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Unit 1: Polynomial and Rational Functions

1. Topic 1.1 · Say it in context$f(t)$ is the temperature of a cup of coffee $t$ minutes after it is poured. On $0\le t\le 30$ the graph is decreasing and concave up. Describe how the temperature changes.

   The temperature drops the whole time. Concave up means the rate of change is increasing, so the negative rate gets closer to 0: the coffee cools quickly at first, then more slowly.

   Review: 1.1 Change in Tandem

2. Topic 1.1 · Know the termWhat is the preimage of an output value?

   The set of all input values that the function rule sends to that output. For $f(x)=x^2$, the preimage of 9 is $\{-3,3\}$.

   Review: 1.1 Change in Tandem

3. Topic 1.2 · Work it out$f(x)=x^2+1$. Find the average rate of change of $f$ on $[1,4]$.

   ${\displaystyle \frac{f(4)-f(1)}{4-1}}={\displaystyle \frac{17-2}{3}}=5$. Over that interval the output rises by an average of 5 units per unit of input.

   Review: 1.2 Rates of Change

4. Topic 1.2 · Say it in context$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a valve opens. The average rate of change of $V$ on $[0,10]$ is $-4$. Interpret it.

   Over the first 10 minutes, the volume of water decreased by an average of 4 liters per minute. The negative sign means volume goes down as time goes up.

   Review: 1.2 Rates of Change

5. Topic 1.3 · Which one fitsInputs $0,1,2,3,4$ give outputs $1,4,9,16,25$. Linear, quadratic, or neither? Justify with rates of change.

   Quadratic. The average rates of change over consecutive unit intervals are $3,5,7,9$. They change at a constant rate of 2, so they follow a linear pattern, which is what a quadratic produces.

   Review: 1.3 Rates of Change in Linear and Quadratic Functions

6. Topic 1.3 · Know the termWhat does the average rate of change of $f$ on $[a,b]$ look like on the graph?

   It is the slope of the secant line through the points $(a,f(a))$ and $(b,f(b))$.

   Review: 1.3 Rates of Change in Linear and Quadratic Functions

7. Topic 1.4 · Spot the mistakeA polynomial has real zeros at $x=-2$ and $x=5$. A student claims it could be increasing on all of $[-2,5]$. What is wrong?

   Between two distinct real zeros of a nonconstant polynomial there must be a local maximum or minimum. The output is 0 at both ends, so it cannot rise the whole way.

   Review: 1.4 Polynomial Functions and Rates of Change

8. Topic 1.4 · Which one fitsWhich must have a global maximum or global minimum: a degree 3 polynomial or a degree 4 polynomial?

   The degree 4 polynomial. Every polynomial of even degree has a global maximum or a global minimum. For a quadratic, it sits at the vertex.

   Review: 1.4 Polynomial Functions and Rates of Change

9. Topic 1.5 · Check the conditionsA degree 5 polynomial with real coefficients has zeros $2$, $-1$ and $3+2i$. How many complex zeros does it have, and what is another one?

   Exactly 5, counting multiplicities, because the degree is 5. Non-real zeros come in conjugate pairs, so $3-2i$ is also a zero.

   Review: 1.5 Polynomial Functions and Complex Zeros

10. Topic 1.5 · Spot the mistakeA student says $f(x)=x^3+x^2$ is an odd function because its degree is odd. Check the claim.

    $f(-x)=-x^3+x^2$, but $-f(x)=-x^3-x^2$. They differ, so $f$ is not odd. The test is $f(-x)=-f(x)$, and degree alone does not settle it.

    Review: 1.5 Polynomial Functions and Complex Zeros

11. Topic 1.5 · Know the termWhat does it mean for a function to be even, on the graph and in the formula?

    The graph is symmetric over the line $x=0$, and $f(-x)=f(x)$ for every input. For example, $p(x)=3x^4$ is even.

    Review: 1.5 Polynomial Functions and Complex Zeros

12. Topic 1.6 · Work it outDescribe the end behavior of $p(x)=-2x^4+7x^3+1$ in limit notation.

    The leading term $-2x^4$ dominates: even degree, negative leading coefficient. So ${\lim }_{x\to \infty }p(x)=-\infty$ and ${\lim }_{x\to -\infty }p(x)=-\infty$.

    Review: 1.6 Polynomial Functions and End Behavior

13. Topic 1.6 · Spot the mistakeA student reads the end behavior of $p(x)=5+3x-x^3$ from the first term, 5, and says the graph levels off. What is the fix?

    End behavior comes from the leading term, the one of highest degree, which is $-x^3$. As $x\to \infty$, $p(x)\to -\infty$, and as $x\to -\infty$, $p(x)\to \infty$.

    Review: 1.6 Polynomial Functions and End Behavior

14. Topic 1.7 · Work it outFind the horizontal asymptote of $r(x)={\displaystyle \frac{6x^2-1}{3x^2+4x}}$.

    Both polynomials have degree 2, so neither dominates. The quotient of the leading terms is ${\displaystyle \frac{6x^2}{3x^2}}=2$, so the horizontal asymptote is $y=2$.

