Read nonlinear graphs and representations

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Lesson progressPractice problems 0/3

Difficulty: Intermediate
Estimated time: 34 minutes
Domains: Advanced Math
Techniques: Nonlinear-graphsFeature-translationInterval-readingSymmetryAsymptotes

What you’ll learn

Tell exactly which feature of a graph a question is asking about.
Turn that feature into a statement about inputs and outputs, like $f(x)=0$ or $f(0)=y$ .
Read zeros, intercepts, maximums and minimums, intervals and symmetry from graphs and tables.
Connect exponential and rational graphs with the lines they approach and the inputs they can’t take.
Tell when what you’re given is enough, and when Desmos can show you a hidden feature.

Prerequisites

Evaluate nonlinear functions and recover inputs0%
Connect an input $x$ with its output $f(x)$ and recover inputs from a given output.

Why this matters on the SAT

Translate the words before reading the graph

One graph can show you lots of numbers at once: where it crosses the axes, where it turns, how high it goes. The question’s wording tells you which feature it wants, and which coordinate of that feature.

Solution to the example

$g(x)=0$ asks for the inputs that give an output of $0$ . On a graph, output $0$ means height $0$ , so look for where the curve meets the $x$ -axis.

The curve meets the $x$ -axis at

(-2,0),\qquad (1,0),\qquad (5,0).

So the zeros are $-2$ , $1$ and $5$ . The greatest is $5$ , so the answer is D.

Look at what the question asked for. The $x$ -intercept is the point $(5,0)$ , but the question wants the value of $x$ , so the answer is $5$ .

SAT example

Each point where the curve meets the

x

-axis has output

0

The complete graph of the polynomial function $g$ is shown. What is the greatest value of $x$ for which $g(x)=0$ ?

A
$-2$
B
$0$
C
$1$
D
$5$

Start with the feature, not the function

You don’t need to decide first whether a graph is a parabola, a cubic or an exponential. Start with what the question wants. The opening question took four moves. Together they’re the feature-first method, and they work on any graph:

Name the target. There, it was a zero. Other questions ask for an intercept, a maximum, an interval, or an input the function can’t take.
Translate it. Turn that feature into a statement about $x$ and $f(x)$ . A zero of $g$ means $g(x)=0$ .
Look where it’s shown. The graph was given, so you read the zeros straight off it. You’d graph an equation only if it hid the feature.
Answer in the right shape. The question wanted a value of $x$ , so the answer was $5$ , not the point $(5,0)$ . Other questions want an output, a point or an interval.

In short: words first, then the graph.

Here’s how the words you’ll see most often translate:

From the question’s words to what you read

The question asks for	What it means	What to read
A zero or root	$f(x)=0$	The $x$ -coordinate of an $x$ -intercept
An $x$ -intercept	$f(x)=0$	The whole point $(x,0)$
The $y$ -intercept	$x=0$	The point $(0,f(0))$
A maximum or minimum value	The greatest or least output	The $y$ -coordinate of the highest or lowest point that’s part of the graph
Where a maximum or minimum happens	The input at that greatest or least output	Its $x$ -coordinate
Solutions to $f(x)=k$	The output equals $k$	The $x$ -coordinates where the graph meets the line $y=k$
Where $f(x)>0$	The output is positive	The intervals where the graph is above the $x$ -axis
Where $f$ is increasing	The output goes up as the input moves right	The intervals where the graph climbs from left to right

Common mistake:

Finding the right point but reporting the wrong part of it. It’s an easy slip, because both numbers are right there and both look like answers. Before you answer, finish this sentence: “The question asks for ___, so I need the $x$ -coordinate, the $y$ -coordinate, the whole point, or an interval.”

Check your understanding:

A graph’s lowest point is $(-3,7)$ . What is the minimum value of the function, and where does that minimum happen?

Read points, levels, and intervals

Every point on a graph is a sentence. The point $(a,b)$ on the graph of $y=f(x)$ says

f(a)=b.

In words: input $a$ gives output $b$ . You’ll see that same fact in three places:

as an equation, $f(a)=b$
as a table row, with input $a$ and output $b$
as a graph point, $(a,b)$

So you can read in either direction. If the question gives you an output, like $f(x)=6$ , look at the horizontal line $y=6$ . Each place it meets the graph gives you one input with output $6$ . If the question gives you an input, like $f(0)$ , go to $x=0$ and read how high the graph is there.

The dashed line

y=6

meets the graph at two inputs. After the vertex, the curve can rise while it is still below the

x

-axis.

