Read points of interest from a graph

Your feedback matters:

Lesson progressPractice problems 0/4

Difficulty: Beginner
Estimated time: 20 minutes
Domains: Algebra Advanced Math
Techniques: GraphingPointsInterpretation

What you’ll learn

Reveal and select Desmos points of interest.
Distinguish x-intercepts, y-intercepts, intersections, and vertices.
Connect roots and zeros to x-intercepts.
Choose the correct coordinate for a quadratic maximum or minimum question.
Use wording and context to select the relevant marked point.
Recognize when the equation already makes graphing unnecessary.

Prerequisites

Introduction to Desmos: your SAT calculator0%
Use the essential Desmos controls and enter equations accurately.

Why this matters on the SAT

Turn a graph into the requested answer

SAT questions often hide the answer in an important point on a graph. Desmos marks visible intercepts, intersections, maximums, and minimums automatically. Once you know which point the question describes, you can often replace several lines of algebra with one graph and one careful click.

SAT example

The number of frog calls heard per minute, $C(t)$ , is modeled by

C(t)=-2t^2+24t+10,

where $t$ is the number of minutes after sunset. For what value of $t$ is $C(t)$ greatest?

A
$3$
B
$5$
C
$6$
D
$7$

Fast Desmos solution

Enter C(x)=-2x^2+24x+10, where $x$ represents $t$ , and select the graph's highest point, $(6,82)$ . The question asks when the number of calls is greatest, so use the horizontal coordinate. The answer is $\boxed{6}$ , choice C.

The point is easy to find. The real SAT skill is deciding whether to report its $x$ -coordinate, its $y$ -coordinate, the full ordered pair, or a value calculated from those coordinates.

Calculator loads as you approach

The same point gives both when the maximum occurs and the maximum value.

When should you read a point from the graph?

Graphing is a strong opening method when a supplied equation or function encodes the requested answer in an important point.

Read a graph point when the question asks for…

an x-intercept, root, or zero;
a y-intercept or initial value;
the point where two graphs meet;
the maximum or minimum value of a parabola;
the input where a parabola’s maximum or minimum occurs; or
an ordered pair or one coordinate of an important point.

Use the shorter method when…

the equation already states the point. For example, $q(x)=3(x-5)^2+2$ has a minimum value of $2$ at $x=5$ .

Desmos uses x as the horizontal graphing variable. If a context uses another input letter, replace it with x while keeping its original meaning.

Check your understanding:

The function $p$ is defined by $p(x)=-4(x+3)^2+11$ . The question asks for the maximum value of $p$ . Should you begin by graphing, and what is the answer?

1. One graph can contain several answers

Graph

f(x)=x^2-4x-5.

In Lesson 1, you selected one intercept and one intersection. Now you will use the same mechanic to distinguish every common point type and extract the requested coordinate.

Select the curve or its expression line. Desmos reveals gray points at the graph’s intercepts and minimum. Select a gray point to keep its coordinates visible.

Question language	Point to use	What the coordinates mean
$x$ -intercept, root, or zero	Where the graph meets the $x$ -axis	The point has $y=0$ ; the root or zero is its $x$ -coordinate
$y$ -intercept or initial value	Where the graph meets the $y$ -axis	The point has $x=0$ ; the initial value is its $y$ -coordinate
Minimum or maximum of a parabola	Its vertex, the lowest or highest point	The $x$ -coordinate says where it occurs; the $y$ -coordinate is the extreme value
Intersection	Where two graphs meet	Both coordinates satisfy both equations

For this graph:

the roots are $x=-1$ and $x=5$ ;
the $y$ -intercept is $(0,-5)$ ; and
the minimum point is $(2,-9)$ .

For a parabola, the vertex is its turning point. It is a minimum when the parabola opens upward and a maximum when the parabola opens downward.

An $x$ -intercept does not have to cross the axis. A graph can touch the $x$ -axis and turn around there. The touching point still has $y=0$ , so its $x$ -coordinate is still a root or zero.

Try it yourself:

Select each point in the calculator. Before reading its label, predict which coordinate must be $0$ or which point must be lowest.

Check your understanding:

The minimum point of $f(x)=x^2-4x-5$ is $(2,-9)$ . What is the minimum value of $f$ , and for what value of $x$ does it occur?

Calculator loads as you approach

Select the graph, then select each marked point and name what it represents.

2. Read the wording before the coordinates

The words in the question determine which part of a point becomes the answer.

If a maximum point is $(6,82)$ :

“For what value of $t$ is the function greatest?” asks for $6$ .
“What is the greatest value of the function?” asks for $82$ .
“What are the coordinates of the maximum?” asks for $(6,82)$ .

The same rule applies to minimum points. For the quadratic models in this lesson, the input is the horizontal coordinate, and the output is the vertical coordinate. In a context question, replace $x$ and $y$ with their meanings and units before answering.

Common mistake:

Selecting the correct point but submitting the wrong coordinate. Complete this sentence before answering: “The question asks for ___, which is the ___-coordinate.”

Intersections require the same decision

An intersection is a point that lies on both graphs. A question may ask for the full point, one coordinate, the smaller or greater coordinate, or an expression built from the coordinates.

Graph

y=\lvert x+1\rvert+2

and

y=5.

The graphs intersect at $(-4,5)$ and $(2,5)$ . If the question asks for the smaller $x$ -coordinate, the answer is $-4$ . If it asks for the $y$ -coordinate of an intersection, the answer is $5$ .