    Review: 1.7 Rational Functions and End Behavior

15. Topic 1.7 · Which one fits$r(x)={\displaystyle \frac{x^3+1}{x^2-4}}$. Horizontal asymptote, slant asymptote, or neither?

    A slant asymptote. The numerator dominates, and the quotient of the leading terms, ${\displaystyle \frac{x^3}{x^2}}=x$, is linear. Long division gives $x$ plus a remainder that goes to 0, so the slant asymptote is $y=x$, and there is no horizontal asymptote.

    Review: 1.7 Rational Functions and End Behavior

16. Topic 1.8 · Check the conditionsTo solve ${\displaystyle \frac{x-2}{x+5}}\ge 0$, which values split the number line into intervals to test, and what is the solution?

    $x=2$, the zero of the numerator, and $x=-5$, the zero of the denominator. Testing each interval gives $x<-5$ or $x\ge 2$. The value $-5$ is excluded because the expression is not defined there.

    Review: 1.8 Rational Functions and Zeros

17. Topic 1.9 · Work it out$r(x)={\displaystyle \frac{x+1}{(x-3)(x+1{)}^2}}$. Where are the vertical asymptotes?

    $x=3$ is a zero of the denominator only. $x=-1$ is a zero of both, but its multiplicity is 2 in the denominator and 1 in the numerator, so it is also an asymptote. Answer: $x=3$ and $x=-1$.

    Review: 1.9 Rational Functions and Vertical Asymptotes

18. Topic 1.10 · Work it outFind the hole in the graph of $r(x)={\displaystyle \frac{x^2-4}{x-2}}$.

    $x=2$ is a zero of multiplicity 1 in both, so there is a hole, not an asymptote. For $x\ne 2$, $r(x)=x+2$, so outputs approach 4. The hole is at $(2,4)$, and ${\lim }_{x\to 2}r(x)=4$.

    Review: 1.10 Rational Functions and Holes

19. Topic 1.10 · Spot the mistakeA student sees the factor $(x-1)$ in both the numerator and denominator of a rational function and says there is a vertical asymptote at $x=1$. When is that wrong?

    When the multiplicity of 1 as a zero of the numerator is greater than or equal to its multiplicity in the denominator. Then the graph has a hole at $x=1$. Only a larger multiplicity in the denominator gives an asymptote.

    Review: 1.10 Rational Functions and Holes

20. Topic 1.11 · Which one fitsWhich form of a rational function shows the x-intercepts, holes and vertical asymptotes fastest, and which form shows end behavior?

    Factored form shows the real zeros of the numerator and denominator, so it reveals x-intercepts, holes, vertical asymptotes and the domain. Standard form shows the leading terms, which give end behavior and any horizontal or slant asymptote.

    Review: 1.11 Equivalent Representations of Polynomial and Rational Expressions

21. Topic 1.11 · Work it outUse polynomial long division to find the slant asymptote of $r(x)={\displaystyle \frac{x^2+3x+5}{x+1}}$.

    $x^2+3x+5=(x+1)(x+2)+3$, so $r(x)=x+2+{\displaystyle \frac{3}{x+1}}$. The remainder term goes to 0 for inputs of large magnitude, so the slant asymptote is $y=x+2$.

    Review: 1.11 Equivalent Representations of Polynomial and Rational Expressions

22. Topic 1.12 · Work it outThe graph of $f$ passes through $(4,1)$. Where does that point go on the graph of $g(x)=3f(x-2)+5$?

    Horizontally, $x-2=4$ gives $x=6$, a shift right 2. Vertically, dilate by 3 and then shift up 5: $3(1)+5=8$. The image is $(6,8)$.

    Review: 1.12 Transformations of Functions

23. Topic 1.12 · Spot the mistakeA student says $g(x)=f(x+3)$ shifts the graph of $f$ right 3 units. Correct it.

    $f(x+h)$ is a horizontal translation by $-h$ units, so the graph moves 3 units left. Check: $g(-3)=f(0)$, so the point that was at input 0 is now at input $-3$.

    Review: 1.12 Transformations of Functions

24. Topic 1.13 · Which one fitsA store's monthly sales rise by about 40 units every month. Which function type should model them, and why?

    Linear. Roughly constant rates of change over equal-length intervals are what a linear model captures.

    Review: 1.13 Function Model Selection and Assumption Articulation

25. Topic 1.13 · Which one fitsAn open box is made by cutting squares of side $x$ from the corners of a sheet. Which function type models its volume, and which models the area of its base?

    Volume is three-dimensional, so a cubic function of $x$ fits it. The base area is two-dimensional, so it is a quadratic function of $x$.

    Review: 1.13 Function Model Selection and Assumption Articulation

26. Topic 1.13 · Check the conditions$n(t)=0.5t^2-3t+8$ models the number of people waiting in line $t$ minutes after a store opens. What restrictions does the context put on this model?

    A range restriction: people come in whole numbers, so outputs should be rounded. A domain restriction: $t\ge 0$, and only as far as the data or the store's hours actually go.

