) - 32) = \frac{5}{9}(8 + 0.25N). +

+ @@ -37,7 +41,12 @@ - + +

+ The function H converts the number of cricket chirps per minute directly + into a temperature in degrees Celsius. +

+
@@ -67,7 +76,14 @@ - + +

+ As an abstract mathematical function, both the domain and range are all real + numbers. In the context of Dolbear's model (domain [40, 180] chirps per + minute), the domain of H is [40, 180] and the corresponding range + is approximately [10, 29.4] degrees Celsius. +

+

diff --git a/source/activities/act-changing-composite-tables-graphs.xml b/source/activities/act-changing-composite-tables-graphs.xml index ecbddb72..7857ecd8 100755 --- a/source/activities/act-changing-composite-tables-graphs.xml +++ b/source/activities/act-changing-composite-tables-graphs.xml @@ -71,7 +71,11 @@ - + +

+ From the graph, q(0) = 2 and p(2) = 1, so p(q(0)) = 1. +

+ @@ -81,7 +85,12 @@ - + +

+ From the graph, p(0) = -\frac{1}{2} and q\!\left(-\frac{1}{2}\right) = 2, + so q(p(0)) = 2. +

+
@@ -89,7 +98,11 @@ - + +

+ From the graph, p(-1) = -1, so (p \circ p)(-1) = p(p(-1)) = p(-1) = -1. +

+
@@ -99,7 +112,11 @@ - + +

+ From the table, g(2) = 0 and f(0) = 6, so (f \circ g)(2) = 6. +

+
@@ -109,7 +126,11 @@ - + +

+ From the table, f(3) = 4 and g(4) = 2, so (g \circ f)(3) = 2. +

+
@@ -119,7 +140,12 @@ - + +

+ From the table, f(0) = 6, but g(6) is not defined in the table, + so g(f(0)) is undefined. +

+
@@ -129,7 +155,14 @@ - + +

+ We need f(g(x)) = 4, which requires g(x) = 1 or g(x) = 3 + (since f(1)=4 and f(3)=4 from the table). From the table, + g(0) = 1 and g(1) = 3, so f(g(x)) = 4 for x = 0 + and x = 1. +

+
@@ -139,7 +172,15 @@ - + +

+ We need q(p(x)) = 1, which (from the graph of q) requires + p(x) = -\frac{4}{3} or p(x) = 2. From the graph of p, + p(x) = 2 at x = 2 and at x \approx -2.5; there are no + x-values for which p(x) = -\frac{4}{3}. So q(p(x)) = 1 + for x = 2 and x \approx -2.5. +

+

diff --git a/source/activities/act-changing-functions-is-it.xml b/source/activities/act-changing-functions-is-it.xml index 7b4ba376..00ad7f1a 100755 --- a/source/activities/act-changing-functions-is-it.xml +++ b/source/activities/act-changing-functions-is-it.xml @@ -37,7 +37,17 @@ - + +

+ The circle is not a function of x, because some values of x + correspond to more than one value of y. For example, when x = 0, + both y = 4 and y = -4 lie on the circle. +

+

+ The curve in the righthand figure is a function of x, because + each value of x in the domain corresponds to exactly one value of y. +

+ @@ -47,7 +57,14 @@ - + +

+ The closing value of the S&P500 can be expressed as a function of the day of + the year, provided the domain is restricted to trading days (excluding weekends and + market holidays). With this restriction, each day in the domain corresponds to + exactly one closing value. +

+
@@ -57,7 +74,13 @@ - + +

+ The odometer reading cannot be viewed as a function of the car's velocity. + The same speed can occur at many different odometer readings throughout a trip, + so a single velocity value can correspond to multiple odometer values. +

+
@@ -112,7 +135,12 @@ - + +

+ The table does not represent a function, because the input values + x = 1 and x = 2 each correspond to more than one output value. +

+

diff --git a/source/activities/act-changing-functions-spherical-tank-draining.xml b/source/activities/act-changing-functions-spherical-tank-draining.xml index e3b7e09b..2324f4c7 100755 --- a/source/activities/act-changing-functions-spherical-tank-draining.xml +++ b/source/activities/act-changing-functions-spherical-tank-draining.xml @@ -34,7 +34,20 @@ - + +

+ At t = 0, the tank is full and the height of the water is 8 m. + At t = 1 minute, 0.5 m of water has drained and the height is 7.5 m. + At t = 2 minutes, 1 m of water has drained and the height is 7 m. +

+

+ The tank has 8 m of water (depth), which is 16 half-meters, with one + half-meter draining each minute. Thus it will take 16 minutes for the tank + to drain completely. The linear height model + h = q(t) = 8 - 0.5t + gives q(t) = 0 when t = 16. +

+ @@ -44,7 +57,14 @@ - + +

+ The domain of each model is determined by the time it takes the tank to drain. + Assuming the tank begins draining at t = 0 minutes and is empty at + t = 16 minutes, the domain of both models is 0 \le t \le 16, + or equivalently [0, 16]. +

