Graphs of Functions

Intuition

In recent chapters you graphed equations like $y = 2x - 1$ and $y = x^2 - 2x - 3$ by finding solution points and plotting them. Now that a rule has a name, $f$ , the same picture becomes something more powerful. The graph of a function collects every input together with the output it produces into one curve, so a single picture holds the whole function at once. Better still, you can run the picture both ways: point to a place on the horizontal axis and the height of the curve there tells you the output. Point to a height instead, and read off which inputs reach it. This lesson shows how to build such a graph and, more useful still, how to read a function’s value, its intercepts, and its domain and range. It shows where the function climbs or falls too, and all of that comes straight off the picture, often with no algebra at all.

Learning goals

Read the graph as every pair $(x, f(x))$
Build a graph from a table of inputs and outputs
Find $f(a)$ by going up to the curve and across
Locate the intercepts, where the graph crosses each axis
Take the domain as horizontal spread and the range as vertical
Say where a function increases, decreases, and turns

The graph of a function is a picture of every input and output

The graph of a function $f$ is the set of all points $(x, f(x))$ as $x$ runs over the domain of $f$ . Read that pairing carefully: the first coordinate of each point is an input, and the second coordinate is the output the function assigns to it. Setting $y = f(x)$ says the same thing in the language of the last chapters, because then each allowed input $x$ gives exactly one point $(x, y)$ with $y = f(x)$ . So graphing a function is nothing new. It is graphing the equation $y = f(x)$ , and every technique you built for lines and parabolas carries straight over. What is new is everything the finished picture now lets you read.

The single most useful fact is this. The second coordinate of every point on the graph is an output. So the height of the graph above an input $a$ , measured against the vertical axis, is exactly $f(a)$ . The function pairs $a$ with one output, so there is exactly one point of the graph directly above (or below) each input in the domain. That single point is why “the height at $x = a$ ” names one definite number. That is the vertical line test from the What Is a Function? lesson, seen from the reading side. Each vertical line meets the graph at most once, and where it does, the height it reaches is the output.

A point lies on the graph exactly when $y = f(x)$ #

Start from the definition: the graph of $f$ is the collection of points $(x, f(x))$ , one for each input $x$ in the domain. Take any point $(a, b)$ and ask whether it belongs to that collection.

Suppose first that $(a, b)$ is on the graph. Then it is one of the listed points, so its first coordinate $a$ is an input. The point’s second coordinate must then be the output paired with $a$ , namely $f(a)$ . Therefore $b = f(a)$ .

Now suppose instead that $b = f(a)$ . Then the point $(a, b)$ is literally $(a, f(a))$ , which is the graph’s point at the input $a$ , so $(a, b)$ is on the graph. The two directions together give the equivalence

(a, b) \text{ is on the graph} \quad \Longleftrightarrow \quad b = f(a).

This one equivalence is the engine of every reading in this lesson. Set $b$ to the height of the graph above $a$ , and it says that height is $f(a)$ . Fix a height $b$ instead, and it says the inputs whose output is $b$ are exactly the $x$ where the horizontal line $y = b$ crosses the graph.

Every point on the graph of f has the form (a, f(a)): its horizontal position is the input a, and its height above the axis is the output f(a). Drop straight down to the horizontal axis to read the input, and straight across to the vertical axis to read the output.

Building a graph from a table

To draw a function’s graph, use the routine you already trust for lines and parabolas. Choose several inputs, evaluate $f$ at each to get the matching output, then plot the points $(x, f(x))$ . Join them with a ruler for a linear rule, or with a smooth curve for a bending one. A small spread of inputs, including a negative, zero, and a few positives, keeps the shape honest and reveals any turning.

Worked example 1 Graph $f(x) = x^2 - 2x - 3$ from a table

The rule squares the input, subtracts twice the input, then subtracts $3$ . Because the highest power of $x$ is a square, you already know the graph will be a parabola. Pick a spread of inputs and evaluate, keeping the parentheses so the signs are safe.

f(-1) = (-1)^2 - 2(-1) - 3 = 1 + 2 - 3 = 0.

\begin{aligned} f(0) &= 0 - 0 - 3 = -3, \\ f(1) &= 1 - 2 - 3 = -4, \\ f(2) &= 4 - 4 - 3 = -3. \end{aligned}

f(3) = 9 - 6 - 3 = 0.

Collect the input-output pairs in a table so each point is easy to plot:

$x$	$-1$	$0$	$1$	$2$	$3$
$f(x)$	$0$	$-3$	$-4$	$-3$	$0$

Plot the five points $(-1, 0)$ , $(0, -3)$ , $(1, -4)$ , $(2, -3)$ , and $(3, 0)$ . They do not lie on a line; they bend into a smooth U. Joining them gives the parabola that is the graph of $f$ , dipping to its lowest point at $(1, -4)$ and rising on both sides.