    Review: 1.13 Function Model Selection and Assumption Articulation

27. Topic 1.14 · Say it in context$C(n)$ is the cost, in dollars, to print $n$ posters. The average rate of change of $C$ on $[100,200]$ is 0.75. Interpret this with units.

    Between 100 and 200 posters, the cost goes up by an average of 0.75 dollars for each additional poster.

    Review: 1.14 Function Model Construction and Application

28. Topic 1.14 · Which one fitsThe brightness of a lamp is inversely proportional to the square of the distance from it. Which function type models brightness as a function of distance?

    A rational function, such as $B(d)={\displaystyle \frac{k}{d^2}}$ for a positive constant $k$. Inverse proportion puts the input in the denominator.

    Review: 1.14 Function Model Construction and Application

Unit 2: Exponential and Logarithmic Functions

1. Topic 2.1 · Work it outAn arithmetic sequence has $a_3=11$ and $a_7=27$. Find $d$ and $a_{10}$.

   $d={\displaystyle \frac{27-11}{7-3}}=4$. Then $a_{10}=a_7+d(10-7)=27+12=39$.

   Review: 2.1 Change in Arithmetic and Geometric Sequences

2. Topic 2.1 · Know the termWhy is the graph of a sequence a set of separate points instead of a curve?

   A sequence is a function from the whole numbers to the real numbers. It has outputs only at whole-number inputs, with nothing in between.

   Review: 2.1 Change in Arithmetic and Geometric Sequences

3. Topic 2.2 · Which one fitsOver equal-length input intervals, table A's outputs go $3,7,11,15$ and table B's go $3,6,12,24$. Which is linear and which is exponential?

   A changes by a constant 4 each step, so it is linear. B changes proportionally, by a constant factor of 2, so it is exponential.

   Review: 2.2 Change in Linear and Exponential Functions

4. Topic 2.2 · Spot the mistakeA student says the sequence $a_n=4+2n$ and the function $f(x)=4+2x$ are the same thing. What is different?

   The rule matches, but the domains differ. The sequence is defined only on whole numbers, so its graph is discrete points. The linear function accepts any real input, so its graph is a line.

   Review: 2.2 Change in Linear and Exponential Functions

5. Topic 2.3 · Check the conditionsFor $f(x)=a{b}^x$ with $a>0$, which values of $b$ give exponential growth and which give exponential decay?

   $b>1$ gives growth and $0<b<1$ gives decay. The base must be positive and cannot equal 1.

   Review: 2.3 Exponential Functions

6. Topic 2.3 · Spot the mistakeA student says $f(x)=3\cdot 2^x$ has a minimum value because its graph flattens out on the left. What is wrong?

   Exponential functions are always increasing or always decreasing, so they have no extrema except on a closed interval. As $x\to -\infty$, $f(x)$ gets arbitrarily close to 0 but never reaches a lowest value.

   Review: 2.3 Exponential Functions

7. Topic 2.4 · Work it outRewrite $f(x)=2^{x+3}$ as a vertical dilation of $y=2^x$.

   By the product property, $2^{x+3}=2^x\cdot 2^3=8\cdot 2^x$. A horizontal translation of an exponential function is the same as a vertical dilation.

   Review: 2.4 Exponential Function Manipulation

8. Topic 2.5 · Say it in contextA town's population is modeled by $P(t)=\mathrm{12,000}(1.03{)}^t$, with $t$ in years. Interpret 1.03.

   It is the growth factor: each year the population is multiplied by 1.03, which is a 3% increase per year.

   Review: 2.5 Exponential Function Context and Data Modeling

9. Topic 2.5 · Work it outAn exponential model $f(x)=a{b}^x$ passes through $(0,50)$ and $(3,400)$. Find it.

   The initial value gives $a=50$. Then $50b^3=400$, so $b^3=8$ and $b=2$. The model is $f(x)=50\cdot 2^x$.

   Review: 2.5 Exponential Function Context and Data Modeling

10. Topic 2.5 · Say it in context$Q(d)=3\cdot 2^d$, with $d$ in days. What weekly growth factor does an equivalent form reveal?

    $Q(d)=3{\left(2^7\right)}^{d/7}=3\cdot {128}^{d/7}$, so the quantity is multiplied by 128 every week.

    Review: 2.5 Exponential Function Context and Data Modeling

11. Topic 2.6 · Say it in contextThe residual plot for a linear model shows a clear U shape. What does that tell you?

    The residuals follow a pattern, so the linear model is not appropriate. A model is justified when its residual plot appears without pattern; a model with curvature, such as quadratic or exponential, may fit better.

    Review: 2.6 Competing Function Model Validation

12. Topic 2.6 · Work it outA model predicts 52 at $x=4$, and the actual value is 47. Find the residual and say whether the model overestimates or underestimates.

    Residual = actual minus predicted $=47-52=-5$. The negative residual means the model overestimates at $x=4$.

    Review: 2.6 Competing Function Model Validation

13. Topic 2.6 · Know the termWhat is a residual, and how is a residual plot set up?

    A residual is the actual value minus the value the model predicts at a given input. A residual plot puts the residuals on the vertical axis and the independent variable on the horizontal axis.