+
@@ -54,7 +74,19 @@ - + +

+ When the tank is full, the depth of the water equals the diameter of the tank, + 2 \times 4 = 8 m. The volume of water in the full tank equals the volume + of the sphere: + \frac{4}{3}\pi (4)^3 = \frac{256\pi}{3} \approx 268 \text{ m}^3. +

+

+ The range of the volume model is 0 \le V \le \frac{256\pi}{3}, going from + completely empty to completely full. The range of the height model is likewise + 0 \le h \le 8. +

+
@@ -76,7 +108,14 @@ - + +

+ The model V = p(t) differs from the abstract function y = r(x) + in its domain and range. The model's domain is restricted to [0, 16] + and its range to \left[0, \frac{256\pi}{3}\right] by the physical context. + The abstract function has an unrestricted domain and an unrestricted range. +

+
@@ -86,7 +125,13 @@ - + +

+ The height function should be linear and decreasing, because the height decreases + by the same amount (0.5 m) every minute. The formula is + q(t) = 8 - 0.5t. +

+

diff --git a/source/activities/act-changing-functions-spherical-tank.xml b/source/activities/act-changing-functions-spherical-tank.xml index f32d3374..12a68ab4 100755 --- a/source/activities/act-changing-functions-spherical-tank.xml +++ b/source/activities/act-changing-functions-spherical-tank.xml @@ -33,7 +33,14 @@ - + +

+ Since depth can't be negative and can't exceed the tank's diameter of 8 m, + the values of h that make sense are 0 \le h \le 8. + The corresponding values of V range from 0 (empty tank) to + f(8) = \frac{256\pi}{3} (full tank), so 0 \le V \le \frac{256\pi}{3}. +

+ @@ -43,7 +50,16 @@ - + +

+ The domain of f in context is [0, 8], since the water depth + can range from 0 m (empty) to 8 m (full diameter). + The range is \left[0, \frac{256\pi}{3}\right]. + Since volumes are non-negative, the codomain can be taken as [0, \infty), + or more broadly as all real numbers if we consider the abstract formula + without physical constraints. +

+
@@ -54,7 +70,19 @@ - + +

+ + f(2) &= \frac{\pi}{3}(4)(10) = \frac{40\pi}{3} \approx 41.9 \text{ m}^3 + f(4) &= \frac{\pi}{3}(16)(8) = \frac{128\pi}{3} \approx 134.0 \text{ m}^3 + f(8) &= \frac{\pi}{3}(64)(4) = \frac{256\pi}{3} \approx 268.1 \text{ m}^3 + + These give the volume of water when the depth is 2, 4, and 8 + meters respectively. The value f(8) = \frac{256\pi}{3} equals the volume + of a sphere of radius 4, confirming that the tank is completely full when + h = 8. +

+
@@ -64,7 +92,15 @@ - + +

+ Neither claim is valid. The domain of f is [0, 8], so + h = 9 is outside the domain — the water cannot be 9 m deep in a + tank whose diameter is only 8 m. Similarly, f(13) falls outside the + domain, and the negative value it produces is further evidence that the formula + should not be applied beyond h = 8. +

+
@@ -74,7 +110,13 @@ - + +

+ No. The maximum value of f on its domain [0, 8] is + f(8) = \frac{256\pi}{3} \approx 268.1 m^3, which is less than + 300. The tank simply cannot hold 300 cubic meters of water. +

+

diff --git a/source/activities/act-changing-inverse-Dolbear.xml b/source/activities/act-changing-inverse-Dolbear.xml index 65c5fe6e..61a0ae57 100755 --- a/source/activities/act-changing-inverse-Dolbear.xml +++ b/source/activities/act-changing-inverse-Dolbear.xml @@ -27,7 +27,12 @@ - + +

+ Subtracting 40 and multiplying by 4: + E(F) = N = 4(F - 40). +

+ @@ -37,7 +42,12 @@ - + +

+ The function E takes a temperature in degrees Fahrenheit as its input and + outputs the corresponding number of snowy tree cricket chirps per minute. +

+
@@ -47,7 +57,15 @@ - + +

+ j(N) = E(D(N)) = 4\!\left(\left(40 + \frac{1}{4}N\right) - 40\right) = 4 \cdot \frac{N}{4} = N. + k(F) = D(E(F)) = 40 + \frac{1}{4}(4(F-40)) = 40 + (F-40) = F. + Both j and k are the identity: composing D and E + in either order returns the original input, confirming that E and D + are inverse functions. +

+
@@ -57,7 +75,14 @@ - + +

+ They express the same relationship between F and N, just solved + for different variables. Each equation can be obtained from the other by + algebraic manipulation, so they describe the same connection between temperature + and chirp rate. +

+

diff --git a/source/activities/act-changing-inverse-does-it.xml b/source/activities/act-changing-inverse-does-it.xml index fd637c8c..42b460ed 100755 --- a/source/activities/act-changing-inverse-does-it.xml +++ b/source/activities/act-changing-inverse-does-it.xml @@ -52,7 +52,12 @@ - + +

+ The function f does not have an inverse, because f(1) = 2 = f(4), + so f^{-1}(2) would need to equal both 1 and 4 — not a function. +

