Read points of interest from a graph

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Learn how to use Desmos points of interest to read SAT intercepts, intersections, roots, vertices, maximums, and minimums without choosing the wrong coordinate.

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

What you’ll learn

Click a graph to show its points of interest, and pick the one you need.
Tell $x$ -intercepts, $y$ -intercepts, intersections, and vertices apart.
See that roots and zeros are the $x$ -coordinates of $x$ -intercepts.
Decide which coordinate answers a maximum or minimum question about a quadratic.
Use the question's wording and context to choose among several marked points.
Spot when the equation already gives you the answer, so you don't need a graph.

Prerequisites

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

Why this matters on the SAT

Turn one graph point into the answer

Many SAT questions come down to one special point on a graph: where it crosses an axis, where two graphs meet, or where a curve turns around. Desmos marks the visible intercepts, intersections, maximums, and minimums for you. Once you know which point the question is describing, one graph and one careful click can replace several lines of algebra.

Here's how that plays out on a typical SAT question.

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?

Fast Desmos solution: Type C(x)=-2x^2+24x+10, with $x$ standing in for $t$ , and click the graph's highest point, $(6,82)$ . The question asks when the calls peak, so you want the horizontal coordinate. The answer is $\boxed{6}$ , choice C.

Finding the point took one click. The real skill is deciding what to report: the $x$ -coordinate, the $y$ -coordinate, the whole ordered pair, or a number you work out from them. The graph finds the point, and the question picks the coordinate.

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One point tells you both when the calls peak and how many there are.

When should you read a point from the graph?

If the question gives you an equation or function, and the answer sits at a special point on its graph, Desmos is a strong place to start.

Graph it and read the point

You need an $x$ -intercept, a root, or a zero, like “Which value of $x$ is a zero of $g$ ?”
You need a $y$ -intercept or a starting value, like a population at time $0$ .
You need the point where two graphs meet, like a line crossing a parabola.
You need the greatest or least value of a parabola, like the most frog calls per minute.
You need the input where that maximum or minimum happens, like the minute when the calls peak.
You need a whole ordered pair, or only one coordinate, of a point like these.

Skip the graph

The equation already shows the point, like $q(x)=3(x-5)^2+2$ , whose minimum value is $2$ at $x=5$ .

Desmos uses $x$ for the horizontal axis. If a model uses another input letter, like $t$ for minutes, type $x$ in its place, and remember that $x$ still means minutes.

Check your understanding:

The function $p$ is defined by $p(x)=-4(x+3)^2+11$ , and a question asks for the maximum value of $p$ . Would you start by graphing? What’s the answer?

One graph can hold several answers

Graph

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

Back in Lesson 1, you clicked one intercept and one intersection. The same click works for every kind of point, so the job now is telling them apart and pulling out the coordinate you need.

Click the curve or its expression line, and gray points appear at the graph's intercepts and its minimum. Click a gray point, and its coordinates stay on screen.

Each marked point answers a different question:

When the question says	Look at	What the coordinates tell you
$x$ -intercept, root, or zero	Where the graph meets the $x$ -axis	Here $y=0$ , and the root or zero is the $x$ -coordinate
$y$ -intercept or initial value	Where the graph meets the $y$ -axis	Here $x=0$ , and the initial value is the $y$ -coordinate
Minimum or maximum of a parabola	Its vertex, the lowest or highest point	The $x$ -coordinate tells you where it happens, and the $y$ -coordinate is the minimum or maximum value
Intersection	Where two graphs meet	The point works in both equations

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

A parabola's vertex is its turning point. It's a minimum when the parabola opens upward and a maximum when it opens downward.

One more thing about $x$ -intercepts: the graph doesn't have to cross the axis. It can touch the $x$ -axis and turn back, the way $y=(x-3)^2$ does at $x=3$ . That point still has $y=0$ , so its $x$ -coordinate is still a root, or zero.

Try it yourself:

Click each point in the calculator. Before you read its label, make a prediction: which coordinate has to be $0$ , or which point has to be the lowest?

Check your understanding:

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

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Click the graph, then click each marked point and say what it shows.

