Digital SAT · Nonlinear Graphs

How to Solve Graph Features on the Digital SAT

Every feature has an address. Zeros sit where the curve crosses the x-axis, the y-intercept is the constant term, and the vertex is the turning point. Some forms of the equation hand a feature straight to you, factored form for zeros and vertex form for the vertex, but you never have to hunt: graph the function in Desmos and click the exact point, and it prints the coordinates.

Written from Perfect1600’s analysis of every graph features question in our bank·Method checked against the current Bluebook test

per test: 2 per test
typical difficulty: Medium
practice questions: 219

How we counted

Frequency reflects how often this question type appears on a full-length Digital SAT. The difficulty mix reflects every question of this type across our bank.

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What the question bank shows

match it

Every feature has a form

Factored for zeros, vertex form for the vertex, the constant for the y-intercept.

61%

Click it on Desmos

Most are settled by graphing and clicking the labeled point.

the hazard

Feature confusion

Reporting the wrong feature or swapping x and y is the frequent error, not the graphing.

How to recognize graph features questions

A function or its graph is given, and one feature is named.
The words zero, root, x-intercept, y-intercept, vertex, maximum, or minimum appear.
Choices are points, tables, or values, and the format is usually multiple choice.
The equation may already be in a form that reveals the feature, such as vertex or factored form.

Why students miss these

The mix-ups are between features and between forms. A y-value gets reported where an x-intercept was asked, or a vertex sign is misread off a shifted equation. Factored form shows the zeros, vertex form shows the turning point, and standard form shows the y-intercept, so hunting for a feature in the wrong form wastes time and invites a sign error. Clicking the point in Desmos sidesteps both.

The step-by-step method

1
Name the feature
Decide exactly what is wanted: a zero, an intercept, the vertex, or a max or min.
2
Match it to a form
Factored form gives zeros, vertex form gives the vertex, the constant term gives the y-intercept.
3
Read or compute
Pull the feature from that form, or set y = 0 for an x-intercept and x = 0 for a y-intercept.
4
Report the right coordinate
Give the point or value asked, and check you did not hand back an x where a y was wanted.

Solving it on Desmos

Graph the function. Type it as y = ... in Desmos to see the curve.
Click the feature. The x-axis crossing for a zero, the y-axis crossing for the y-intercept, the turning point for the vertex; each gets labeled coordinates.
Take the value asked. Read the x, the y, or the full point from the labeled dot.

Full Desmos walkthrough for graph features→

When to use it: Because Desmos labels each feature exactly, clicking is the surest route on nearly all of these. Read from the equation only when a form hands the feature over at a glance, like the y-intercept from the constant.

See it live: open a real question in the same Desmos calculator you get on Perfect1600, with the equations already typed in.

Worked examples

Easy example

f(x) = (x - 2)^2

Which of the following tables gives three values of x and their corresponding values of

f(x)

for the function f?

x	f(x)
1	4
2	3
3	4

x	f(x)
1	1
2	0
3	1

x	f(x)
1	1
2	0
3	5

x	f(x)
1	0
2	1
3	4

A: At $x = 1$ , the function f gives $f(1) = 1$ , not 4 as shown, so this table does not represent it.

B: Every value in this table satisfies the function f, so it correctly represents the relationship.

C: At $x = 3$ , the function f gives $f(3) = 1$ , not 5 as shown, so this table does not represent it.

D: At $x = 1$ , the function f gives $f(1) = 1$ , not 0 as shown, so this table does not represent it.

Explanation

Substitute each x-value into the function f and compare the result with the value in the table. The correct table is the one whose every value matches; a table with even one value that does not satisfy the function f is incorrect.

Medium example

Consider the function

f

given by

f(x)=x^2-64

. What is the positive

x

-intercept of the graph of

f

-8

0

8

64

A: Incorrect. $-8$ is the negative $x$ -intercept.

B: Incorrect. $f(0)=-64\ne0$ .

C: Correct. $x^2-64=0$ gives $x=\pm8$ ; the positive one is 8.

D: Incorrect. 64 is the constant, not an $x$ -intercept.

Explanation

x^2 = 64 gives x = +-8; positive is 8.

Hard example

Consider the function

f

given by

f(x)=-2(x+3)^2+8

. What is the maximum value of

f

-3

-2

3

8

A: Incorrect. $-3$ is the $x$ -coordinate of the vertex, not the value.

B: Incorrect. $-2$ is the leading coefficient, not the value.

C: Incorrect. 3 is part of the vertex $x$ -coordinate, not the maximum.

D: Correct. In $-2(x+3)^2+8$ , the parabola opens down, so the maximum value is 8.

Explanation

Vertex form: max value 8 (at x = -3).

The common traps

Pattern	What it does	The tell
x and y swapped	Reported a y-value where an x-intercept was asked.	An x-intercept has y = 0; a y-intercept has x = 0. Check which coordinate is wanted.
Sign off a shift	Misread the vertex sign from a shifted equation like (x + 3) squared.	In vertex form the x-coordinate takes the opposite sign of the number inside.
Feature in the wrong form	Tried to read a zero from standard form or the y-intercept from factored form.	Match feature to form, or simply graph and click.
Answered a different feature	Gave the vertex when a zero was asked, or a max value instead of where it occurs.	Check whether the location or the value of the feature is wanted.

Try it: two real questions

Question 1easy

$f(x) = x^2 - 3$ Which of the following tables gives three values of x and their corresponding values of $f(x)$ for the function f?

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

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

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

x	f(x)
0	0
1	1
2	4

Question 2easy

In the $xy$ -plane, the graph of $f(x) = (x - 2)(x + 6)$ is a parabola. What is the $x$ -coordinate of the vertex of this parabola?

Question 3 is ready when you are

Keep going with more questions of this type, each with the same step-by-step reasoning. Free to start.

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Often confused with

Quadratic & Exponential Functions

Exponential Growth and Decay

Common questions

How do you find the zeros of a function on the SAT?

Set the function to zero and solve, or read them from factored form where each factor gives a zero. On the digital test, graph it in Desmos and click where the curve crosses the x-axis.

How do I find the vertex of a parabola on the Digital SAT?

Vertex form, y = a(x - h) squared + k, places the vertex at (h, k). Otherwise graph the function and click the turning point for its exact coordinates.

What is the difference between an x-intercept and a y-intercept?

An x-intercept is where the graph crosses the x-axis, so y = 0. A y-intercept is where it crosses the y-axis, so x = 0, and it equals the constant term in y = ... form.

Can Desmos find features of a graph for me?

Yes. Graph the function and click the feature; it prints the exact coordinates, which removes the sign and form errors these questions punish.

How do I read a feature from the equation without graphing?

Match it to a form: factored shows zeros, vertex form shows the vertex, the constant term is the y-intercept. If the form does not hand it over, graphing is quicker.

Practice graph features the way it is tested

Start with the free 16-question diagnostic, then drill this type with step-by-step reasoning on every question.

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What the question bank shows

How to recognize graph features questions

Why students miss these

The step-by-step method

Name the feature

Match it to a form

Read or compute

Report the right coordinate

Solving it on Desmos

Worked examples

The common traps

Try it: two real questions

Question 3 is ready when you are

Often confused with

Related reading

Common questions

How do you find the zeros of a function on the SAT?

How do I find the vertex of a parabola on the Digital SAT?

What is the difference between an x-intercept and a y-intercept?

Can Desmos find features of a graph for me?

How do I read a feature from the equation without graphing?

Practice graph features the way it is tested