+ @@ -87,7 +92,12 @@ - + +

+ The function g does have an inverse, since all five output values + are distinct. For example, g^{-1}(4) = 0 and g^{-1}(0) = 1. +

+
@@ -97,7 +107,14 @@ - + +

+ The function p(t) = 7 - \frac{3}{5}t is linear and one-to-one, so it + does have an inverse. Solving y = 7 - \frac{3}{5}t for t: + p^{-1}(y) = \frac{5}{3}(7-y). + For example, p^{-1}(7) = 0 and p^{-1}(4) = 5. +

+
@@ -107,7 +124,13 @@ - + +

+ The function q(t) = 7 - \frac{3}{5}t^4 does not have an inverse. + Since q(1) = q(-1) = 7 - \frac{3}{5}, the value q^{-1}\!\left(\frac{32}{5}\right) + could be either 1 or -1. +

+
@@ -127,7 +150,17 @@ - + +

+ The function r(t) passes the horizontal line test, so it does + have an inverse. For example, r^{-1}(2) = -1 and r^{-1}(0) = 0. +

+

+ The function s(t) does not have an inverse, because it fails the + horizontal line test: a horizontal line between y=1 and y=2 crosses + the graph in more than one place. +

+

diff --git a/source/activities/act-changing-inverse-rainfall.xml b/source/activities/act-changing-inverse-rainfall.xml index e9a1cb62..77768cb4 100755 --- a/source/activities/act-changing-inverse-rainfall.xml +++ b/source/activities/act-changing-inverse-rainfall.xml @@ -30,7 +30,13 @@ - + +

+ g(3) = \frac{4}{3+2} + 1 = \frac{4}{5} + 1 = \frac{9}{5} = 1.8 \text{ cm/hr.} + At t = 3 hours into the storm, the rain is falling at a rate of 1.8 + centimeters per hour. +

+ @@ -40,7 +46,15 @@ - + +

+ Using g(3) = \frac{9}{5} and g(5) = \frac{4}{7} + 1 = \frac{11}{7}: + AV_{[3,5]} = \frac{\frac{11}{7} - \frac{9}{5}}{2} = \frac{\frac{55-63}{35}}{2} = \frac{-8/35}{2} = -\frac{4}{35} \approx -0.114 \text{ cm/hr}^2. + Between hours 3 and 5, the rate of rainfall decreased on average by about + 0.114 cm/hr per hour. Since g is a decreasing function, we expect + the rainfall rate to continue decreasing throughout the storm. +

+
@@ -50,7 +64,15 @@ - + +

+ At the endpoints: g(0) = \frac{4}{2}+1 = 3 and + g(10) = \frac{4}{12}+1 = \frac{4}{3}. Since g is strictly + decreasing on [0,10], the range is \left[\frac{4}{3}, 3\right]. + The function has an inverse because it is strictly decreasing (one-to-one) on + this domain — it passes the horizontal line test. +

+
@@ -60,7 +82,17 @@ - + +

+ Setting \frac{9}{5} = \frac{4}{t+2} + 1: + + \frac{4}{5} &= \frac{4}{t+2} + t + 2 &= 5 \implies t = 3. + + So g^{-1}\!\left(\frac{9}{5}\right) = 3: the rainfall rate equals + 1.8 cm/hr at exactly 3 hours into the storm. +

+
@@ -70,7 +102,15 @@ - + +

+ No. Setting g(t) = 1 gives \frac{4}{t+2} + 1 = 1, so + \frac{4}{t+2} = 0, which has no solution. Since \frac{4}{t+2} > 0 + for all t in [0,10], we have g(t) > 1 throughout the storm. + (The range of g is \left[\frac{4}{3}, 3\right], and + 1 \notin \left[\frac{4}{3}, 3\right].) +

+

diff --git a/source/activities/act-changing-linear-Kilimanjaro.xml b/source/activities/act-changing-linear-Kilimanjaro.xml index ebb32d6f..26838d17 100755 --- a/source/activities/act-changing-linear-Kilimanjaro.xml +++ b/source/activities/act-changing-linear-Kilimanjaro.xml @@ -35,7 +35,14 @@ - + +

+ Using the points (0, 1951) and (7, 1555), the slope is + \frac{1555-1951}{7-0} = \frac{-396}{7} \approx -56.6 m^2/year. + Since t=0 gives A = 1951, the model is: + f(t) = 1951 - \frac{396}{7} t \approx 1951 - 56.6t. +

+ @@ -45,7 +52,13 @@ - + +

+ The slope \approx -56.6 m^2/year means the ice cover decreases by + about 56.6 square meters per year. The A-intercept of 1951 m^2 + represents the ice cover in the year 2000 (t=0). +

+
@@ -57,7 +70,13 @@ - + +

+ f(17) \approx 1951 - 56.6(17) \approx 989 \text{ m}^2. + In the year 2017, the model predicts that the ice field near the summit of + Mt. Kilimanjaro will have an area of approximately 989 square meters. +