Some questions ask about a whole stretch of inputs, called an interval:

$f(x)>0$ where the graph is above the $x$ -axis
$f(x)<0$ where it’s below the $x$ -axis
$f$ is increasing where the graph goes up as you move from left to right
$f$ is decreasing where it goes down as you move from left to right

The graph switches between going up and going down at a turning point. On a parabola, that turning point is the vertex. Other polynomials can have several turning points, but you read them the same way.

Common mistake:

Mixing up where the graph is with which way it’s going. Positive means above the $x$ -axis. Increasing means climbing as $x$ moves right. Look at the figure: just right of the vertex, the curve is climbing but still below the axis, so the function is increasing and negative at the same time.

Check your understanding:

The horizontal line $y=6$ meets the graph of $y=f(x)$ at $(-1,6)$ and $(5,6)$ . What are the solutions to $f(x)=6$ ?

Example: connect a table with its graph

Worked example

Each table row becomes one graph point. Equal outputs at

x=0

and

x=4

show symmetry around

x=2

The table and graph below represent the same quadratic function $q$ .

Which statement about the minimum value of $q$ , the $x$ -intercepts, or the $y$ -intercept must be true?

A
The $x$ -intercepts are $(0,-1)$ and $(0,5)$ .
B
The minimum value of $q$ is $-9$ , and it occurs at $x=2$ .
C
The $y$ -intercept is $(0,-9)$ .
D
The function is positive for $-1<x<5$ .

Step 1

Translate each feature

That’s a long question, but it’s really asking: which of these four sentences is true? Before you check them, turn the features the question names into inputs and outputs:

the minimum value is the least output
an $x$ -intercept has output $0$
the $y$ -intercept has input $0$

Step 2

Find the lowest point

The lowest point on the graph is $(2,-9)$ , and the table agrees: $q(2)=-9$ . So the minimum value is $-9$ , and it happens at $x=2$ .

Step 3

Check the intercept choices

The rows $q(-1)=0$ and $q(5)=0$ have output $0$ , so the $x$ -intercepts are

(-1,0)\quad\text{and}\quad(5,0).

Choice A has the coordinates flipped. The row $q(0)=-5$ gives the $y$ -intercept $(0,-5)$ , not the $(0,-9)$ in choice C.

Step 4

Check the sign and choose

Between the zeros $-1$ and $5$ , the curve is below the $x$ -axis, so $q(x)<0$ on $-1<x<5$ . Choice D has the sign backward.

Only choice B is true.

The table hides one more feature: symmetry. The inputs $0$ and $4$ give the same output:

q(0)=q(4)=-5.

A parabola is its own mirror image across its axis of symmetry, so equal outputs come from inputs the same distance from that axis. The axis sits halfway between them:

\frac{0+4}{2}=2,

and that’s exactly the vertex’s $x$ -coordinate. This works for any quadratic: two different inputs with the same output always have their midpoint on the axis of symmetry.

Check your understanding:

A quadratic function $p$ has $p(-2)=p(8)$ . What is the $x$ -coordinate of its axis of symmetry?

Asymptotes and excluded inputs

Some graphs are built around a line that the curve gets closer and closer to. That line is called an asymptote.

A horizontal asymptote marks a height the curve gets close to. For simple rational functions like

r

, a vertical asymptote marks an input the function can’t take.

Take the exponential function

e(x)=2^x+1.

Since $2^x$ is always positive, every output is above $1$ . As $x$ gets more and more negative, $2^x$ shrinks toward $0$ , so the outputs get closer and closer to $1$ . The horizontal asymptote is $y=1$ . The graph still has an ordinary $y$ -intercept. Plug in $0$ :

e(0)=2^0+1=2,

so the $y$ -intercept is $(0,2)$ .

Now the rational function

r(x)=\frac{4}{x-2}+1.

Try $x=2$ : the denominator becomes $0$ , and you can’t divide by $0$ . So $2$ is an input $r$ can’t take. The math word for this is excluded from the domain, where the domain is the set of inputs a function can take. On the graph, you see it as a vertical asymptote at $x=2$ . Far to the left and far to the right, the curve also gets closer and closer to the horizontal line $y=1$ .

Neither dashed line in the figure is an intercept:

the vertical asymptote $x=2$ isn’t part of the graph at all, because $2$ isn’t an input $r$ can take
the horizontal asymptote $y=1$ only marks the height the curve gets close to
the real $x$ -intercept is $(-2,0)$ , because $r(-2)=\dfrac{4}{-4}+1=0$

Common mistake:

Calling a dashed asymptote an intercept. It’s tempting, because the dashed line is drawn right there with the curve. But an intercept is a point on the curve. A vertical asymptote sits at an input the function can’t take, like $x=2$ for $r$ , so the curve can’t meet the $x$ -axis there.

