AP Precalculus Unit 1: Polynomial and Rational Functions

How two quantities change together; average rates of change and what they say about increasing, decreasing and concavity; the patterns of change in linear and quadratic functions; zeros (real and complex), multiplicity and end behavior of polynomials; zeros, vertical asymptotes, holes and end behavior of rational functions; equivalent forms (factoring, long division, the binomial theorem); transformations; and choosing, building and judging a function model.

CED topics in this unit: 1.1, 1.2, 1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9, 1.10, 1.11, 1.12, 1.13, 1.14.

Key formulas and facts

• Average rate of change of $f$ over $[a,b]$: $\dfrac{f(b)-f(a)}{b-a}$, the slope of the secant line. Units: output units per input unit.
• Linear functions: equal input steps give equal output differences. Quadratic functions: equal input steps give average rates of change that change by a constant amount (constant second differences).
• Concave up on an interval: the rate of change is increasing there. Concave down: the rate of change is decreasing. A point of inflection is where the rate of change switches between increasing and decreasing.
• A degree-$n$ polynomial has exactly $n$ complex zeros counted with multiplicity, at most $n-1$ turning points, and nonreal zeros in conjugate pairs $a\pm bi$ when its coefficients are real.
• Multiplicity: at a zero of odd multiplicity the graph crosses the $x$-axis; at a zero of even multiplicity it touches and turns back.
• End behavior of a polynomial comes from its leading term $a_nx^n$: for example, $\lim_{x\to\infty}(-2x^4+x)=-\infty$.
• Rational function $r=\frac{p}{q}$ in lowest terms: zeros where $p=0$; vertical asymptotes where $q=0$. A factor that cancels completely from the denominator gives a hole instead.
• Horizontal asymptote: degree of $p$ less than degree of $q$ gives $y=0$; equal degrees give $y=$ (ratio of leading coefficients); degree of $p$ one more than degree of $q$ gives a slant asymptote (the quotient from long division).
• Binomial theorem: $(a+b)^n=\sum_{k=0}^{n}\binom nk a^{n-k}b^k$. Remainder theorem: dividing $p(x)$ by $x-c$ leaves remainder $p(c)$.
• Transformations of $y=f(x)$: $f(x-h)+k$ shifts right $h$ and up $k$; $af(x)$ is a vertical dilation by $|a|$ (a reflection over the $x$-axis if $a<0$); $f(bx)$ is a horizontal dilation by $\frac1{|b|}$ (a reflection over the $y$-axis if $b<0$).

Worked examples

• Average rate of change: for $f(x)=x^2-3x$ on $[2,5]$, $\frac{f(5)-f(2)}{5-2}=\frac{10-(-2)}{3}=4$.
• Second differences: the outputs $1, 4, 11, 22, 37$ at $x=0,1,2,3,4$ have first differences $3, 7, 11, 15$ and second differences all $4$, so a quadratic fits: $y=2x^2+x+1$.
• Complex zeros: $p(x)=x^3-x^2+4x-4=(x-1)(x^2+4)$ has zeros $1$, $2i$ and $-2i$.
• Sign chart: $\frac{(x+1)(x-2)^2}{x-4}<0$ on $(-1,2)\cup(2,4)$; the sign does not change at $x=2$ (even multiplicity) and $x=2$ itself gives $0$, so it is left out.
• Hole and asymptote: $r(x)=\frac{x^2-x-6}{x^2-9}=\frac{(x-3)(x+2)}{(x-3)(x+3)}$ has a hole at $\left(3,\frac56\right)$, a vertical asymptote $x=-3$ and a horizontal asymptote $y=1$.
• Slant asymptote: $\frac{2x^2-x+3}{x+1}=2x-3+\frac{6}{x+1}$, so $y=2x-3$.
• Binomial theorem: the $x^2$ term of $(3x+2)^4$ is $\binom42(3x)^2(2)^2=216x^2$.
• Transformation: if $(4,-2)$ is on $y=f(x)$, then $(2,1)$ is on $y=-\frac12f(2x)$.

Common traps

• Calling every zero of a denominator a vertical asymptote: check whether the factor cancels first (then it is a hole).
• Giving a zero of a rational function where the denominator is also zero: the function is undefined there.
• Reading $f(x+3)$ as a shift right: it moves the graph $3$ units left.
• Deciding end behavior from a large middle coefficient: only the leading term matters as $x\to\pm\infty$.
• Writing an average rate of change without units, or reversing the order of the subtraction in the numerator and denominator.
• Forgetting that a real-coefficient polynomial with zero $3+i$ also has zero $3-i$, which raises the least possible degree.
• Extending a model outside the inputs where it makes sense (negative heights, negative volumes, times after an event ends).

Practice questions

127 multiple-choice and 4 free-response questions, each tagged with its CED topics and with "No calculator" or "Graphing calculator". Try each one before opening the answer. Each line of a free-response scoring guide is one point, and every free-response question here is out of 6. 33 of the 127 multiple-choice questions here need a graphing calculator; on the May 2027 exam, Part A of the multiple choice is 29 questions with no calculator and Part B is 13 questions with a graphing calculator. A free account opens the worked answers and scoring guides of 20 questions a day; Pro opens every one.

Question 1 · Multiple choice

The graph of $y=f(x)$ is shown, where $f(x)=x^3-6x^2+9x$. On which interval is $f$ decreasing?