    Review: 2.6 Competing Function Model Validation

14. Topic 2.7 · Work it out$f(x)=x^2+1$ and $g(x)=3x-2$. Find $f(g(1))$ and $g(f(1))$.

    $g(1)=1$, so $f(g(1))=f(1)=2$. Also $f(1)=2$, so $g(f(1))=g(2)=4$. The results differ because composition is not commutative.

    Review: 2.7 Composition of Functions

15. Topic 2.7 · Check the conditions$f(x)=\sqrt{x}$ and $g(x)=x-5$. What is the domain of $f(g(x))$?

    Only inputs whose output from $g$ is in the domain of $f$ work. That needs $x-5\ge 0$, so the domain is $x\ge 5$.

    Review: 2.7 Composition of Functions

16. Topic 2.7 · Know the termWhat is the identity function, and what does it do in a composition?

    It is $f(x)=x$. Composing it with any function $g$ gives back $g$: $g(f(x))=f(g(x))=g(x)$, the way adding 0 or multiplying by 1 leaves a number unchanged.

    Review: 2.7 Composition of Functions

17. Topic 2.8 · Work it outFind the inverse of $f(x)={\displaystyle \frac{x-4}{3}}$.

    Swap $x$ and $y$: $x={\displaystyle \frac{y-4}{3}}$. Solve: $y=3x+4$, so $f^{-1}(x)=3x+4$. Check: $f(f^{-1}(x))={\displaystyle \frac{3x+4-4}{3}}=x$.

    Review: 2.8 Inverse Functions

18. Topic 2.8 · Check the conditionsWhy does $f(x)=x^2$ need a restricted domain to have an inverse function, and what is one choice?

    Each output must come from a unique input, but $f(3)=f(-3)=9$. Restricting the domain to $x\ge 0$ makes $f$ invertible, with $f^{-1}(x)=\sqrt{x}$.

    Review: 2.8 Inverse Functions

19. Topic 2.9 · Work it outEvaluate ${\log }_{2}32$ and $\log 0.01$.

    $2^5=32$, so ${\log }_{2}32=5$. With no base written the base is 10, and ${10}^{-2}=0.01$, so $\log 0.01=-2$.

    Review: 2.9 Logarithmic Expressions

20. Topic 2.9 · Say it in contextOn a base 10 logarithmic scale, two marks are 3 units apart. How do the quantities they stand for compare?

    Each unit is a multiplicative change by the base, so the larger quantity is ${10}^3=1000$ times the smaller one.

    Review: 2.9 Logarithmic Expressions

21. Topic 2.10 · Know the termExponential growth versus logarithmic growth: how do the outputs change as the inputs change?

    Exponential: outputs change multiplicatively as inputs change additively. Logarithmic: outputs change additively as inputs change multiplicatively, so each doubling of the input adds the same amount to the output.

    Review: 2.10 Inverses of Exponential Functions

22. Topic 2.11 · Check the conditionsWhat are the domain and range of $f(x)=2{\log }_{3}x$, and what happens near $x=0$?

    Domain: $x>0$. Range: all real numbers. The graph is vertically asymptotic to $x=0$: as $x\to 0^{+}$, $f(x)\to -\infty$.

    Review: 2.11 Logarithmic Functions

23. Topic 2.11 · Spot the mistakeA student says a logarithmic function levels off to a horizontal asymptote as $x\to \infty$. Is that right?

    No. It grows slowly, but its end behavior is unbounded: ${\lim }_{x\to \infty }a{\log }_{b}x$ is $\infty$ or $-\infty$. It also has no extrema except on a closed interval.

    Review: 2.11 Logarithmic Functions

24. Topic 2.12 · Work it outRewrite ${\log }_2(8x)$ as a vertical translation of ${\log }_{2}x$.

    By the product property, ${\log }_2(8x)={\log }_{2}8+{\log }_{2}x=3+{\log }_{2}x$. A horizontal dilation of a logarithmic function is the same as a vertical translation.

    Review: 2.12 Logarithmic Function Manipulation

25. Topic 2.13 · Work it outSolve $5\cdot 2^x=30$.

    Divide by 5: $2^x=6$. Then $x={\log }_{2}6={\displaystyle \frac{\ln 6}{\ln 2}}\approx 2.585$.

    Review: 2.13 Exponential and Logarithmic Equations and Inequalities

26. Topic 2.13 · Spot the mistakeSolving $\log x+\log (x-3)=1$ produces $x=5$ and $x=-2$. What must you check before reporting both?

    Check for extraneous solutions. $\log (-2)$ is not defined, so $x=-2$ is rejected. The only solution is $x=5$, and it works: $\log 5+\log 2=\log 10=1$.

    Review: 2.13 Exponential and Logarithmic Equations and Inequalities

27. Topic 2.14 · Which one fitsA sound meter's reading goes up by 10 each time the sound's intensity is multiplied by 10. Which function type models the reading as a function of intensity?

    Logarithmic. The output rises by equal steps while the input is multiplied by a constant factor, which is the pattern a logarithmic model describes.

