Exercises: Interpret Key Features of Graphs

What is the $y$-intercept of the graph of $f(x) = 3x - 12$?

The graph of a function passes through the points $(-3, 0)$, $(0, 5)$, and $(2, 0)$. Which are the $x$-intercepts?

The function $f$ has these values: $f(1) = 5$, $f(2) = 8$, $f(3) = 11$. What can you say about $f$ on the interval $[1, 3]$?

A company's monthly profit (in dollars) is $P(x) = 4x - 200$, where $x$ is the number of items sold. Find the $x$-intercept and enter the value of $x$ at that point.

A ball is kicked upward. Its height in meters is $h(t) = -5t^2 + 20t$, where $t$ is time in seconds. The $y$-intercept tells you the height when the ball is kicked. What is the $y$-intercept (in meters)?

The graph below shows a function $f$. On the interval $[0, 3]$, the graph goes from $f(0) = 2$ up to $f(3) = 8$. On the interval $[3, 6]$, the graph goes from $f(3) = 8$ down to $f(6) = 1$. On which interval is $f$ decreasing?

A function $g$ has $g(x) > 0$ for $1 < x < 5$ and $g(x) < 0$ for $x < 1$ and $x > 5$. What are the $x$-intercepts of $g$?

A graph shows two peaks: one at $(2, 15)$ and one at $(6, 20)$. A student says the relative maximum is at $(6, 20)$ because it is the absolute highest point visible. Is the student correct?

The profit function $P(x) = 2x - 10$ (in thousands of dollars) models a small business, where $x$ is items sold. The $x$-intercept is at $x = 5$. What does this mean in context?

The graph of function $f$ has these features:

• From $x = 0$ to $x = 4$: the output increases from 2 to 10.
• From $x = 4$ to $x = 8$: the output decreases from 10 to 3.
• All output values are positive throughout.

$f$ is increasing on the interval $[\_\_\_, \_\_\_]$ and decreasing on the interval $[\_\_\_, \_\_\_]$.
$f$ is positive on the interval $[\_\_\_, \_\_\_]$.

A graph of a function $f$ is shown. The graph has a local peak at $(3, 12)$ and a local valley at $(7, 4)$. Which statement is correct?

A student views the graph of $f(x) = x^3$ on the screen from $x = -2$ to $x = 2$. On that window, the graph appears to stop at $(2, 8)$ on the right and $(-2, -8)$ on the left. The student says: "The end behavior is: $f$ ends at 8 on the right and at $-8$ on the left." What is wrong?

The function $f(x) = x^2$ is symmetric about the $y$-axis. If a context models the kinetic energy of a car when it deviates $x$ mph from the speed limit (negative $x$ means below the limit, positive $x$ means above), explain what the symmetry tells you about the situation.

A soccer ball is kicked upward. Its height in meters is $h(t) = -5t^2 + 20t$, where $t$ is time in seconds. The vertex of this parabola is at $t = 2$ seconds.

What is the maximum height of the ball (in meters)? Evaluate $h(2)$.

What error did the student make?

Explain what end behavior means and why the student's description is incorrect.

A graph of $f$ has two peaks: a relative maximum at $(2, 10)$ and another at $(6, 15)$. A student says "the relative maximum of $f$ is at $(6, 15)$ because that is the only one that is a maximum." Is the student correct? Explain what a relative maximum means and identify all relative maxima.

A cup of hot coffee cools over time. It starts at 180°F, cools rapidly at first, then more slowly, and approaches room temperature (72°F) without ever going below it. Describe the key features of this function and sketch it (describe the sketch in words).