+
@@ -68,7 +87,16 @@ - + +

+ Setting f(t) = 0: + + 0 &\approx 1951 - 56.6t + t &\approx \frac{1951}{56.6} \approx 34.5. + + The model predicts the ice will vanish around the year 2034. +

+
@@ -80,7 +108,12 @@ - + +

+ A reasonable domain is 0 \le t \le 34.5 (from the year 2000 until the ice + disappears). The corresponding range is 0 \le A \le 1951. +

+

diff --git a/source/activities/act-changing-linear-finding-eqs.xml b/source/activities/act-changing-linear-finding-eqs.xml index 75ad7cd6..42322d66 100755 --- a/source/activities/act-changing-linear-finding-eqs.xml +++ b/source/activities/act-changing-linear-finding-eqs.xml @@ -28,7 +28,11 @@ - + +

+ y - (-17) = \frac{3}{7}(x - (-11)), \quad \text{i.e.,} \quad y + 17 = \frac{3}{7}(x+11). +

+ @@ -38,7 +42,13 @@ - + +

+ The slope is m = \frac{-1-5}{3-(-2)} = -\frac{6}{5}. Using the point + (3,-1): + y + 1 = -\frac{6}{5}(x-3). +

+
@@ -48,7 +58,13 @@ - + +

+ Solving 2x - 3y = 5 for y gives y = \frac{2}{3}x - \frac{5}{3}, + so the slope is m = \frac{2}{3}. Using the point (4,9): + y - 9 = \frac{2}{3}(x-4). +

+
@@ -99,7 +115,17 @@ - + +

+ The function appears to be linear because the rate of change is constant: + for each increase of 1 in x, f(x) decreases by 2, + giving slope m = -2. Using the point (1,7): + + y - 7 &= -2(x-1) + f(x) &= 9 - 2x. + +

+
@@ -110,11 +136,19 @@ Plot of a linear function h. -->

ADD ALT TEXT TO THIS IMAGE

+ -->
- + +

+ The graph has slope m = -\frac{1}{3} and passes through (4,1), so: + + y - 1 &= -\frac{1}{3}(x-4) + h(x) &= \frac{7}{3} - \frac{1}{3}x. + +

+

diff --git a/source/activities/act-changing-linear-in-context.xml b/source/activities/act-changing-linear-in-context.xml index f0466841..4c1acaf5 100755 --- a/source/activities/act-changing-linear-in-context.xml +++ b/source/activities/act-changing-linear-in-context.xml @@ -28,7 +28,11 @@ - + +

+ f(t) = 28750 + 825t. +

+ @@ -38,7 +42,12 @@ - + +

+ The slope is -465 people per year. This means the town's population + decreases by 465 people each year. +

+
@@ -48,7 +57,13 @@ - + +

+ p(t) = \frac{32\pi}{3} - 1.2t. + The tank empties when p(t) = 0, i.e., t = \frac{32\pi}{3 \cdot 1.2} \approx 27.9 minutes. + A reasonable domain is [0, 27.9]. +

+
@@ -58,7 +73,12 @@ - + +

+ Since q(t) = 0.65t is a linear function with slope 0.65, the water + level rises at a constant rate of 0.65 feet per minute. +

+
@@ -68,7 +88,18 @@ - + +

+ The slope is \frac{4600 - 10200}{10-5} = \frac{-5600}{5} = -1120 dollars per year. + Using the point (5, 10200): + + C - 10200 &= -1120(t-5) + L(t) &= 15800 - 1120t. + + The car reaches zero value when t \approx 14.1, so a reasonable domain is + [0, 14.1]. The slope means the car loses \$1120 in value each year. +

+

diff --git a/source/activities/act-changing-quadratic-falling-ball.xml b/source/activities/act-changing-quadratic-falling-ball.xml index f10d09cf..f150d14a 100755 --- a/source/activities/act-changing-quadratic-falling-ball.xml +++ b/source/activities/act-changing-quadratic-falling-ball.xml @@ -28,7 +28,13 @@ - + +

+ Using the standard model s(t) = -16t^2 + v_0 t + s_0 with initial height + s_0 = 37 ft and initial velocity v_0 = 41 ft/s: + s(t) = -16t^2 + 41t + 37. +

+ @@ -49,7 +55,11 @@ - + +

+ From the graph, the balloon appears to land at approximately t \approx 3.3 seconds. +

+
@@ -60,7 +70,18 @@ - + +

+ Setting s(t) = 0 and applying the quadratic formula with a=-16, + b=41, c=37: + + t &= \frac{-41 \pm \sqrt{41^2 - 4(-16)(37)}}{2(-16)} + &= \frac{-41 \pm \sqrt{1681 + 2368}}{-32} + &= \frac{-41 \pm \sqrt{4049}}{-32}. + + Taking the root that gives t > 0: t = \frac{-41 - \sqrt{4049}}{-32} \approx 3.27 seconds. +

+
@@ -70,7 +91,13 @@ - + +

+ The vertex occurs at t = -\frac{b}{2a} = -\frac{41}{2(-16)} = \frac{41}{32} \approx 1.28 seconds. + The maximum height is + s\!\left(\frac{41}{32}\right) = -16 \cdot \frac{1681}{1024} + 41 \cdot \frac{41}{32} + 37 = \frac{4049}{64} \approx 63.3 \text{ feet.} +