    Review: 2.14 Logarithmic Function Context and Data Modeling

28. Topic 2.14 · Work it outA model $f(x)=a{\log }_{2}x$ passes through $(8,12)$. Find $a$ and predict $f(32)$.

    $12=a{\log }_{2}8=3a$, so $a=4$. Then $f(32)=4{\log }_{2}32=4\cdot 5=20$.

    Review: 2.14 Logarithmic Function Context and Data Modeling

29. Topic 2.15 · Say it in contextData plotted on a semi-log plot, with the y-axis logarithmically scaled, fall close to a line. What kind of model fits the original data?

    An exponential model. With the y-axis logarithmically scaled, data with exponential characteristics appear linear.

    Review: 2.15 Semi-log Plots

30. Topic 2.15 · Work it outOn a base 10 semi-log plot, data from $y=a{b}^x$ follow the line $\log y=0.3x+2$. Find $a$ and $b$.

    The slope is $\log b$, so $b={10}^{0.3}\approx 1.995$. The intercept is $\log a$, so $a={10}^2=100$. The model is $y=100({10}^{0.3}{)}^x\approx 100(1.995{)}^x$.

    Review: 2.15 Semi-log Plots

Unit 3: Trigonometric and Polar Functions

1. Topic 3.1 · Work it outA periodic function has period 6, and $f(2)=5$. Find $f(14)$ and $f(-4)$.

   $14=2+2(6)$ and $-4=2-6$, so both inputs differ from 2 by whole periods. That makes $f(14)=f(-4)=5$.

   Review: 3.1 Periodic Phenomena

2. Topic 3.1 · Say it in contextThe water height at a harbor repeats every 12.4 hours. What does 12.4 mean for the function $h(t)$?

   It is the period, the smallest positive $k$ with $h(t+k)=h(t)$ for all $t$. Every feature of one 12.4-hour cycle, including its highs, lows and fastest rise, repeats in every cycle.

   Review: 3.1 Periodic Phenomena

3. Topic 3.2 · Work it outThe terminal ray of $\theta$ meets the unit circle at $(-0.6,0.8)$. Find $\sin \theta$, $\cos \theta$ and $\tan \theta$.

   $\sin \theta =0.8$, the y-coordinate. $\cos \theta =-0.6$, the x-coordinate. $\tan \theta ={\displaystyle \frac{0.8}{-0.6}}=-{\displaystyle \frac{4}{3}}$, the slope of the terminal ray.

   Review: 3.2 Sine, Cosine, and Tangent

4. Topic 3.3 · Work it outFind the exact values of $\sin {\displaystyle \frac{5\pi }{6}}$ and $\cos {\displaystyle \frac{5\pi }{6}}$.

   The terminal ray is in Quadrant II and makes an angle of ${\displaystyle \frac{\pi }{6}}$ with the negative x-axis. So $\sin {\displaystyle \frac{5\pi }{6}}={\displaystyle \frac{1}{2}}$ and $\cos {\displaystyle \frac{5\pi }{6}}=-{\displaystyle \frac{\sqrt{3}}{2}}$.

   Review: 3.3 Sine and Cosine Function Values

5. Topic 3.4 · Say it in contextAs $\theta$ increases from 0 to $\pi$, describe the outputs of $\sin \theta$ using the unit circle.

   The point moves counterclockwise from $(1,0)$ over the top to $(-1,0)$. Its y-coordinate rises from 0 to 1 at $\theta ={\displaystyle \frac{\pi }{2}}$, then falls back to 0.

   Review: 3.4 Sine and Cosine Function Graphs

6. Topic 3.4 · Know the termWhat are the domain and range of $f(\theta )=\cos \theta$?

   Domain: all real numbers, since every angle measure has a cosine. Range: $[-1,1]$; the outputs oscillate between $-1$ and $1$, taking every value in between.

   Review: 3.4 Sine and Cosine Function Graphs

7. Topic 3.5 · Work it outA sinusoidal graph has a maximum of 14 and a minimum of 2. Find the amplitude and the midline.

   Amplitude is half the difference: ${\displaystyle \frac{14-2}{2}}=6$. The midline is the mean of the maximum and minimum: $y={\displaystyle \frac{14+2}{2}}=8$.

   Review: 3.5 Sinusoidal Functions

8. Topic 3.5 · Spot the mistakeA student says $\cos \theta$ is an odd function, just like $\sin \theta$. Correct it.

   The graph of $y=\cos \theta$ has reflective symmetry over the y-axis, so cosine is even. Sine is the odd one: its graph has rotational symmetry about the origin.

   Review: 3.5 Sinusoidal Functions

9. Topic 3.6 · Work it outFor $f(\theta )=3\sin \left(2(\theta -{\displaystyle \frac{\pi }{4}})\right)-1$, give the amplitude, period, midline and phase shift.

   Amplitude 3. Period $\left|{\displaystyle \frac{1}{2}}\right|\cdot 2\pi =\pi$. Midline $y=-1$. Here $c=-{\displaystyle \frac{\pi }{4}}$, so the phase shift is $-c={\displaystyle \frac{\pi }{4}}$, to the right.