1. $(0,3)$
2. $(1,3)$
3. $(3,\infty)$
4. $(-\infty,1)$

Question 2 · Multiple choice

The table shows the distance $d$, in miles, that a cyclist is from home $t$ hours after leaving. Which statement best describes how $d$ changes as $t$ increases from $0$ to $4$?

1. $d$ increases by the same amount each hour
2. $d$ increases, but by a smaller amount each hour
3. $d$ decreases each hour
4. $d$ increases, and it increases by a larger amount each hour

Question 3 · Multiple choice

Let $g(x)=(x+2)(x-1)(x-5)$. For which values of $x$ is $g(x)<0$?

1. $(-2,1)\cup(5,\infty)$
2. $(-2,5)$
3. $(-\infty,-2)\cup(1,5)$
4. $(1,5)$

Question 4 · Multiple choice

Let $h(x)=8-\frac{x^2}{2}$. Which statement describes $h$ on the interval $[0,4]$?

1. $h$ is decreasing, and its rate of change is decreasing
2. $h$ is decreasing, and its rate of change is increasing
3. $h$ is increasing, and its rate of change is decreasing
4. $h$ is increasing, and its rate of change is increasing

Question 5 · Multiple choice

The graph of $y=f(x)$, where $f(x)=-x^2+2x+3$, is shown. On which interval are both $f(x)>0$ and $f$ increasing?

1. $(1,3)$
2. $(-1,3)$
3. $(-\infty,1)$
4. $(-1,1)$

Question 6 · Multiple choice

The table gives values of a quadratic function $f$. Between which consecutive $x$-values in the table must $f$ have a zero?

1. Between $-1$ and $0$, and between $2$ and $3$
2. Between $0$ and $1$, and between $1$ and $2$
3. Between $-1$ and $0$ only
4. Between $2$ and $3$ only

Question 7 · Multiple choice

The height, in feet, of a ball $t$ seconds after it is thrown is $h(t)=-16t^2+48t+4$ for $0\le t\le 3.1$. During which time interval is the ball rising?

1. $(1.5,3.1)$
2. $(0,3)$
3. $(0,1.5)$
4. $(0,3.1)$

Question 8 · Multiple choice

Let $f(x)=0.2x^3-1.4x^2+2x+4$. The function $f$ is increasing on $(-\infty,a]$ and on $[b,\infty)$, and decreasing on $[a,b]$. To three decimal places, what is $a$?

1. $0.880$
2. $3.786$
3. $2.333$
4. $4.812$

Question 9 · Multiple choice

Let $g(x)=x^4-3x^3+x+2$. On the interval $1\le x\le3$, the graph of $g$ changes from decreasing to increasing at exactly one value $x=c$. To three decimal places, what is $c$?

1. $-4.318$
2. $2.198$
3. $1.500$
4. $3.000$

Question 10 · Multiple choice

The height of a weather balloon, in meters, is modeled by $h(t)=-0.3t^3+2.2t^2+1.5t+5$ for $0\le t\le7$, where $t$ is in minutes. To the nearest thousandth of a minute, at what time is the balloon highest?

1. $30.106$
2. $2.444$
3. $5.209$
4. $7.000$

Question 11 · Multiple choice

What is the average rate of change of $f(x)=x^3-2x$ over the interval $[1,3]$?

1. $22$
2. $11$
3. $13$
4. $\frac{25}{3}$

Question 12 · Multiple choice

The table gives the temperature $T$, in degrees Fahrenheit, of an oven $m$ minutes after it is switched on. What is the average rate of change of $T$ from $m=2$ to $m=9$, in degrees per minute?

1. $3$
2. $\frac{15}{7}$
3. $\frac{7}{15}$
4. $\frac{61}{9}$

Question 13 · Multiple choice

The graph of $y=g(x)$, where $g(x)=4-(x-2)^2$, is shown. What is the average rate of change of $g$ over $[0,3]$?

1. $1$
2. $3$
3. $-1$
4. $\frac{1}{3}$

Question 14 · Multiple choice

Let $g(x)=\sqrt{x^2+5}$. What is the average rate of change of $g$ over $[1,4]$?

1. $2.133$
2. $0.533$
3. $1.000$
4. $0.711$

Question 15 · Multiple choice

The population $P$ of a town, in thousands, is modeled by a function of time $t$ in years. The average rate of change of $P$ over $[2,6]$ is $3.5$ thousand people per year, and $P(2)=40$. What is $P(6)$?

1. $43.5$
2. $47$
3. $54$
4. $61$

Question 16 · Multiple choice

For $f(x)=x^2+kx$, the average rate of change of $f$ over $[1,3]$ is $10$. What is the value of $k$?

1. $2$
2. $10$
3. $6$
4. $8$

Question 17 · Multiple choice

Let $h(x)=2^x+x^2$. What is the average rate of change of $h$ over the interval $[0.5,2.5]$?

1. $10.243$
2. $6.786$
3. $4.763$
4. $5.121$

Question 18 · Multiple choice

Let $r(x)=\dfrac{3x+1}{x^2+2}$. What is the average rate of change of $r$ over $[1.3,\,3.7]$?

1. $-0.232$
2. $0.232$
3. $-0.557$
4. $-0.111$

Question 19 · Multiple choice

The temperature of a liquid, in degrees Celsius, $t$ minutes into an experiment is modeled by $C(t)=0.02t^3-0.5t^2+4t+10$. What is the average rate of change of the temperature, in degrees per minute, from $t=2.5$ to $t=8.5$?