+
@@ -83,7 +110,16 @@ - + +

+ Using s(1.5) = 62.5, s(2) = 55, s(2.5) = 39.5, s(3) = 16: + AV_{[1.5,2]} = \frac{55-62.5}{0.5} = -15 \text{ ft/s}, + AV_{[2,2.5]} = \frac{39.5-55}{0.5} = -31 \text{ ft/s}, + AV_{[2.5,3]} = \frac{16-39.5}{0.5} = -47 \text{ ft/s.} + These values represent the average downward velocity of the balloon over each + half-second interval; the balloon is falling faster and faster as it descends. +

+

diff --git a/source/activities/act-changing-quadratic-parameters.xml b/source/activities/act-changing-quadratic-parameters.xml index 2aa55eed..2d35eb25 100755 --- a/source/activities/act-changing-quadratic-parameters.xml +++ b/source/activities/act-changing-quadratic-parameters.xml @@ -31,7 +31,13 @@ - + +

+ If a > 0 the parabola opens upward; if a < 0 it opens downward. + The larger |a| is, the narrower the parabola; the closer |a| is to + zero, the wider the parabola. +

+ @@ -41,7 +47,13 @@ - + +

+ With a = 1 and c = 0, changing b shifts the vertex + diagonally: making b > 0 moves the vertex left and down, while + making b < 0 moves it right and down. +

+
@@ -51,7 +63,13 @@ - + +

+ With a = 1 and b = 0, changing c moves the vertex straight + up or down along the y-axis. The vertex is at (0, c), and c + is the y-intercept of the function. +

+
@@ -63,7 +81,15 @@ - + +

+ The parameter a has the most clearly understandable effect: it controls + whether the parabola opens up or down and how wide it is. The parameter c + has a simple effect as well (pure vertical translation). The parameter b + has the most complicated effect because changing it moves the vertex diagonally + rather than purely up, down, or in terms of width. +

+
@@ -74,7 +100,16 @@ - + +

+ Yes. Since (1,12) and (2,12) have the same output, the axis of + symmetry is x = 1.5 and the vertex has x-coordinate 1.5. + Since the parabola must pass through (0,8), which is below the values at + x=1 and x=2, the parabola opens downward. With c = q(0) = 8 + and using vertex form or solving the system, the formula is + q(x) = -2x^2 + 6x + 8. +

+

diff --git a/source/activities/act-changing-quadratic-properties.xml b/source/activities/act-changing-quadratic-properties.xml index 11df9e0b..ea52eed6 100755 --- a/source/activities/act-changing-quadratic-properties.xml +++ b/source/activities/act-changing-quadratic-properties.xml @@ -27,7 +27,15 @@ - + +

+ Three points determine a unique quadratic, so exactly one such function exists. + Using the factored form q(x) = a(x+5)(x-10) and the y-intercept + q(0) = -1: + a(5)(-10) = -50a = -1 \implies a = \frac{1}{50}. + Thus q(x) = \frac{1}{50}x^2 - \frac{1}{10}x - 1. +

+ @@ -37,7 +45,13 @@ - + +

+ We can guarantee exactly two x-intercepts. Since the vertex (-3,-4) + is below the x-axis and the parabola opens upward, both arms must cross the + x-axis. +

+
@@ -47,7 +61,16 @@ - + +

+ Yes. Using vertex form q(x) = a(x+3)^2 - 4 and the point (-1,-3): + + -3 &= a(-1+3)^2 - 4 = 4a - 4 + a &= \frac{1}{4}. + + Thus q(x) = \frac{1}{4}(x+3)^2 - 4. +

+
@@ -57,7 +80,17 @@ - + +

+ The vertex (-1,9) is above the x-axis and a=-3 < 0, so the + parabola opens downward and has two x-intercepts. Setting p(x)=0: + + -3(x+1)^2 + 9 &= 0 + (x+1)^2 &= 3 + x &= -1 \pm \sqrt{3}. + +

+
@@ -67,7 +100,15 @@ - + +

+ The discriminant is b^2 - 4ac = 10^2 - 4(-2)(-20) = 100 - 160 = -60 < 0, + so there are no real x-intercepts. Equivalently, the vertex is at + x = \frac{-10}{2(-2)} = 2.5 with w(2.5) = -7.5 < 0, and since + a = -2 < 0 the parabola opens downward and lies entirely below the + x-axis. +

+

diff --git a/source/activities/act-changing-transformations-combined.xml b/source/activities/act-changing-transformations-combined.xml index f82caf17..8fb0bb34 100755 --- a/source/activities/act-changing-transformations-combined.xml +++ b/source/activities/act-changing-transformations-combined.xml @@ -37,7 +37,14 @@ - + +

+ The function p(x) = -\frac{1}{2}f(x-1)+2 is obtained from f by + shifting right 1 unit, applying a vertical compression by \frac{1}{2} and + a reflection across the x-axis, then shifting up 2 units. + The point (-2, 2) on f moves to (-1, 1) on p. +