   Review: 3.6 Sinusoidal Function Transformations

10. Topic 3.6 · Spot the mistakeA student says $g(\theta )=\sin (3\theta )$ has period $6\pi$. What went wrong?

    The period changes by a factor of $\left|{\displaystyle \frac{1}{b}}\right|$, so it becomes ${\displaystyle \frac{1}{3}}\cdot 2\pi ={\displaystyle \frac{2\pi }{3}}$. Multiplying the input by 3 makes the cycles three times as short, not three times as long.

    Review: 3.6 Sinusoidal Function Transformations

11. Topic 3.7 · Work it outA city's temperature peaks at 30 degrees at 3 p.m. and bottoms out at 18 degrees at 3 a.m. every day. Find the amplitude, midline and period for a sinusoidal model.

    Amplitude: ${\displaystyle \frac{30-18}{2}}=6$ degrees. Midline: $y=24$ degrees. Consecutive maxima are 24 hours apart, so the period is 24 hours.

    Review: 3.7 Sinusoidal Function Context and Data Modeling

12. Topic 3.7 · Say it in context$h(t)=20-18\cos \left({\displaystyle \frac{\pi }{5}}t\right)$ gives a Ferris wheel rider's height in meters after $t$ minutes. Interpret the 20 and the period.

    20 m is the midline, the height of the center of the wheel. The period is ${\displaystyle \frac{2\pi }{\pi /5}}=10$, so one full turn of the wheel takes 10 minutes.

    Review: 3.7 Sinusoidal Function Context and Data Modeling

13. Topic 3.8 · Work it outOn $0\le \theta \le 2\pi$, where does $f(\theta )=\tan \theta$ have vertical asymptotes, and what is its period?

    Asymptotes are where $\cos \theta =0$: $\theta ={\displaystyle \frac{\pi }{2}}$ and $\theta ={\displaystyle \frac{3\pi }{2}}$. The period is $\pi$, because the slope of the terminal ray repeats every half revolution.

    Review: 3.8 The Tangent Function

14. Topic 3.9 · Work it outEvaluate $\arcsin (-{\displaystyle \frac{1}{2}})$ and $\arccos (-{\displaystyle \frac{1}{2}})$.

    Arcsine outputs lie in $[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$, so $\arcsin (-{\displaystyle \frac{1}{2}})=-{\displaystyle \frac{\pi }{6}}$. Arccosine outputs lie in $[0,\pi ]$, so $\arccos (-{\displaystyle \frac{1}{2}})={\displaystyle \frac{2\pi }{3}}$.

    Review: 3.9 Inverse Trigonometric Functions

15. Topic 3.9 · Spot the mistakeA student says $\arcsin (\sin \pi )=\pi$. Fix it.

    $\sin \pi =0$, and $\arcsin 0=0$. Sine is invertible only on the restricted domain $[-{\displaystyle \frac{\pi }{2}},{\displaystyle \frac{\pi }{2}}]$, and $\pi$ is outside it.

    Review: 3.9 Inverse Trigonometric Functions

16. Topic 3.10 · Work it outSolve $2\sin \theta =1$ for $0\le \theta <2\pi$.

    $\sin \theta ={\displaystyle \frac{1}{2}}$. Arcsine gives ${\displaystyle \frac{\pi }{6}}$. Sine is also positive in Quadrant II, which gives ${\displaystyle \frac{5\pi }{6}}$. Solutions: ${\displaystyle \frac{\pi }{6}}$ and ${\displaystyle \frac{5\pi }{6}}$.

    Review: 3.10 Trigonometric Equations and Inequalities

17. Topic 3.10 · Spot the mistakeSolving $\cos \theta =0.3$ with no domain restriction, a student finds $\theta \approx 1.266$ and stops. What is missing?

    Trigonometric equations usually have infinitely many solutions. Cosine is even and periodic, so $\theta \approx \pm 1.266+2\pi k$ for any integer $k$. A context often limits which of these count.

    Review: 3.10 Trigonometric Equations and Inequalities

18. Topic 3.11 · Work it outFind $\sec {\displaystyle \frac{\pi }{3}}$ and $\cot {\displaystyle \frac{\pi }{4}}$.

    $\cos {\displaystyle \frac{\pi }{3}}={\displaystyle \frac{1}{2}}$, so $\sec {\displaystyle \frac{\pi }{3}}=2$. $\tan {\displaystyle \frac{\pi }{4}}=1$, so $\cot {\displaystyle \frac{\pi }{4}}=1$.

    Review: 3.11 The Secant, Cosecant, and Cotangent Functions

19. Topic 3.11 · Spot the mistakeA student says $\csc \theta =0.5$ for some angle. Why is that impossible?

    Cosecant is the reciprocal of sine, and sine values lie between $-1$ and $1$. So cosecant values are never between them: the range is $(-\infty ,-1]\cup [1,\infty )$.

    Review: 3.11 The Secant, Cosecant, and Cotangent Functions

20. Topic 3.12 · Work it out$\sin \theta ={\displaystyle \frac{3}{5}}$ and $\theta$ is in Quadrant II. Find $\cos \theta$.