1. $2.970$
2. $0.495$
3. $0.349$
4. $0.315$

Question 20 · Multiple choice

Which type of function best fits the data in the table?

1. Linear
2. Quadratic
3. Exponential
4. Cubic, but not quadratic

Question 21 · Multiple choice

The table gives values of a quadratic function $q$ at equally spaced inputs. What is $q(6)$?

1. $40$
2. $49$
3. $47$
4. $53$

Question 22 · Multiple choice

A linear function $f$ has an average rate of change of $4$ over every interval, and $f(2)=7$. What is $f(10)$?

1. $39$
2. $32$
3. $47$
4. $40$

Question 23 · Multiple choice

The average rates of change of a quadratic function $f$ over $[0,1]$, $[1,2]$ and $[2,3]$ are $7$, $3$ and $-1$. What is the average rate of change of $f$ over $[3,4]$?

1. $-5$
2. $-1$
3. $-4$
4. $-3$

Question 24 · Multiple choice

Let $f(x)=-2x^2+12x+1$. For any $k$, the average rate of change of $f$ over $[k+1,k+2]$ minus the average rate of change over $[k,k+1]$ is

1. $-2$
2. $12$
3. $-4$
4. $4$

Question 25 · Multiple choice

A linear function $g$ has $g(-1)=4$ and $g(3)=-8$. Which of the following is $g(x)$?

1. $-3x+4$
2. $-3x+1$
3. $3x+7$
4. $-x+3$

Question 26 · Multiple choice

A quadratic function $h$ has $h(0)=1$. Its average rate of change over $[0,2]$ is $6$, and over $[2,4]$ it is $-2$. What is the maximum value of $h$?

1. $13$
2. $\frac{25}{2}$
3. $9$
4. $\frac{27}{2}$

Question 27 · Multiple choice

A quadratic regression is fit to the data in the table. For the regression model, the average rates of change over consecutive intervals of length $1$ change by the same amount each time. To three decimal places, what is that amount?

1. $0.677$
2. $3.594$
3. $1.354$
4. $3.600$

Question 28 · Multiple choice

A linear regression is fit to the data in the table. To three decimal places, what is the rate of change of the regression line?

1. $9.650$
2. $0.528$
3. $1.925$
4. $1.895$

Question 29 · Multiple choice

A quadratic regression is fit to the data in the table. To three decimal places, what is the average rate of change of the regression model over $[2,5]$?

1. $-0.038$
2. $-0.067$
3. $-0.114$
4. $-1.620$

Question 30 · Multiple choice

The graph of the polynomial $p(x)=x^4-4x^2+2$ is shown. How many relative extrema does $p$ have?

1. $2$
2. $3$
3. $4$
4. $1$

Question 31 · Multiple choice

The graph of $p(x)=-x^3+3x+1$ is shown for $-2\le x\le 2$. What is the absolute maximum value of $p$ on $[-2,2]$?

1. $3$
2. $1$
3. $5$
4. $-1$

Question 32 · Multiple choice

Let $p(x)=x^3-6x^2+5$. At what value of $x$ does the graph of $p$ have a point of inflection?

1. $0$
2. $4$
3. $6$
4. $2$

Question 33 · Multiple choice

On which interval is the graph of $q(x)=x^3+3x^2-9x$ concave down?

1. $(-1,\infty)$
2. $(-3,1)$
3. $(-\infty,-1)$
4. $(-\infty,-3)$

Question 34 · Multiple choice

For a function $f$, the average rates of change over $[0,1]$, $[1,2]$, $[2,3]$ and $[3,4]$ are $5$, $2$, $-1$ and $-4$. Which statement is consistent with this information?

1. The graph of $f$ is concave down on $[0,4]$
2. The graph of $f$ is concave up on $[0,4]$
3. $f$ is linear on $[0,4]$
4. $f$ is decreasing on $[0,4]$

Question 35 · Multiple choice

What is the absolute maximum value of $p(x)=-x^4+8x^2-3$?

1. $-3$
2. $16$
3. $29$
4. $13$

Question 36 · Multiple choice

What is the greatest number of relative extrema a polynomial function of degree $5$ can have?

1. $5$
2. $4$
3. $3$
4. $6$

Question 37 · Multiple choice

Let $p(x)=0.5x^3-2x^2-3x+7$. To three decimal places, what is the relative maximum value of $p$?

1. $-0.610$
2. $7.972$
3. $-6.713$
4. $7.000$

Question 38 · Multiple choice

The graph of $q(x)=x^4-2x^3-5x^2+x$ has two points of inflection. To three decimal places, what is the $x$-coordinate of the one farther to the right?

1. $-0.541$
2. $2.471$
3. $1.541$
4. $-12.010$

Question 39 · Multiple choice

Let $p(x)=x^4-4.2x^2+x+3$. To three decimal places, what is the absolute minimum value of $p$ on the interval $[-2,2]$?

1. $0.008$
2. $-1.505$
3. $0.200$
4. $-2.888$

Question 40 · Multiple choice

A polynomial $p$ with real coefficients has degree $4$. Two of its zeros are $2$ and $3+i$. Which of the following must also be a zero of $p$?