+ @@ -47,7 +54,14 @@ - + +

+ The function q(x) = 2g(x+0.5) - 0.75 is obtained from g by + shifting left 0.5 units, stretching vertically by a factor of 2, then + shifting down 0.75 units. + The point (1.5, 1.5) on g moves to (1, 2.25) on q. +

+
@@ -57,7 +71,17 @@ - + +

+ Algebraically: + r(x) = \frac{1}{2}(-f(x-1)-4) = -\frac{1}{2}f(x-1) - 2. + Since p(x) = -\frac{1}{2}f(x-1)+2, we have r(x) \ne p(x) + — they differ by 4 in output. + As transformations, both shift f right 1 unit and apply a vertical + compression and reflection by -\frac{1}{2}, but p then shifts + the result up 2 units while r shifts it down 2 units. +

+
@@ -70,7 +94,12 @@ - + +

+ s(x) = -2.5\,g(x+1.25) + 1.75. + The point (1.5, 1.5) on g moves to (0.25, -2) on s. +

+

diff --git a/source/activities/act-changing-transformations-translations.xml b/source/activities/act-changing-transformations-translations.xml index a8af2e3b..d801707f 100755 --- a/source/activities/act-changing-transformations-translations.xml +++ b/source/activities/act-changing-transformations-translations.xml @@ -37,7 +37,16 @@ - + +

+ The graph of g(x) = r(x) + 2 is r shifted vertically up 2 units; + the point (-2, -1) moves to (-2, 1). + The graph of h(x) = r(x+1) is r shifted horizontally to the left + 1 unit; the point (-2, -1) moves to (-3, -1). + The graph of f(x) = r(x+1) + 2 combines both transformations — a shift + left 1 unit and up 2 units — moving (-2, -1) to (-3, 1). +

+ @@ -47,7 +56,16 @@ - + +

+ The graph of k(x) = s(x) - 1 is s shifted down 1 unit; + the point (-2, -3) moves to (-2, -4). + The graph of j(x) = s(x-2) is s shifted right 2 units; + the point (-2, -3) moves to (0, -3). + The graph of m(x) = s(x-2) - 1 combines both — a shift right 2 units + and down 1 unit — moving (-2, -3) to (0, -4). +

+
@@ -57,7 +75,17 @@ - + +

+ Substituting q(x) = x^2: + + p(x) &= q(x+3) - 4 = (x+3)^2 - 4 + &= x^2 + 6x + 9 - 4 = x^2 + 6x + 5. + + The function p is a translation of q three units to the left + and four units down. +

+

diff --git a/source/activities/act-changing-transformations-vert-stretch.xml b/source/activities/act-changing-transformations-vert-stretch.xml index d6f88d94..76e28837 100755 --- a/source/activities/act-changing-transformations-vert-stretch.xml +++ b/source/activities/act-changing-transformations-vert-stretch.xml @@ -37,7 +37,15 @@ - + +

+ The graph of g(x) = 3r(x) is a vertical stretch of r by a factor + of 3; the point (-2, -1) moves to (-2, -3). + The graph of h(x) = \frac{1}{3}r(x) is a vertical compression of r + by a factor of \frac{1}{3}; the point (-2, -1) moves to + (-2, -\frac{1}{3}). +

+ @@ -47,7 +55,15 @@ - + +

+ The graph of k(x) = -s(x) is a reflection of s across the + x-axis; the point (-2, -3) moves to (-2, 3). + The graph of j(x) = -\frac{1}{2}s(x) is a vertical compression by + \frac{1}{2} combined with a reflection across the x-axis; + the point (-2, -3) moves to (-2, \frac{3}{2}). +

+
@@ -61,7 +77,14 @@ - + +

+ For m(x) = 2r(x+1) - 1: the point (-2, -1) on r moves + to (-3, -3) on m. + For n(x) = \frac{1}{2}s(x-2) + 2: the point (-2, -3) on s + moves to (0, \frac{1}{2}) on n. +

+
@@ -71,7 +94,16 @@ - + +

+ The function m(x) = 2r(x+1) - 1 results from three elementary + transformations of r: a horizontal shift left 1 unit (replacing x + with x+1), a vertical stretch by a factor of 2 (multiplying by 2), and + a vertical shift down 1 unit (subtracting 1). + The vertical stretch and the horizontal shift may be applied in either order, + but both must be applied before the vertical shift. +

+

diff --git a/source/previews/PA-changing-aroc.xml b/source/previews/PA-changing-aroc.xml index ba44e289..bd30defd 100755 --- a/source/previews/PA-changing-aroc.xml +++ b/source/previews/PA-changing-aroc.xml @@ -28,7 +28,15 @@ - + +

+ First compute s(1.5) = 64 - 16(1.5-1)^2 = 60 and + s(2.5) = 64 - 16(2.5-1)^2 = 28. Then + + AV_{[1.5,2.5]} &= \frac{s(2.5)-s(1.5)}{2.5-1.5} = \frac{28-60}{1} = -32. + +