    Pythagorean identity: ${\cos }^2\theta =1-{\displaystyle \frac{9}{25}}={\displaystyle \frac{16}{25}}$. Cosine is negative in Quadrant II, so $\cos \theta =-{\displaystyle \frac{4}{5}}$.

    Review: 3.12 Equivalent Representations of Trigonometric Functions

21. Topic 3.12 · Work it outUse a sum identity to find $\sin {\displaystyle \frac{7\pi }{12}}$ exactly.

    ${\displaystyle \frac{7\pi }{12}}={\displaystyle \frac{\pi }{3}}+{\displaystyle \frac{\pi }{4}}$, so $\sin {\displaystyle \frac{7\pi }{12}}={\displaystyle \frac{\sqrt{3}}{2}}\cdot {\displaystyle \frac{\sqrt{2}}{2}}+{\displaystyle \frac{1}{2}}\cdot {\displaystyle \frac{\sqrt{2}}{2}}={\displaystyle \frac{\sqrt{6}+\sqrt{2}}{4}}$.

    Review: 3.12 Equivalent Representations of Trigonometric Functions

22. Topic 3.13 · Work it outConvert the polar point $(4,{\displaystyle \frac{2\pi }{3}})$ to rectangular coordinates.

    $x=4\cos {\displaystyle \frac{2\pi }{3}}=-2$ and $y=4\sin {\displaystyle \frac{2\pi }{3}}=2\sqrt{3}$, so the point is $(-2,2\sqrt{3})$.

    Review: 3.13 Trigonometry and Polar Coordinates

23. Topic 3.13 · Say it in contextWhere is the polar point $(-2,{\displaystyle \frac{\pi }{6}})$?

    A negative $r$ means move 2 units along the line containing the terminal ray of ${\displaystyle \frac{\pi }{6}}$, but in the opposite direction. It is the same point as $(2,{\displaystyle \frac{7\pi }{6}})$.

    Review: 3.13 Trigonometry and Polar Coordinates

24. Topic 3.14 · Know the termIn the graph of a polar function $r=f(\theta )$, what do the input and the output stand for?

    The input $\theta$ is an angle measure from the polar axis. The output $r$ is a radius: the signed displacement from the origin along the line of that angle.

    Review: 3.14 Polar Function Graphs

25. Topic 3.14 · Work it outFor $r=2+2\cos \theta$, find $r$ at $\theta =0$, ${\displaystyle \frac{\pi }{2}}$ and $\pi$, and describe each point.

    At $\theta =0$, $r=4$: the point 4 units out on the polar axis. At ${\displaystyle \frac{\pi }{2}}$, $r=2$: 2 units up the y-axis. At $\pi$, $r=0$, so the curve passes through the origin.

    Review: 3.14 Polar Function Graphs

26. Topic 3.15 · Say it in contextOn an interval, $r=f(\theta )$ is negative and decreasing. Is the point on the graph getting closer to the origin or farther away?

    Farther. Negative and decreasing means $r$ moves away from 0, so the distance from the origin increases.

    Review: 3.15 Rates of Change in Polar Functions

27. Topic 3.15 · Work it out$f\left({\displaystyle \frac{\pi }{6}}\right)=3$ and $f\left({\displaystyle \frac{\pi }{2}}\right)=5$ for $r=f(\theta )$. Find the average rate of change of $r$ with respect to $\theta$ and interpret it.

    ${\displaystyle \frac{5-3}{\pi /2-\pi /6}}={\displaystyle \frac{2}{\pi /3}}={\displaystyle \frac{6}{\pi }}\approx 1.91$. Over that interval the signed radius grows by about 1.91 units per radian, on average.

    Review: 3.15 Rates of Change in Polar Functions

Unit 4: Functions Involving Parameters, Vectors, and Matrices

1. Topic 4.1 · Know the term · Not on the AP ExamWhat is a parametric function in the plane?

   A pair of equations, $x(t)$ and $y(t)$, that make both coordinates depend on a single independent variable $t$, the parameter. Together they form $f(t)=(x(t),y(t))$.

   Review: 4.1 Parametric Functions

2. Topic 4.2 · Work it out · Not on the AP ExamA particle moves with $x(t)=t^2-4$ and $y(t)=t+1$ for $t\ge 0$. When and where does it cross the y-axis?

   It is on the y-axis when $x(t)=0$, so $t^2=4$ and, with $t\ge 0$, $t=2$. Then $y(2)=3$, so the point is $(0,3)$.

   Review: 4.2 Parametric Functions Modeling Planar Motion

3. Topic 4.3 · Say it in context · Not on the AP ExamOn an interval, $x(t)$ is decreasing and $y(t)$ is increasing. Which way is the particle moving?

   Left, because $x$ is decreasing, and up, because $y$ is increasing. So it moves up and to the left.

   Review: 4.3 Parametric Functions and Rates of Change

4. Topic 4.4 · Work it out · Not on the AP ExamParametrize the circle of radius 3 centered at $(1,-2)$, traced counterclockwise once.