1. $-3+i$
2. $-2$
3. $3-i$
4. $-3-i$

Question 41 · Multiple choice

Let $p(x)=(x-1)^2(x+3)^3$. Which statement describes the graph of $p$ at $x=-3$?

1. It touches the $x$-axis without crossing
2. It crosses the $x$-axis
3. It has a vertical asymptote
4. It does not meet the $x$-axis

Question 42 · Multiple choice

How many distinct real zeros does $p(x)=x^4-10x^2+9$ have?

1. $2$
2. $0$
3. $3$
4. $4$

Question 43 · Multiple choice

Find every zero, real or nonreal, of the cubic $p(x)=x^3-3x^2+9x-27$.

1. $3$ only
2. $-3,\ 3i,\ -3i$
3. $3,\ 3i,\ -3i$
4. $3,\ -3,\ 3i$

Question 44 · Multiple choice

Which polynomial has zeros only at $x=1$ (multiplicity $2$) and $x=-4$ (multiplicity $1$), and leading coefficient $1$?

1. $(x-1)^2(x+4)$
2. $(x+1)^2(x-4)$
3. $(x-1)(x+4)^2$
4. $(x-1)(x+4)$

Question 45 · Multiple choice

A polynomial of degree $6$ with real coefficients has exactly two distinct real zeros, each of multiplicity $1$. How many nonreal zeros does it have, counted with multiplicity?

1. $4$
2. $2$
3. $6$
4. $3$

Question 46 · Multiple choice

For what value of $k$ is $x+2$ a factor of $x^3+kx^2-4x-12$?

1. $-3$
2. $5$
3. $1$
4. $3$

Question 47 · Multiple choice

The polynomial $p(x)=x^3+2x-7$ has exactly one real zero. To three decimal places, what is it?

1. $1.569$
2. $1.913$
3. $3.500$
4. $-1.569$

Question 48 · Multiple choice

To three decimal places, what is the greatest real zero of $p(x)=x^4-5x^2+3x+1$?

1. $-2.461$
2. $1.700$
3. $1.000$
4. $2.236$

Question 49 · Multiple choice

The polynomial $p(x)=2x^3-x^2-8x+3$ has one zero between $x=1$ and $x=3$. To three decimal places, what is that zero?

1. $0.371$
2. $-1.948$
3. $2.078$
4. $1.500$

Question 50 · Multiple choice

Find $\displaystyle\lim_{x\to-\infty}\left(4x^2-3x^5\right)$.

1. $-\infty$
2. $\infty$
3. $-2$
4. $0$

Question 51 · Multiple choice

Which describes the end behavior of $p(x)=(3-x)(x+1)^2(2x-1)$?

1. $\lim_{x\to-\infty}p(x)=\infty$ and $\lim_{x\to\infty}p(x)=\infty$
2. $\lim_{x\to-\infty}p(x)=-\infty$ and $\lim_{x\to\infty}p(x)=\infty$
3. $\lim_{x\to-\infty}p(x)=-\infty$ and $\lim_{x\to\infty}p(x)=-\infty$
4. $\lim_{x\to-\infty}p(x)=\infty$ and $\lim_{x\to\infty}p(x)=-\infty$

Question 52 · Multiple choice

The graph of a polynomial $p$ is shown. Which of the following could be $p(x)$?

1. $-(x+2)(x-1)^2$
2. $(x+2)(x-1)^2$
3. $(x-2)(x+1)^2$
4. $(x+2)^2(x-1)$

Question 53 · Multiple choice

A polynomial $p$ satisfies $\displaystyle\lim_{x\to-\infty}p(x)=-\infty$ and $\displaystyle\lim_{x\to\infty}p(x)=-\infty$. Which of the following could be $p(x)$?

1. $-x^4+x$
2. $x^3-5$
3. $-x^3$
4. $2x^4$

Question 54 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{(2x-1)^3(4-x^2)}{x^5}$?

1. $8$
2. $-2$
3. $-8$
4. $4$

Question 55 · Multiple choice

Let $f(x)=0.01x^3-50x^2$. What is $\displaystyle\lim_{x\to\infty}f(x)$?

1. $-\infty$
2. $0$
3. $-50$
4. $\infty$

Question 56 · Multiple choice

Let $p(x)=x^4-30x^3$. What is the least number $N$ such that $p(x)>0$ for all $x>N$?

1. $29$
2. $31$
3. $0$
4. $30$

Question 57 · Multiple choice

Which statement about $g(x)=-3x^6+100x^5$ is true?

1. $g(x)\to\infty$ as $x\to\infty$ and as $x\to-\infty$
2. $g(x)\to\infty$ as $x\to\infty$, and $g(x)\to-\infty$ as $x\to-\infty$
3. $g(x)\to-\infty$ as $x\to\infty$ and as $x\to-\infty$
4. $g(x)\to-\infty$ as $x\to\infty$, and $g(x)\to\infty$ as $x\to-\infty$

Question 58 · Multiple choice

Let $p(x)=0.05x^4-2x$ and $q(x)=3x^3+10$. As $x\to\infty$, $p(x)$ eventually exceeds $q(x)$: there is a number $c$ such that $p(x)>q(x)$ for every $x>c$, and $p(c)=q(c)$. To three decimal places, what is $c$?

1. $60.000$
2. $20.000$
3. $15.000$
4. $60.012$

Question 59 · Multiple choice

What is the horizontal asymptote of the graph of $r(x)=\dfrac{6x^2-1}{3x^2+x}$?