+ @@ -39,7 +47,13 @@ - + +

+ The units are feet per second. The value AV_{[1.5,2.5]} = -32 means that + between t = 1.5 and t = 2.5 seconds, the ball's height decreases + at an average rate of 32 feet per second. +

+
@@ -60,7 +74,14 @@ - + +

+ Without additional context specifying when the ball is launched or lands, the + domain and range of the model are not fully determined by the equation alone. + The formula s(t) = 64 - 16(t-1)^2 is a classical free-fall equation, + and negative values of t or s are not necessarily unphysical. +

+
@@ -74,7 +95,17 @@ - + +

+ The slope is m = AV_{[1.5,2.5]} = -32. Using the point (1.5, 60) + to find the vertical intercept: + + 60 &= -32(1.5) + b + b &= 108. + + The equation of the line is y = -32t + 108. +

+
@@ -85,7 +116,12 @@ - + +

+ AV_{[1.5,2.5]} is the slope of the secant line connecting the points + (1.5, s(1.5)) and (2.5, s(2.5)) on the graph of s. +

+
@@ -96,7 +132,26 @@ - + +

+ Interval [0.25, 0.75]: + s(0.25) = 55, s(0.75) = 63, so + AV_{[0.25,0.75]} = \frac{63-55}{0.5} = 16 \text{ ft/sec.} + The secant line through (0.25, 55) with slope 16 is y = 16t + 51. +

+

+ Interval [0.5, 1.5]: + s(0.5) = 60 and s(1.5) = 60, so + AV_{[0.5,1.5]} = \frac{60-60}{1} = 0 \text{ ft/sec.} + The secant line is the horizontal line y = 60. +

+

+ Interval [1, 3]: + s(1) = 64, s(3) = 0, so + AV_{[1,3]} = \frac{0-64}{2} = -32 \text{ ft/sec.} + The secant line through (3, 0) with slope -32 is y = -32t + 96. +

+
diff --git a/source/previews/PA-changing-combining.xml b/source/previews/PA-changing-combining.xml index c72e14fb..c96b2d87 100755 --- a/source/previews/PA-changing-combining.xml +++ b/source/previews/PA-changing-combining.xml @@ -68,7 +68,12 @@ - + +

+ From the table, f(3) = 20 and g(3) = 2, so + h(3) = f(3) + g(3) = 20 + 2 = 22. +

+
@@ -78,7 +83,12 @@ - + +

+ From the graph, p(-1) = \frac{1}{5} and q(-1) = \frac{7}{2}, so + r(-1) = p(-1) - q(-1) = \frac{1}{5} - \frac{7}{2} = \frac{2}{10} - \frac{35}{10} = -\frac{33}{10}. +

+
@@ -88,7 +98,18 @@ - + +

+ Yes. Setting r(x) = 0 requires p(x) = q(x). + On the relevant interval, p(x) = -\frac{2}{5}x - \frac{1}{5} and + q(x) = 4x + 12, so: + + -\frac{2}{5}x - \frac{1}{5} &= 4x + 12 + -\frac{22}{5}x &= \frac{61}{5} + x &= -\frac{61}{22}. + +

+
@@ -98,7 +119,12 @@ - + +

+ From the table, f(0) = 5 and g(0) = 9, so + k(0) = f(0) \cdot g(0) = 5 \cdot 9 = 45. +

+
@@ -108,7 +134,12 @@ - + +

+ From the graph, p(1) = -\frac{3}{5} and q(1) = \frac{5}{2}, so + s(1) = \frac{p(1)}{q(1)} = \frac{-3/5}{5/2} = -\frac{3}{5} \cdot \frac{2}{5} = -\frac{6}{25}. +

+
@@ -118,7 +149,12 @@ - + +

+ Yes: s(x) is undefined wherever q(x) = 0. + From the graph, q(-3) = 0, so s(-3) is undefined. +

+
diff --git a/source/previews/PA-changing-composite.xml b/source/previews/PA-changing-composite.xml index be75c29b..0408ec83 100755 --- a/source/previews/PA-changing-composite.xml +++ b/source/previews/PA-changing-composite.xml @@ -27,7 +27,15 @@ - + +

+ + r(t) &= p(q(t)) = p(t^2 - 1) + &= 3(t^2 - 1) - 4 + &= 3t^2 - 7. + +

+
@@ -37,7 +45,14 @@ - + +

+ In the introductory example, the inner function g(t) = 3t-4 is linear + and the outer function f(x) = x^2 - 1 is quadratic. In part (a), the + roles are reversed: the inner function q(t) = t^2-1 is quadratic and the + outer function p(x) = 3x-4 is linear. +

+
@@ -47,7 +62,11 @@ - + +

+ q(s(z)) = \left(\frac{1}{z+4}\right)^2 - 1 = \frac{1}{(z+4)^2} - 1. +

+
@@ -57,7 +76,12 @@ - + +

+ One natural decomposition: f(x) = \sqrt{x} and g(t) = 2t^2 + 5, + so that f(g(t)) = \sqrt{2t^2 + 5} = h(t). +

+
diff --git a/source/previews/PA-changing-inverse-F-C.xml b/source/previews/PA-changing-inverse-F-C.xml index ec046ad7..f98707e9 100755 --- a/source/previews/PA-changing-inverse-F-C.xml +++ b/source/previews/PA-changing-inverse-F-C.xml @@ -27,7 +27,13 @@ - + +

+ Subtracting 32 from both sides: F - 32 = \frac{9}{5}C. + Multiplying both sides by \frac{5}{9}: + C = \frac{5}{9}(F-32). +

+
@@ -35,22 +41,36 @@ Note that the equation C = \frac{5}{9}(F-32) expresses C as a function of F. Call this function h so that C = h(F) = \frac{5}{9}(F-32).

