Linear, Quadratic & Polynomial Functions (College Board AP® Precalculus): Exam Questions

A bicycle rental company charges a flat fee plus a constant rate per hour for bicycle rentals. The cost of a $2$-hour rental is $14, and the cost of a $5$-hour rental is $26.

The total cost, in dollars, of a bicycle rental can be modeled by the function $C$ given by $C\left(t\right)=a+bt$, where $C\left(t\right)$ is the total cost, in dollars, and $t$ is the number of hours of the rental.

Use the given data to write two equations that can be used to find the values for constants $a$ and $b$ in the expression for $C\left(t\right)$.

The table gives values for a function $f$ at selected values of $x$.

Based on the table, which of the following function types best models function $f$: linear, quadratic, exponential, or logarithmic?

The polynomial function $f$ is given by $f\left(x\right)=-2x^4+5x^2-1$.

Determine the end behavior of $f$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

The function $g$ is given by $g\left(x\right)=-0.1371x^4+x^3-1.2x^2$.

(i) Find all values of $x$, as decimal approximations, for which $g\left(x\right)=11$, or indicate that there are no such values.

(ii) Determine the end behavior of $g$ as $x$ increases without bound. Express your answer using the mathematical notation of a limit.

The polynomial function $p$ is given by $p\left(x\right)=x^3-4x^2+4x$. Rewrite $p\left(x\right)$ as a product of linear factors.

Find all real zeros of $p$, and state the multiplicity of each. For each zero, state whether the graph of $y=p\left(x\right)$ crosses the $x$-axis or is tangent to the $x$-axis at that point.

Determine the end behavior of $p$. Express your answer using the mathematical notation of a limit.

Use Pascal's Triangle to rewrite ${\left(x-2\right)}^4$ in the form $a{x}^4+b{x}^3+c{x}^2+dx+e$, where $a$, $b$, $c$, $d$, and $e$ are integers to be found.

The polynomial function $p$ is odd, and $p$ has a relative minimum value of $-7$ at $x=3$. Find the value of $p\left(-3\right)$, and state whether $p\left(-3\right)$ is a relative maximum or a relative minimum of $p$. Give a reason for your answer.

The height of a ball, in meters, $t$ seconds after it is dropped from a building is given in the table.

Find the average rate of change of $h$ over the interval $\left[0,1\right]$, and the average rate of change of $h$ over the interval $\left[3,4\right]$. Show the work that leads to your answers.

Based on the table, determine which type of function best models $h$: linear, quadratic, cubic, or quartic. Give a reason for your answer based on the relationship between the change in the output values of $h$ and the change in the input values of $h$.

Compare the average rates of change you found in part (a). Interpret the meaning of this comparison in the context of the problem.

The polynomial function $p$ is given by $p\left(x\right)=x^3-x^2-6x$. Rewrite $p\left(x\right)$ as a product of linear factors.

Find all values of $x$ for which $p\left(x\right)>0$.

Maria opens a savings account with an initial deposit, and then makes equal weekly deposits of the same amount each week. After 2 weeks, the account has $250 in it. After 8 weeks, the account has $670 in it.

Let $A\left(t\right)$ be the amount of money in Maria's savings account, in dollars, $t$ weeks after she opened it. Construct a linear model for $A\left(t\right)$.

Interpret the meaning of the slope of your linear model in the context of this problem. Include appropriate units in your interpretation.

The water level in a reservoir, in meters, is modeled by the function $L$, where $L\left(t\right)=-t^3+11t^2+100$ for $0\le t\le 10$. $L\left(t\right)$ is the water level in meters, and $t$ is the number of days since the start of the rainy season.

Find the maximum water level during this period, and the time at which the maximum occurs. Express both values as decimal approximations.

Find the average rate of change of $L$ over the interval $\left[0,5\right]$. Include appropriate units in your answer. Show the work that leads to your answer.

Interpret the meaning of your answer to part (b) in the context of the problem.

A polynomial function $p$ has degree 3 with real coefficients. Two of the zeros of $p$ are $x=4$ and $x=-1+2i$. The graph of $p$ passes through the point $\left(0,-20\right)$. Construct an expression for $p\left(x\right)$ in standard form. Show the work that leads to your answer.

The polynomial function $f$ is given by $f\left(x\right)=x^4-6x^3+7x^2+12x-18$.

(i) Find all values of $x$, as decimal approximations, for which $f\left(x\right)=0$, or indicate that there are no such values.

(ii) Determine the end behavior of $f$ as $x$ decreases without bound. Express your answer using the mathematical notation of a limit.

(iii) For each zero of $f$ found in (i), determine whether the multiplicity of the zero is even or odd. Give a reason for your answer.