   $x(t)=1+3\cos t$ and $y(t)=-2+3\sin t$ for $0\le t\le 2\pi$. It is a transformation of $(\cos t,\sin t)$.

   Review: 4.4 Parametrically Defined Circles and Lines

5. Topic 4.5 · Work it out · Not on the AP ExamFrom $x^2+y^2=25$, write a function whose graph is the top half of the circle.

   Solve for $y$ and keep the positive root: $y=\sqrt{25-x^2}$ for $-5\le x\le 5$. The negative root gives a second function, the bottom half.

   Review: 4.5 Implicitly Defined Functions

6. Topic 4.6 · Which one fits · Not on the AP ExamWhich conic section is ${\displaystyle \frac{(x-1{)}^2}{9}}-{\displaystyle \frac{(y+2{)}^2}{4}}=1$, and which way does it open?

   A hyperbola centered at $(1,-2)$. The $x$ term is the positive one, so it opens left and right. Its asymptotes are $y+2=\pm {\displaystyle \frac{2}{3}}(x-1)$.

   Review: 4.6 Conic Sections

7. Topic 4.7 · Work it out · Not on the AP ExamParametrize the ellipse ${\displaystyle \frac{(x-2{)}^2}{9}}+{\displaystyle \frac{(y-1{)}^2}{4}}=1$.

   $x(t)=2+3\cos t$ and $y(t)=1+2\sin t$ for $0\le t\le 2\pi$. Substituting gives ${\cos }^{2}t+{\sin }^{2}t=1$, so every point satisfies the equation.

   Review: 4.7 Parametrization of Implicitly Defined Functions

8. Topic 4.8 · Work it out · Not on the AP Exam$\overrightarrow{u}=⟨3,-4⟩$. Find the magnitude of $\overrightarrow{u}$ and a unit vector in the same direction.

   Magnitude: $\sqrt{9+16}=5$. Multiply by the reciprocal, ${\displaystyle \frac{1}{5}}$: $⟨{\displaystyle \frac{3}{5}},-{\displaystyle \frac{4}{5}}⟩$.

   Review: 4.8 Vectors

9. Topic 4.9 · Say it in context · Not on the AP ExamAt $t=1$, a particle's velocity vector is $⟨-2,3⟩$. Which way is it moving, and how fast?

   The first component is negative, so it moves left; the second is positive, so it moves up. Its speed is the magnitude: $\sqrt{4+9}=\sqrt{13}\approx 3.606$.

   Review: 4.9 Vector-Valued Functions

10. Topic 4.10 · Work it out · Not on the AP ExamMultiply $\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]\left[\begin{array}{c}5\\ 6\end{array}\right]$.

    Each entry is a row dotted with the column: $1(5)+2(6)=17$ and $3(5)+4(6)=39$. The product is $\left[\begin{array}{c}17\\ 39\end{array}\right]$.

    Review: 4.10 Matrices

11. Topic 4.11 · Work it out · Not on the AP ExamFind $\det (A)$ and $A^{-1}$ for $A=\left[\begin{array}{cc}4& 7\\ 2& 6\end{array}\right]$.

    $\det (A)=4(6)-7(2)=10$. It is not 0, so the inverse exists: $A^{-1}={\displaystyle \frac{1}{10}}\left[\begin{array}{cc}6& -7\\ -2& 4\end{array}\right]$.

    Review: 4.11 The Inverse and Determinant of a Matrix

12. Topic 4.12 · Check the conditions · Not on the AP ExamCan $L(⟨x,y⟩)=⟨x+2,3y⟩$ be a linear transformation?

    No. A linear transformation maps the zero vector to the zero vector, but $L(⟨0,0⟩)=⟨2,0⟩$.

    Review: 4.12 Linear Transformations and Matrices

13. Topic 4.13 · Work it out · Not on the AP ExamWrite the matrix that rotates every vector by ${\displaystyle \frac{\pi }{2}}$ counterclockwise about the origin.

    Use $\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$ with $\theta ={\displaystyle \frac{\pi }{2}}$: $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$. It sends $⟨1,0⟩$ to $⟨0,1⟩$.

    Review: 4.13 Matrices as Functions

14. Topic 4.14 · Work it out · Not on the AP ExamEach year 10% of a city's renters buy homes and 5% of its owners become renters. Now there are 1000 renters and 500 owners. Predict next year's numbers.

    Renters: $0.90(1000)+0.05(500)=925$. Owners: $0.10(1000)+0.95(500)=575$. As a product: $\left[\begin{array}{cc}0.90& 0.05\\ 0.10& 0.95\end{array}\right]\left[\begin{array}{c}1000\\ 500\end{array}\right]=\left[\begin{array}{c}925\\ 575\end{array}\right]$.

    Review: 4.14 Matrices Modeling Contexts

15. Topic 4.14 · Know the term · Not on the AP ExamIn a transition matrix model, what is a steady state, and how can you find it?

    A distribution between states that does not change from one step to the next. Multiply the state vector by the transition matrix again and again; the results approach it.

    Review: 4.14 Matrices Modeling Contexts