- Find the simplest expression that you can for the composite function j(C) = h(g(C)). + Find the simplest expression that you can for the composite function j(C) = h(g(C)).

- + +

+ + j(C) &= h(g(C)) = \frac{5}{9}\!\left(\frac{9}{5}C + 32 - 32\right) + &= \frac{5}{9} \cdot \frac{9}{5}C = C. + +

+

- Find the simplest expression that you can for the composite function k(F) = g(h(F)). + Find the simplest expression that you can for the composite function k(F) = g(h(F)).

- + +

+ + k(F) &= g(h(F)) = \frac{9}{5}\!\left(\frac{5}{9}(F-32)\right) + 32 + &= (F-32) + 32 = F. + +

+
@@ -60,7 +80,15 @@ - + +

+ The function g converts Celsius to Fahrenheit, and h converts + Fahrenheit back to Celsius. Composing them in either order simply undoes the + conversion: applying both functions in sequence returns the original input. + This is why j(C) = C and k(F) = F — the two functions are + inverses of each other. +

+
diff --git a/source/previews/PA-changing-linear-3-ex.xml b/source/previews/PA-changing-linear-3-ex.xml index 4c76a343..1c1d1325 100755 --- a/source/previews/PA-changing-linear-3-ex.xml +++ b/source/previews/PA-changing-linear-3-ex.xml @@ -24,7 +24,14 @@ - + +

+ AV_{[-3,-1]} = \frac{f(-1)-f(-3)}{-1-(-3)} = \frac{10-16}{2} = -3, + AV_{[2,5]} = \frac{f(5)-f(2)}{5-2} = \frac{-8-1}{3} = -3, + AV_{[4,10]} = \frac{f(10)-f(4)}{10-4} = \frac{-23-(-5)}{6} = -3. + All three average rates of change equal -3. +

+
@@ -118,7 +125,15 @@ - + +

+ Reading values from the table: + AV_{[-5,-2]} = \frac{g(-2)-g(-5)}{-2-(-5)} = \frac{-1.25-(-2.75)}{3} = \frac{1.5}{3} = 0.5, + AV_{[-1,1]} = \frac{g(1)-g(-1)}{1-(-1)} = \frac{0.25-(-0.75)}{2} = \frac{1}{2} = 0.5, + AV_{[0,4]} = \frac{g(4)-g(0)}{4-0} = \frac{1.75-(-0.25)}{4} = \frac{2}{4} = 0.5. + All three average rates of change equal 0.5. +

+
@@ -135,7 +150,15 @@ - + +

+ Reading values from the graph: + AV_{[-5,-2]} = \frac{h(-2)-h(-5)}{-2-(-5)} = \frac{3-4}{3} = -\frac{1}{3}, + AV_{[-1,1]} = \frac{h(1)-h(-1)}{1-(-1)} = \frac{2-\frac{8}{3}}{2} = -\frac{1}{3}, + AV_{[0,4]} = \frac{h(4)-h(0)}{4-0} = \frac{1-\frac{7}{3}}{4} = -\frac{1}{3}. + All three average rates of change equal -\frac{1}{3}. +

+
@@ -145,12 +168,19 @@ - + +

+ All three functions have a constant average rate of change: the average rate of + change is the same regardless of which interval is chosen. They differ in their + specific rate of change (-3, 0.5, and -\frac{1}{3} respectively) + and in how they were presented (formula, table, graph). +

+

- For the function y = f(x) = 7 - 3x from (a), find the simplest expression you can for + For the function y = f(x) = 7 - 3x from (a), find the simplest expression you can for AV_{[a,b]} = \frac{f(b)-f(a)}{b-a} @@ -159,7 +189,15 @@ - + +

+ + AV_{[a,b]} &= \frac{f(b)-f(a)}{b-a} = \frac{(7-3b)-(7-3a)}{b-a} + &= \frac{3a-3b}{b-a} = \frac{3(a-b)}{b-a} = -3. + + For any a \ne b, AV_{[a,b]} = -3. +

+
diff --git a/source/previews/PA-changing-quadratic.xml b/source/previews/PA-changing-quadratic.xml index b0e44115..3950fa1c 100755 --- a/source/previews/PA-changing-quadratic.xml +++ b/source/previews/PA-changing-quadratic.xml @@ -25,7 +25,7 @@

- Execute appropriate computations to complete both of the following tables: values of the function h on the left, average rates of change for h on the right. + Execute appropriate computations to complete both of the following tables: values of the function h on the left, average rates of change for h on the right.

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