Check your understanding:

For $s(x)=\dfrac{6}{x+3}-2$ , which input is excluded from the domain, and what is the vertical asymptote?

Use Desmos only when the feature is hidden

Before you open Desmos, look at what you’re given. Often the feature you need is already on the page, so don’t redraw a graph you already have.

Read it directly when…

you’re given a clear graph or table, like the graph of $g$ in the opening question.
a marked point already shows the coordinate you need, like a lowest point labeled $(2,-9)$ .
a short equation gives the answer in one quick step, like $e(0)=2^0+1=2$ , or seeing that $x=2$ makes the denominator of $r$ equal $0$ .
the answer comes from symmetry, like $q(0)=q(4)$ putting the axis at $x=2$ .

Use Desmos when…

you’re given only an equation, and you can’t see its roots, its highest or lowest point, or where it’s positive or negative.
you need the solutions to $f(x)=k$ , and the line $y=k$ might meet the graph more than once.
you want to see how many inputs work, or which one is bigger, before you pin down their exact values.

When the answer has to be exact, let Desmos show you the pattern and let algebra confirm the numbers.

Say you’re given the polynomial

p(x)=x^3-3x^2-6x+8,

and the question asks: On which interval is $p(x)<0$ ?

The question wants to know where $p$ is negative, and the equation doesn’t show that, so graph it. Enter:

p(x)=x^3-3x^2-6x+8

Then read the points you need with the three-step point-reading flow:

Select the curve. Gray points of interest appear, such as its intercepts and turning points.
Select the gray points you need. Here, that’s each $x$ -intercept. Their coordinates appear: $(-2,0)$ , $(1,0)$ and $(4,0)$ .
Take only the part the question asks for. Here that’s the inputs, $x=-2$ , $x=1$ and $x=4$ , not the whole points.

Those three inputs split the $x$ -axis into four regions. Now look for where the curve is below the $x$ -axis:

x<-2 \quad\text{and}\quad 1<x<4.

In interval notation, that’s

\boxed{(-\infty,-2)\cup(1,4)}.

Are the boundaries exactly $-2$ , $1$ and $4$ , or just close? The factored form settles it:

p(x)=(x+2)(x-1)(x-4)

is $0$ at exactly those three inputs. Desmos shows you the pattern, and the factors confirm the values are exact.

Try it yourself:

Before you select the intercepts, trace the curve from left to right. Where is it above the axis, and where is it below? Then select the points and see whether your boundaries match.

Calculator loads as you approach

Find the three x-intercepts, then see where the curve dips below the x-axis.

For quadratics, picking the form that shows the feature you need comes next, in Use the three forms of a quadratic function.

Practice problems

Your turn. For each one, decide first whether you need Desmos at all.

Read an exponential table

Practice problem

Selected values of an exponential function $f$ are shown.

$x$	$f(x)$
$-1$	$\dfrac32$
$0$	$2$
$1$	$3$
$2$	$5$

Which statement about the $y$ -intercept or the graph point corresponding to $f(1)$ must be true?

Calculator loads as you approach

The table already shows both features you need. Plot the points in Desmos only if you want a check.

Find where a cubic is negative

Practice problem

The function $g$ is defined by

g(x)=x^3-2x^2-5x+6.

On which interval is $g(x)<0$ ?

Calculator loads as you approach

Graph

g

, find its x-intercepts, and see where the curve is below the x-axis.

Use table symmetry to connect the zeros

Practice problem

A quadratic function $h$ is represented by the table.

$x$	$h(x)$
$1$	$13$
$3$	$1$
$7$	$1$
$9$	$13$

The equation $h(x)=0$ has two real solutions, $r$ and $s$ . What is the value of $r+s$ ?

Your answer

Calculator loads as you approach

You can’t get the full equation from the table right away. Try symmetry first: look for two inputs with the same output.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Words first: name the feature, then read or graph.
A point $(a,b)$ means $f(a)=b$ .
A zero is an $x$ -value where $f(x)=0$ . An $x$ -intercept is the whole point $(x,0)$ , and the $y$ -intercept is $(0,f(0))$ .
At a highest or lowest point, $x$ tells you where the maximum or minimum happens, and $y$ is its value.
Positive and negative are about where the graph is. Increasing and decreasing are about which way it’s going.
For a quadratic, two inputs with equal outputs have their midpoint on the axis of symmetry.
For simple rational functions like $r$ , a vertical asymptote marks an input the function can’t take.
Read a graph or table you’re given. When an equation hides the feature, graph it and take only the coordinate or interval you need.

Next lesson

Use the three forms of a quadratic function

Choose standard, factored, or vertex form when a quadratic equation needs to display a particular feature.

Start next lesson

Practice

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202 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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