1. $y=2$
2. $y=6$
3. $y=\frac{1}{3}$
4. $y=0$

Question 60 · Multiple choice

What is $\displaystyle\lim_{x\to\infty}\frac{4x^3-x}{2x^2+7}$?

1. $2$
2. $\infty$
3. $0$
4. $-\infty$

Question 61 · Multiple choice

Which line is a slant asymptote of the graph of $r(x)=\dfrac{x^2+3x+1}{x-1}$?

1. $y=x+3$
2. $y=x+4$
3. $y=x-4$
4. $y=x+2$

Question 62 · Multiple choice

What is $\displaystyle\lim_{x\to-\infty}\frac{5-2x}{x+3}$?

1. $\frac{5}{3}$
2. $2$
3. $-\infty$
4. $-2$

Question 63 · Multiple choice

Which of the following functions has a graph with horizontal asymptote $y=0$?

1. $\frac{x^2+1}{x^2-4}$
2. $\frac{x^3}{x^2+1}$
3. $\frac{x+1}{x^2-4}$
4. $\frac{2x}{x-4}$

Question 64 · Multiple choice

The graph of $r(x)=\dfrac{ax^2+5}{2x^2-x}$ has horizontal asymptote $y=3$. What is $a$?

1. $6$
2. $3$
3. $\frac{3}{2}$
4. $5$

Question 65 · Multiple choice

Let $r(x)=\dfrac{2x^2+1}{x^2+x+3}$. Its graph has the horizontal asymptote $y=2$, but $r$ takes values above $2$. To three decimal places, what is the maximum value of $r$?

1. $2.217$
2. $2.000$
3. $-5.098$
4. $0.328$

Question 66 · Multiple choice

Let $r(x)=\dfrac{5x^2-3x}{x^2+1}$, whose graph has horizontal asymptote $y=5$. For large $x>0$, $r(x)$ is close to $5$. To three decimal places, what is the greatest $x$ for which $r(x)=4.9$?

1. $-1.553$
2. $31.553$
3. $8.888$
4. $49.000$

Question 67 · Multiple choice

Find every value of $x$ at which $r(x)=\dfrac{x^2-16}{x^2+x-12}$ equals $0$.

1. $x=4$ and $x=-4$
2. $x=4$ only
3. $x=-4$ only
4. $x=3$ and $x=4$

Question 68 · Multiple choice

For which values of $x$ is $\dfrac{(x-1)(x+4)}{x-3}\ge0$?

1. $(-\infty,-4]\cup[1,3)$
2. $[-4,1]\cup[3,\infty)$
3. $[-4,1]\cup(3,\infty)$
4. $[1,3)$

Question 69 · Multiple choice

What is the $x$-intercept of the graph of $r(x)=\dfrac{2x+5}{x^2+1}$?

1. $-\frac{5}{2}$
2. $\frac{5}{2}$
3. $-\frac{2}{5}$
4. $5$

Question 70 · Multiple choice

Which function has zeros at $x=-1$ and $x=4$ and at no other $x$?

1. $\frac{(x+1)(x-4)}{x-4}$
2. $\frac{(x-1)(x+4)}{x+2}$
3. $\frac{(x+1)(x-4)}{x^2-1}$
4. $\frac{(x+1)(x-4)}{x-2}$

Question 71 · Multiple choice

For which values of $x$ is $\dfrac{x^2-4x}{x^2-16}<0$?

1. $(-4,0)$
2. $(-4,0)\cup(0,4)$
3. $(0,4)$
4. $(-\infty,-4)$

Question 72 · Multiple choice

Let $r(x)=\dfrac{(x-2)^2(x+1)}{(x-5)(x+3)}$. At which zero of $r$ does $r(x)$ not change sign?

1. $x=-1$
2. $x=2$
3. $x=5$
4. $x=-3$

Question 73 · Multiple choice

Find every value of $x$ at which $r(x)=\dfrac{x^3-4x}{x^2+4x+4}$ equals $0$.

1. $x=-2$, $x=0$ and $x=2$
2. $x=0$ only
3. $x=0$ and $x=2$
4. $x=-2$ and $x=2$

Question 74 · Multiple choice

To three decimal places, what is the greatest zero of $r(x)=\dfrac{x^3-3x+1}{x^2+4}$?

1. $0.347$
2. $-1.879$
3. $1.532$
4. $2.000$

Question 75 · Multiple choice

The function $r(x)=\dfrac{x^3-2x-5}{x-1}$ has exactly one real zero. To three decimal places, what is it?

1. $1.000$
2. $-2.095$
3. $1.710$
4. $2.095$

Question 76 · Multiple choice

Find all the vertical asymptotes of the graph of $y=\dfrac{x+3}{x^2-x-12}$.

1. $x=-3$ and $x=4$
2. $x=-3$ only
3. There are none
4. $x=4$ only

Question 77 · Multiple choice

As $x$ approaches $3$ from the left, what does $\dfrac{x+2}{x-3}$ approach?

1. $\infty$
2. $-\infty$
3. $5$
4. $0$

Question 78 · Multiple choice

What is $\displaystyle\lim_{x\to-1}\frac{1}{(x+1)^2}$?

1. $-\infty$
2. $0$
3. $\infty$
4. $1$

Question 79 · Multiple choice

Let $r(x)=\dfrac{x-1}{(x-4)^2(x+2)}$. Which statement about the behavior of $r$ near $x=4$ is true?