The graph of f(x) = x squared minus 2x minus 3, built from the five table points. It is a parabola with its lowest point at (1, -4). Every point on the curve has the form (x, f(x)), and the two x-intercepts and the y-intercept are marked.

Reading a value off the graph

Two questions come up again and again, and the graph answers both by tracing a straight path to an axis.

Finding $f(a)$ from the picture. Locate the input $a$ on the horizontal axis, move straight up or down until you meet the curve, then read the height there against the vertical axis. That height is $f(a)$ , because the point you land on is $(a, f(a))$ .

Solving $f(x) = b$ from the picture. This asks the reverse question: which inputs produce the output $b$ . Locate the value $b$ on the vertical axis, move straight across to the curve, and read the input below each meeting point. Every such input $x$ satisfies $f(x) = b$ . A horizontal line can cross a curve more than once, so an equation $f(x) = b$ can have several solutions, one solution, or none. The picture shows exactly which case you are in.

Reading the graph of f(x) = x squared minus 2x minus 3 at the height y = -3. The horizontal line y = -3 meets the curve at two points, above x = 0 and x = 2, so f(x) = -3 has the two solutions x = 0 and x = 2. Read the other way, going up from x = 2 to the curve gives the height f(2) = -3.

Worked example 2 Reading $f(2)$ and solving $f(x) = -3$ from the graph

Use the graph of $f(x) = x^2 - 2x - 3$ above, without any algebra.

To find $f(2)$ , start at $x = 2$ on the horizontal axis and move straight down to the curve. You land on the point $(2, -3)$ , whose height is $-3$ , so

f(2) = -3.

To solve $f(x) = -3$ , do the reverse. Find $-3$ on the vertical axis and slide across the horizontal line $y = -3$ . It meets the parabola at two points, the ones sitting above $x = 0$ and above $x = 2$ . Reading the input beneath each gives the two solutions

f(x) = -3 \quad \Longrightarrow \quad x = 0 \ \text{ or } \ x = 2.

The same value $-3$ is an output of the function at two different inputs, which is why the horizontal line crosses twice. A quick check confirms both: $f(0) = -3$ and $f(2) = 4 - 4 - 3 = -3$ .

Check your understanding

The graph of a function $g$ passes through the points $(-2, 5)$ , $(-1, 0)$ , $(0, -3)$ , $(1, -4)$ , and $(2, -3)$ . What is $g(-2)$ ?

5

-2

0

-3

Intercepts and where the graph crosses the axes

Two special readings come up so often that they have their own names.

The y-intercept is the point where the graph crosses the vertical axis. Every point on that axis has $x = 0$ , so the y-intercept is the single point $(0, f(0))$ , and you find it by evaluating $f(0)$ . A function has at most one y-intercept, because $0$ is one input with one output.

The x-intercepts are the points where the graph crosses the horizontal axis. Every point on that axis has $y = 0$ , so an x-intercept is a point $(a, 0)$ with $f(a) = 0$ . Finding the x-intercepts therefore means solving the equation $f(x) = 0$ , which is the $b = 0$ case of the reading above. These inputs are also called the zeros or roots of the function, and there may be several, one, or none.

Worked example 3 Find the intercepts of $f(x) = x^2 - 2x - 3$

For the y-intercept, evaluate the rule at $x = 0$ :

f(0) = 0 - 0 - 3 = -3,

so the y-intercept is the point $(0, -3)$ .

For the x-intercepts, set the output to $0$ and solve $f(x) = 0$ . The rule factors, which is the fastest route to its zeros:

x^2 - 2x - 3 = 0 \quad \Longrightarrow \quad (x - 3)(x + 1) = 0.

A product is zero exactly when one of its factors is zero, so $x = 3$ or $x = -1$ . The x-intercepts are $(3, 0)$ and $(-1, 0)$ . All three points match the graph: it crosses the vertical axis once, at $-3$ , and the horizontal axis twice, at $-1$ and $3$ .

Domain and range from a graph

The domain is the set of allowed inputs and the range is the set of outputs the function actually produces, and a graph puts both on display. Sweep your eye left to right across the graph: the $x$ -values it covers are the domain. Sweep bottom to top: the $y$ -values it covers are the range. A graph that runs on forever to the left and right, like a full parabola, has domain “all numbers.” By contrast, a graph drawn only between two endpoints has a domain that stops at those endpoints.

Reading domain and range from a graph. This curve is the same rule but defined only for inputs from -1 to 3, so its domain is -1 is less than or equal to x is less than or equal to 3. The heights it reaches run from the low point -4 up to 0, so its range is -4 is less than or equal to y is less than or equal to 0.

Worked example 4 Read the domain and range from the graph

The curve above is drawn only between its two endpoints, $(-1, 0)$ on the left and $(3, 0)$ on the right, so those endpoints fence in the inputs.

Sweeping left to right, the graph occupies every horizontal position from $x = -1$ to $x = 3$ , and nothing outside. That horizontal spread is the domain:

\text{domain} \colon \quad -1 \le x \le 3.