Lesson 4 develops the full systems-at-intersections workflow. Here, focus on selecting the right point and reading the requested coordinate.

Check your understanding:

The graphs above intersect at $(-4,5)$ and $(2,5)$ . What is the greater $x$ -coordinate, and what is the $y$ -coordinate of either intersection?

Calculator loads as you approach

Both intersections have the same y-coordinate, but different x-coordinates.

3. Choose the point that fits the context

Worked example

The height $h(t)$ , in meters, of a model rocket

$t$ seconds after launch

is modeled by

h(t)=-4(t+1)(t-7).

According to the model, how many seconds after launch does the rocket hit the ground?

A
$-1$
B
$3$
C
$7$
D
$64$

Step 1

Translate the event into a point

At ground level, the height is $0$ . On the graph of $h$ , that means an $x$ -intercept.

Step 2

Graph and select the intercepts

Enter h(x)=-4(x+1)(x-7), where $x$ represents $t$ . Select its graph, then select the marked points on the $x$ -axis. Desmos shows intercepts at

(-1,0)\qquad\text{and}\qquad(7,0).

Calculator loads as you approach

The graph contains two mathematical roots, but the context determines which one answers the question.

Step 3

Apply the time condition

The question asks for a time after launch, so $t$ must be nonnegative. The intercept at $x=-1$ represents $t=-1$ , which does not describe a time after launch. Use the other intercept:

t=7.

Step 4

Submit the requested quantity

The rocket hits the ground $7$ seconds after launch, so the answer is C.

The maximum point $(3,64)$ answers different questions: the rocket reaches a maximum height of $64$ meters at $t=3$ seconds.

Common mistake:

Choosing every visible intercept without applying the context. A graph can contain mathematically valid points that the real situation excludes.

4. Make the point visible

If the point you need is not visible:

Select the curve or its expression line so Desmos shows its gray points of interest.
Pan or zoom until the relevant part of the graph is on screen.
Hide unrelated graphs if several points overlap.
Select the gray point to keep its coordinates visible.
Reread the prompt before transferring a value.

Graph-point labels are numerical. Follow any rounding direction in the prompt, and verify with substitution or algebra when the question requires an exact value that the label does not clearly show.

Try it yourself:

The graph has important points just beyond the right edge. Pan right, select the curve, and reveal its minimum and both $x$ -intercepts.

Calculator loads as you approach

Pan right to reveal the minimum at 12 comma negative 4 and the second x-intercept.

Finish the solution

The function is already graphed. Select its $x$ -intercepts, then choose the listed value that is a zero.

Connect a zero to an x-intercept

Finish the solution

The function $g$ is defined by

g(x)=x^3-4x^2-7x+10.

Which of the following values of $x$ is a zero of $g$ ?

First steps

The function is entered and its graph is visible.
Select the curve to reveal its $x$ -intercepts.

Finish it

Calculator loads as you approach

Select the x-intercepts and compare their x-coordinates with the choices.

Practice problems

SAT practice problems

Identify the required point type before entering the equation, then submit only the quantity the question asks for.

Read a y-intercept

Practice problem

The graph of

3x+2y=12

intersects the $y$ -axis at which point?

Calculator loads as you approach

Graph the equation, select the line, and identify its y-intercept.

Find where a model is greatest

Practice problem

A music streaming service models the number of new subscribers $S(d)$ , in thousands, $d$ days after a promotion begins by

S(d)=-(d-6)(d-22)+300.

For what value of $d$ is $S(d)$ greatest?

Calculator loads as you approach

Replace the contextual input with x, then select the parabola’s maximum.

Choose one intersection x-value

Practice problem

The graphs of

y=12

and

y=(x-1)^2+3

intersect at points $(x,y)$ in the $xy$ -plane. Which choice is one possible value of $x$ ?

Calculator loads as you approach

Graph both equations and compare the x-coordinates of their intersections.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Select a graph or expression line to reveal its gray points of interest, then select the point you need.
If a context function uses an input letter such as $t$ or $d$ , replace it with $x$ in Desmos and keep its contextual meaning.
Roots and zeros are $x$ -coordinates of $x$ -intercepts.
A $y$ -intercept has $x=0$ and often represents an initial value.
At a quadratic minimum or maximum, the $x$ -coordinate says where it occurs and the $y$ -coordinate is the extreme value.
An intersection lies on both graphs, but the question may ask for only one coordinate.
Use wording, units, signs, and contextual feasibility to choose among several visible points.
Read graph decimals as numerical candidates and verify when an exact value is required.
Skip graphing when a familiar equation form already states the answer.

Next lesson

Solve one-variable equations

Turn equations into graphs and select the correct solution when more than one is visible.

Start next lesson

Free Practice

What you’ll learn

Prerequisites

Turn a graph into the requested answer

Fast Desmos solution

When should you read a point from the graph?

Read a graph point when the question asks for…

Use the shorter method when…

1. One graph can contain several answers

2. Read the wording before the coordinates

Intersections require the same decision

3. Choose the point that fits the context

Translate the event into a point

Graph and select the intercepts

Apply the time condition

Submit the requested quantity

4. Make the point visible

Finish the solution

Connect a zero to an x-intercept

First steps

Finish it

Practice problems

SAT practice problems

Read a y-intercept

Find where a model is greatest

Choose one intersection x-value

4 practice examples left

Quick recap

Solve one-variable equations

Free question bank practice problems