Read the wording before the coordinates

Same point, different questions, different answers. Say a maximum point is $(6,82)$ :

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

Minimum points work the same way. For quadratic models like these, the input is the horizontal coordinate and the output is the vertical one. In a word problem, turn $x$ and $y$ back into what they mean, with units, before you answer. In the frog-call model, $6$ means $6$ minutes after sunset, and $82$ means $82$ calls per minute.

Common mistake:

A common slip is clicking the right point and then submitting the wrong coordinate. After each click, pause and finish this sentence: “The question asks for ___, which is the ___-coordinate.”

Intersections need the same choice

An intersection is a point on both graphs. Here too, the question might want the whole point, one coordinate, the smaller or greater coordinate, or a value you build from the coordinates.

Graph

y=\lvert x+1\rvert+2

and

y=5.

The graphs meet 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 $5$ either way, because both points sit on the line $y=5$ .

The full systems-at-intersections workflow comes later, in Lesson 4.

Check your understanding:

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

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

Example: Pick 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

Turn “hits the ground” into a point

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

Step 2

Graph it and click the intercepts

Type h(x)=-4(x+1)(x-7), with $x$ standing in for $t$ . Click the graph, then click the marked points on the $x$ -axis. Desmos shows intercepts at

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

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The graph has two roots, but only the story tells you which one answers the question.

Step 3

Check each time against the story

Two intercepts, but the rocket lands only once. So which one counts? The question asks for a time after launch, so $t$ can't be negative. The intercept at $x=-1$ means $t=-1$ , a second before launch, so it doesn't fit. That leaves the other intercept:

t=7.

Step 4

Answer what the question asks

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

The graph's highest point, $(3,64)$ , answers different questions: the rocket reaches its maximum height of $64$ meters at $t=3$ seconds. That's where choices B and D come from. They're real numbers from the graph, just not the landing time.

Common mistake: Picking $-1$ , choice A. It's a true root of the equation, but not a moment in the rocket's flight. A graph can show points that are correct math and still impossible in the real situation.

When the point isn't visible

Can't see the point you need? Work through these steps:

Click the curve or its expression line so its gray points of interest appear.
Pan or zoom until the part of the graph you need is on screen.
If points overlap, hide the graphs you don't need.
Click the gray point so its coordinates stay visible.
Reread the question before you copy a value.

Point labels are decimals, so an exact answer like $\sqrt{2}$ shows up only as a rounded number. If the question tells you how to round, round that way. If it needs an exact value that the label doesn't clearly show, check it by substituting or with a little algebra.

Try it yourself:

Some important points on this graph sit just past the right edge of the view. Pan right, click the curve, and find its minimum and both $x$ -intercepts.

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Pan right to find the minimum at (12, -4) and the second x-intercept.

Finish the solution

Zeros work the same way on any graph, not only on parabolas. This one is a cubic, and the first two steps are done for you.

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 typed in and graphed.
Click the curve to show its $x$ -intercepts.

Finish it

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Click the x-intercepts and compare them with the choices.

Practice problems

Three questions, three kinds of points. Before you type anything, name the point you're looking for.

Practice problem

The graph of

3x+2y=12

intersects the $y$ -axis at which point?

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Find where the line meets the y-axis.

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

Type x in place of d, then click the parabola’s highest point.

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 $x$ ?

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Graph both and check the x-coordinates where they meet.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

The graph finds the point, and the question picks the coordinate.
Click a graph or its expression line to show its gray points of interest, then click the one you need.
Desmos graphs in $x$ , so type $x$ for an input letter like $t$ or $d$ , and keep its meaning in mind.
Roots and zeros are the $x$ -coordinates of $x$ -intercepts.
A $y$ -intercept has $x=0$ , and it's often the starting value.
At a quadratic's minimum or maximum, the $x$ -coordinate says where it happens and the $y$ -coordinate is the value.
An intersection is on both graphs, but the question may want only one coordinate.
When several points show, let the wording, units, signs, and the situation pick one.
Point labels are decimals, so check the value when the question needs it exact.
If the equation already shows the answer, skip the graph.

Next lesson

Solve one-variable equations

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

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Practice

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

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