Sweeping bottom to top, the lowest the curve reaches is the valley at height $-4$ , and the highest is $0$ , shared by the two endpoints. That vertical spread is the range:

\text{range} \colon \quad -4 \le y \le 0.

Notice the domain is read along the horizontal axis and the range along the vertical axis. If instead the parabola were drawn with arrows running off both sides, the inputs would never stop and the domain would be all numbers. With those arrows, the range would be $y \ge -4$ , every height from the low point upward.

Check your understanding

A graph is drawn only between the endpoints $(-3, 1)$ and $(4, 1)$ , and over that stretch its lowest height is $-2$ and its highest height is $5$ . What is the range?

-3 \le y \le 4

-2 \le y \le 5

1 \le y \le 5

-2 \le y \le 1

Increasing and decreasing

The last thing a picture shows at a glance is the function’s direction. Read the graph the way you read a sentence, left to right, in the direction of increasing $x$ . Where the curve rises as you move right, the outputs are getting larger, and the function is increasing there. Where the curve falls as you move right, the outputs are getting smaller, and the function is decreasing.

Made precise, a function is increasing on an interval if larger inputs give larger outputs across it (if $a < b$ then $f(a) < f(b)$ ). A function is decreasing on an interval if larger inputs give smaller outputs across it (if $a < b$ then $f(a) > f(b)$ ). A point where the graph stops falling and starts rising, or the reverse, is a turning point. At the bottom of a valley the output is smaller than at any nearby input, a minimum. At the top of a hill the output is larger than at any nearby input, a maximum.

The graph of f(x) = x squared minus 2x minus 3 falls as x moves from the left up to the turning point at (1, -4), then rises after it. So f is decreasing for x less than 1 and increasing for x greater than 1, and the turning point (1, -4) is its minimum.

Worked example 5 Where is $f(x) = x^2 - 2x - 3$ increasing?

Trace the graph from left to right. Starting far to the left, the curve comes down steadily, so the outputs are shrinking and $f$ is decreasing. It keeps falling until it reaches the lowest point at $(1, -4)$ , then it turns and climbs. From that turning point onward the outputs grow, so $f$ is increasing.

Reading the two stretches off the horizontal axis, the turning happens at $x = 1$ , so

f \text{ is decreasing for } x < 1 \quad \text{and increasing for } x > 1.

The turning point $(1, -4)$ is the minimum, the single lowest output the function ever produces. This is the same value you would get by hunting for the vertex algebraically, but here you simply saw it on the picture.

Check your understanding

The graph of a function falls as $x$ goes from $-4$ up to $2$ , then rises as $x$ goes from $2$ onward. On which interval is the function increasing?

x < 2

x > 2

-4 < x < 2

D all

x

Common mistakes

Confusing input and output when reading. To find $f(a)$ , start on the horizontal axis at the input $a$ and go to the curve. To solve $f(x) = b$ , start on the vertical axis at the output $b$ and go across to the curve. Reversing them answers the wrong question.
Expecting $f(x) = b$ to have exactly one solution. A horizontal line can meet a curve several times, so one output can come from several inputs. Read off every crossing, not just the first.
Mixing up the two intercepts. The y-intercept comes from setting $x = 0$ and computing $f(0)$ ; the x-intercepts come from setting $y = 0$ and solving $f(x) = 0$ . Zero the coordinate of the axis you are crossing, which is the opposite letter.
Swapping domain and range. The domain is the horizontal spread of the graph (the inputs), read along the x-axis; the range is the vertical spread (the outputs), read along the y-axis.
Judging direction from right to left. Increasing and decreasing are always read moving left to right, in the direction of increasing $x$ . A valley looks like it “goes up” on its left side only if you read backward.
Reading a height off the wrong axis. The value $f(a)$ is a height, measured against the vertical axis. The horizontal axis only tells you the input $a$ .

Key takeaways

The graph of a function $f$ is the set of all points $(x, f(x))$ ; a point $(a, b)$ lies on it exactly when $f(a) = b$ .
Build a graph by making a table of inputs and outputs, plotting the points, and connecting them with a line or a smooth curve.
To read $f(a)$ , go from $a$ on the horizontal axis to the curve and read the height. To solve $f(x) = b$ , go from $b$ on the vertical axis across to the curve and read the inputs; there may be several.
The y-intercept is $(0, f(0))$ ; the x-intercepts (the zeros of $f$ ) are the solutions of $f(x) = 0$ .
The domain is the horizontal spread of the graph (its inputs) and the range is the vertical spread (its outputs).
A function is increasing where its graph rises left to right and decreasing where it falls; a turning point is a maximum or a minimum.

Graphs of Functions

Learning goals

The graph of a function is a picture of every input and output

Building a graph from a table

Reading a value off the graph

Intercepts and where the graph crosses the axes

Domain and range from a graph

Increasing and decreasing

Common mistakes

Practice

Multiple Choice Questions (MCQ)

Free Response Questions (FRQ)