1. $r(x)\to-\infty$ as $x\to4^-$ and $r(x)\to\infty$ as $x\to4^+$
2. $r(x)\to\infty$ as $x\to4^-$ and $r(x)\to-\infty$ as $x\to4^+$
3. $r(x)\to-\infty$ as $x\to4^-$ and as $x\to4^+$
4. $r(x)\to\infty$ as $x\to4^-$ and as $x\to4^+$

Question 80 · Multiple choice

Which function has a vertical asymptote at $x=3$ and a hole at $x=-1$?

1. $\frac{x+1}{(x+1)(x-3)}$
2. $\frac{x-1}{(x+1)(x-3)}$
3. $\frac{(x+1)(x-3)}{x+1}$
4. $\frac{x-3}{(x+1)(x-3)}$

Question 81 · Multiple choice

The graph of a rational function $r$ is shown, with its asymptotes dashed. Which of the following could be $r(x)$?

1. $\frac{2x}{x+1}$
2. $\frac{2x}{x-1}$
3. $\frac{x}{x-1}$
4. $\frac{2(x-1)}{x}$

Question 82 · Multiple choice

What is $\displaystyle\lim_{x\to3^+}\frac{x^2-9}{x^2-6x+9}$?

1. $-\infty$
2. $1$
3. $\infty$
4. $6$

Question 83 · Multiple choice

The graph of $r(x)=\dfrac{x+1}{x^3-x-3}$ has exactly one vertical asymptote. To three decimal places, what is its $x$-coordinate?

1. $1.672$
2. $-1.000$
3. $3.000$
4. $1.442$

Question 84 · Multiple choice

The graph of $r(x)=\dfrac{2x-1}{x^2-3.4x+1.7}$ has two vertical asymptotes. To three decimal places, what is the $x$-coordinate of the one farther to the right?

1. $0.609$
2. $2.791$
3. $0.500$
4. $1.700$

Question 85 · Multiple choice

The graph of $r(x)=\dfrac{x^2-4}{x^2-x-2}$ has a hole. What is the $y$-coordinate of the hole?

1. $0$
2. $2$
3. $-\frac{4}{3}$
4. $\frac{4}{3}$

Question 86 · Multiple choice

Which function has a hole in its graph at $x=5$?

1. $\frac{x^2-25}{x-5}$
2. $\frac{x+5}{x^2-25}$
3. $\frac{x}{x-5}$
4. $\frac{x^2+5}{x+5}$

Question 87 · Multiple choice

The graph of $r(x)=\dfrac{x^2+2x-15}{x^2-9}$ has one hole. Where is it?

1. $\left(3,\frac{4}{3}\right)$
2. $\left(-3,\frac{1}{3}\right)$
3. $\left(3,0\right)$
4. $\left(-5,0\right)$

Question 88 · Multiple choice

The graph of $g(x)=\dfrac{x^2+kx+6}{x+2}$ has a hole at $x=-2$. What is $k$?

1. $-5$
2. $5$
3. $3$
4. $6$

Question 89 · Multiple choice

The graph of $f(x)=\dfrac{x^3-1}{x-1}$ has a hole at $x=1$. What is the $y$-coordinate of the hole?

1. $0$
2. $1$
3. $2$
4. $3$

Question 90 · Multiple choice

Let $r(x)=\dfrac{(x+3)^2}{(x+3)(x-1)}$. Which statement about the graph of $r$ at $x=-3$ is true?

1. It has a vertical asymptote $x=-3$
2. It crosses the $x$-axis at $(-3,0)$
3. It has a hole at $(-3,0)$
4. It has a hole at $\left(-3,\frac{3}{2}\right)$

Question 91 · Multiple choice

For $x\ne1$, which expression is equal to $\dfrac{3x^3-7x^2+9x-4}{x-1}$?

1. $3x^2-4x+5-\frac{1}{x-1}$
2. $3x^2-10x+19-\frac{23}{x-1}$
3. $3x^2-4x+5+\frac{1}{x-1}$
4. $3x^2-4x+6$

Question 92 · Multiple choice

What is the coefficient of $x^3$ in the expansion of $(2x-1)^5$?

1. $-80$
2. $40$
3. $10$
4. $80$

Question 93 · Multiple choice

What is the coefficient of $x^2$ in the expansion of $(x-3)^4$?

1. $54$
2. $-54$
3. $6$
4. $-108$

Question 94 · Multiple choice

Which is a factored form of $x^3-27$?

1. $(x-3)(x^2-3x+9)$
2. $(x-3)(x^2+3x+9)$
3. $(x-3)^3$
4. $(x+3)(x^2-3x+9)$

Question 95 · Multiple choice

Which form of $p(x)=2x^2+2x-12$ shows its zeros directly?

1. $2(x-3)(x+2)$
2. $2(x+3)(x-2)$
3. $(2x-4)(x+2)$
4. $2\left(x+\frac{1}{2}\right)^2-12$

Question 96 · Multiple choice

Simplify $\dfrac{x^2+7x+12}{x^2-16}$ for $x\ne\pm4$.

1. $\frac{x+3}{x-4}$
2. $\frac{x+3}{x+4}$
3. $\frac{x+4}{x-3}$
4. $\frac{7x+12}{-16}$

Question 97 · Multiple choice

When $p(x)=x^4+3x^3-7x+1$ is divided by $x+2$, what is the remainder?

