Math Core

A graph packs a whole function into one picture. With a little practice you can read off where it crosses the axes, where it rises and falls, and its highest and lowest points, without doing any algebra. These key features are how people describe and compare functions.

Intercepts

Definition

Intercepts

A $y$ -intercept is a point where the graph crosses the $y$ -axis. It happens at $x = 0$ , so its value is $f(0)$ .

An $x$ -intercept is a point where the graph crosses the $x$ -axis. It happens where $f(x) = 0$ . The $x$ -values of the $x$ -intercepts are also called the zeros of the function.

A function has at most one $y$ -intercept (by the vertical line test, the line $x = 0$ can cross it only once), but it can have any number of $x$ -intercepts.

We'll use the graph of $f(x) = x^2 - 2x - 3$ throughout this lesson.

The graph of f(x) = x² − 2x − 3 with its intercepts and lowest point marked.Open in grapher →

The $y$ -intercept is $(0, -3)$ . Check: $f(0) = 0 - 0 - 3 = -3$ .
The $x$ -intercepts are $(-1, 0)$ and $(3, 0)$ , so the zeros are $-1$ and $3$ . Check: $f(3) = 9 - 6 - 3 = 0$ .

Increasing and decreasing

Read a graph the way you read a sentence: left to right.

Increasing, decreasing, constant

A function is increasing where its graph goes up as you move right.
It is decreasing where its graph goes down as you move right.
It is constant where its graph is flat.

Describe these intervals using $x$ -values.

The graph of $f$ falls until it reaches its lowest point at $x = 1$ , then rises. So $f$ is decreasing for $x < 1$ and increasing for $x > 1$ .

Common mistake

Intervals of increase and decrease are described with $x$ -values, not $y$ -values. It's tempting to say " $f$ increases for $y > -4$ ," but the question is where along the $x$ -axis the graph rises. The answer is $x > 1$ .

Maximum and minimum

Definition

Maximum and minimum

The maximum of a function is its greatest output, the $y$ -value of the highest point on the graph. The minimum is its least output, the $y$ -value of the lowest point.

The lowest point of $f$ is $(1, -4)$ , so the minimum value is $-4$ , reached at $x = 1$ . The arms rise forever, so $f$ has no maximum. Notice that a function often switches from decreasing to increasing exactly at a minimum, and from increasing to decreasing at a maximum.

Positive and negative

A function is positive where its graph is above the $x$ -axis ( $f(x) > 0$ ) and negative where it is below ( $f(x) < 0$ ). The zeros are where it switches.

For our $f$ , the graph dips below the $x$ -axis between its zeros. So $f$ is negative for $-1 < x < 3$ and positive for $x < -1$ or $x > 3$ .

Solving equations with a graph

The graph can also answer questions like "when is $f(x) = 5$ ?" Draw the horizontal line $y = 5$ and read the $x$ -value of each crossing.

Worked example: Reading solutions from a graph

Use the graph to solve $f(x) = 5$ for $f(x) = x^2 - 2x - 3$ .

The line y = 5 crosses the graph of f at x = −2 and x = 4.Open in grapher →

The line $y = 5$ crosses the graph at $(-2, 5)$ and $(4, 5)$ , so the solutions are $x = -2$ and $x = 4$ .

Check with the rule: $f(-2) = 4 + 4 - 3 = 5$ and $f(4) = 16 - 8 - 3 = 5$ .

Putting it together

Worked example: Describing a graph completely

Describe the key features of $g(x) = -|x - 2| + 3$ .

The graph of g(x) = −|x − 2| + 3, an upside-down V.Open in grapher →

$y$ -intercept: $(0, 1)$ , since $g(0) = -2 + 3 = 1$ .
Zeros: $-1$ and $5$ .
Maximum: $3$ , at $x = 2$ . There is no minimum.
Increasing for $x < 2$ ; decreasing for $x > 2$ .
Positive for $-1 < x < 5$ ; negative for $x < -1$ or $x > 5$ .
Domain: all real numbers. Range: $y \le 3$ .

Graphs of real situations

In a real situation, every feature has a meaning. Always read the axis labels to know what the input and output measure.

Worked example: A launched ball

A ball is tossed up from a balcony. Its height in meters after $t$ seconds is $h(t) = -5t^2 + 20t + 25$ , graphed below until it hits the ground.

Height of the ball (meters) against time (seconds).Open in grapher →

The $y$ -intercept $(0, 25)$ : the ball starts $25$ meters above the ground.
The maximum $(2, 45)$ : after $2$ seconds the ball reaches its greatest height, $45$ meters.
The zero $t = 5$ : the ball hits the ground after $5$ seconds.
The height is increasing for $0 < t < 2$ (going up) and decreasing for $2 < t < 5$ (coming down).

Check the peak: $h(2) = -5(4) + 40 + 25 = 45$ . Check the landing: $h(5) = -125 + 100 + 25 = 0$ .

Tip

When you read a value off a graph, check it with the rule if you have one. A point that looks like $(2, 45)$ might really be $(2.1, 44.9)$ , and substituting tells you for sure.

Practice

Problems 1 to 6 use the graph of $f(x) = (x + 1)^2 - 4$ below.

The graph of y = f(x).Open in grapher →

Practice 1

Find $f(2)$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

List the zeros of $f$ .

Separate answers with commas, e.g. 2, -5

Practice 3

What is the $y$ -coordinate of the $y$ -intercept?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

What is the minimum value of $f$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Where is $f$ increasing?

Practice 6

Use the graph to solve $f(x) = 5$ .

Separate answers with commas, e.g. 2, -5

Practice 7

Which statement about the function $g(x) = -|x - 2| + 3$ from the lesson is false?

Practice 8

A drone's altitude in meters $t$ minutes after takeoff is $A(t) = -2|t - 6| + 12$ for $0 \le t \le 12$ . For how many minutes is the drone at an altitude of $8$ meters or more?

Altitude of the drone (meters) against time (minutes), with the line y = 8 dashed.Open in grapher →

Enter a number. Fractions like 3/4 and sqrt(2) are OK.