1. $-7$
2. $15$
3. $7$
4. $-33$

Question 98 · Multiple choice

Which expression is equivalent to $\dfrac{3x^2+2x}{x+1}$ for $x\ne-1$?

1. $3x+1-\frac{1}{x+1}$
2. $3x-1-\frac{1}{x+1}$
3. $3x+5+\frac{5}{x+1}$
4. $3x-1+\frac{1}{x+1}$

Question 99 · Multiple choice

Which expression is equivalent to $(1+x)^3-(1-x)^3$?

1. $6x$
2. $2x^3$
3. $6x^2+2$
4. $2x^3+6x$

Question 100 · Multiple choice

If $f(x)=x^2$, where is the vertex of the graph of $y=f(x+4)-5$?

1. $\left(4,-5\right)$
2. $\left(-5,-4\right)$
3. $\left(-4,-5\right)$
4. $\left(-4,5\right)$

Question 101 · Multiple choice

The domain of $f$ is $[-2,6]$. What is the domain of $g(x)=f(2x)$?

1. $[-4,12]$
2. $[-1,3]$
3. $[0,8]$
4. $[-2,6]$

Question 102 · Multiple choice

The range of $f$ is $[1,5]$. What is the range of $g(x)=-2f(x)+3$?

1. $[-7,1]$
2. $[1,13]$
3. $[-7,-1]$
4. $[-1,7]$

Question 103 · Multiple choice

The point $(2,5)$ is on the graph of $y=f(x)$. Which point must be on the graph of $y=3f(x+1)$?

1. $\left(3,15\right)$
2. $\left(1,8\right)$
3. $\left(1,15\right)$
4. $\left(3,6\right)$

Question 104 · Multiple choice

The graph of $f(x)=|x|$ is reflected over the $x$-axis, then translated $4$ units left and $1$ unit up. Which is the new function?

1. $-|x+4|+1$
2. $-|x-4|+1$
3. $|x+4|-1$
4. $-|x+4|-1$

Question 105 · Multiple choice

Let $f(x)=x^3$ and $g(x)=8x^3$. For what value of $b$ is $g(x)=f(bx)$ for all $x$?

1. $8$
2. $\frac{1}{2}$
3. $\frac{1}{8}$
4. $2$

Question 106 · Multiple choice

What is the vertex of the graph of $h(x)=-2(x+1)^2+7$?

1. $\left(1,7\right)$
2. $\left(-1,7\right)$
3. $\left(-1,-7\right)$
4. $\left(-2,7\right)$

Question 107 · Multiple choice

The table gives values of $f$. Let $g(x)=f(x-1)-4$. What is $g(2)$?

1. $-4$
2. $1$
3. $-3$
4. $-2$

Question 108 · Multiple choice

Let $f(x)=x^3-5x$ and $g(x)=f\big(2(x-1)\big)+3$. To three decimal places, what is the $x$-coordinate of the relative minimum of $g$?

1. $1.291$
2. $3.582$
3. $1.645$
4. $0.645$

Question 109 · Multiple choice

Let $f(x)=x^4-3.5x^2+x$ and $h(x)=-0.5f(x+2)$. To three decimal places, what is the absolute maximum value of $h$?

1. $-4.419$
2. $0.889$
3. $8.839$
4. $2.210$

Question 110 · Multiple choice

Which type of function is most appropriate to model the data in the table?

1. Quadratic with a negative leading coefficient
2. Cubic
3. Linear
4. Rational with a vertical asymptote

Question 111 · Multiple choice

The table gives the number of different handshakes $H$ when each of $n$ people shakes hands once with each other person. Which type of function models $H$ exactly?

1. Linear
2. Quadratic
3. Exponential
4. Cubic

Question 112 · Multiple choice

A drone's height $h$, in meters, is recorded every second. Which type of model fits these data exactly?

1. Quadratic
2. Linear
3. Exponential
4. Cubic, but not quadratic

Question 113 · Multiple choice

An open box is made from a $12$-inch by $12$-inch sheet by cutting a square of side $x$ inches from each corner and folding up the sides. Its volume is $V(x)=x(12-2x)^2$. What domain makes sense for this model?

1. $(0,12)$
2. $[0,\infty)$
3. $(0,3)$
4. $(0,6)$

Question 114 · Multiple choice

The average cost per item, in dollars, of making $n$ items is $C(n)=\dfrac{500+4n}{n}$. As $n$ increases without bound, what value does $C(n)$ approach?

1. $0$
2. $4$
3. $500$
4. $504$

Question 115 · Multiple choice

Which type of function fits the data in the table exactly?

1. Quadratic
2. Linear
3. Cubic
4. Exponential

Question 116 · Multiple choice

A cubic regression $y=ax^3+bx^2+cx+d$ is fit to the data in the table. To three decimal places, what is $a$?

1. $0.274$
2. $0.300$
3. $2.174$
4. $-2.578$

Question 117 · Multiple choice

The data in the table decrease and then increase, so a quadratic model is chosen. Use a quadratic regression to predict $y$ when $x=7.5$.

1. $26.240$
2. $39.393$
3. $35.670$
4. $26.617$

Question 118 · Multiple choice

The data in the table are roughly linear. Use a linear regression to predict $y$ when $x=10$.

1. $21.880$
2. $22.900$
3. $21.737$
4. $19.459$

Question 119 · Multiple choice

A quadratic model passes through $(0,4)$, $(1,7)$ and $(2,14)$. What does the model predict at $x=5$?

1. $59$
2. $39$
3. $64$
4. $54$

Question 120 · Multiple choice

Use a quadratic regression on the data in the table to predict $y$ when $x=7$.

1. $49.130$
2. $45.420$
3. $55.780$
4. $51.480$

Question 121 · Multiple choice

A phone plan costs $C(m)=20$ dollars for $0\le m\le100$ minutes and $C(m)=20+0.15(m-100)$ dollars for $m>100$. What is $C(240)$?

1. $41$
2. $36$
3. $56$
4. $21$

Question 122 · Multiple choice

Use a linear regression on the data in the table to predict $y$ when $x=12$.

1. $27.500$
2. $25.880$
3. $26.970$
4. $28.100$

Question 123 · Multiple choice

A company's monthly profit, in thousands of dollars, is modeled by $P(t)=-0.5t^2+6t+20$ for $0\le t\le12$, where $t$ is months since a product launch. At what time is the modeled profit greatest?

1. $t=12$
2. $t=6$
3. $t=3$
4. $t=20$

Question 124 · Multiple choice

The volume of water in a tank, in gallons, is modeled by $V(t)=1200-80t$, where $t$ is hours after a drain is opened. Which domain is appropriate for this model?

1. $[0,1200]$
2. $[0,\infty)$
3. $[0,15]$
4. $[15,\infty)$

Question 125 · Multiple choice

An open box is made from a $20$-inch by $14$-inch sheet of cardboard by cutting a square of side $x$ inches from each corner and folding up the sides, so $V(x)=x(20-2x)(14-2x)$ for $0<x<7$. To three decimal places, what is the greatest possible volume, in cubic inches?

1. $2.704$
2. $336.000$
3. $333.926$
4. $339.013$

Question 126 · Multiple choice

The table shows a company's monthly sales, in thousands of dollars, $x$ months after a product launch. Use a cubic regression to predict the sales when $x=5.5$.

1. $26.069$
2. $21.669$
3. $18.573$
4. $33.967$

Question 127 · Multiple choice

A company's monthly profit, in thousands of dollars, is modeled by $P(x)=-0.02x^3+0.9x^2+5x-40$ for $0\le x\le40$, where $x$ is the number of units sold, in hundreds. To three decimal places, how many hundred units maximize the profit?

1. $386.566$
2. $32.559$
3. $15.000$
4. $30.000$

Question 128 · Free response (6 points) · Function concepts

Let $f(x)=x^3-6x^2+4x+5$. A graphing calculator is allowed.

1. (A(i)) Find the average rate of change of $f$ over $[0,3]$. [1 point]
2. (A(ii)) Find all real zeros of $f$. Give decimal answers correct to three places. [1 point]
3. (B) Find the interval on which $f$ is decreasing. Give a reason for your answer. [2 points]
4. (C) Use the average rates of change of $f$ over $[0,0.5]$ and over $[0.5,1]$ to decide whether the rate of change of $f$ is increasing or decreasing on $[0,1]$. What does this tell you about the graph of $f$ there? [2 points]

Question 129 · Free response (6 points) · Modeling a non-periodic context

A ball is launched upward from a platform. Its height $h$, in feet, above the ground was recorded every half second, as shown in the table. A graphing calculator is allowed.

1. (A(i)) Find the average rates of change of $h$ over $[0,0.5]$, $[0.5,1]$ and $[1,1.5]$, and explain why a quadratic model is appropriate. [1 point]
2. (A(ii)) Find the quadratic function $h(t)=at^2+bt+c$ that fits all four data points exactly. [1 point]
3. (B(i)) Use the model to find the time when the ball reaches the ground. Give your answer correct to three decimal places. [1 point]
4. (B(ii)) Find the average rate of change of $h$ over $[0,2]$, with units. [1 point]
5. (C) Find $h(3)$ from the model and use it to explain why the model is not valid at $t=3$. [2 points]

Question 130 · Free response (6 points) · Function concepts

Let $r(x)=\dfrac{2x^2-2x-12}{x^2-4x+3}$. A graphing calculator is allowed.

1. (A(i)) Find the $x$-coordinate of each vertical asymptote and of each hole in the graph of $r$. [1 point]
2. (A(ii)) Find the coordinates of the hole. [1 point]
3. (B) Find $\displaystyle\lim_{x\to\infty}r(x)$ and $\displaystyle\lim_{x\to1^+}r(x)$. [2 points]
4. (C) Find all values of $x$ for which $r(x)\ge0$. [2 points]

Question 131 · Free response (6 points) · Modeling a non-periodic context

A tank holds $40$ liters of a salt solution containing $12$ grams of salt. A solution with $0.8$ gram of salt per liter is pumped in. After $t$ liters have been added, the concentration of salt in the tank, in grams per liter, is modeled by $C(t)=\dfrac{12+0.8t}{40+t}$. A graphing calculator is allowed.

1. (A(i)) Find $C(10)$ and say what it means in context. [1 point]
2. (A(ii)) Find the average rate of change of $C$ over $[0,20]$. Give units. [1 point]
3. (B) Find $\displaystyle\lim_{t\to\infty}C(t)$ and explain what it means in context. [2 points]
4. (C) How many liters must be added for the concentration to reach $0.6$ gram per liter? [2 points]