SAT/Desmos guide/Linear functions

How to graph linear functions on Desmos

Lines are the backbone of SAT Algebra, and Desmos turns slope-and-intercept questions into a quick look at a graph. Type the equation, or drop in two points, and the numbers you need are right there.

Enter a line as y = mx + b — the slope is m and the y-intercept is b, both readable straight from the equation. Click where the line crosses the x-axis to read the x-intercept. To get the slope through two points, type the points as (x, y) and use rise over run, or graph a line through them and read its m. Every feature comes off the screen.

Practice this on real questions All Desmos techniques

A line through two points: Make the mathematical object in this brief the hero: a straight rising line on a grid: two ringed points it passes through, a small right-triangle "rise over run" sketch between them, and ringed marks where the line meets the x-axis and y-axis

mThe slope — the coefficient of x in y = mx + b.

bThe y-intercept, where the line crosses the y-axis.

rise/runThe slope between two points, vertical change over horizontal.

1 lineSlope, both intercepts and the equation, all from one graph.

Step by step

Type the line, read the numbers

Whether you're handed an equation or a pair of points, the line goes on the screen and the slope and intercepts follow.

Step 1

Graph the line

Enter it in slope-intercept form, e.g. y = 2x + 3. The slope is the number on x and the y-intercept is the constant — both visible before you even look at the graph.

Step 2

Read the intercepts

The y-intercept is b. For the x-intercept, click where the line crosses the x-axis; Desmos shows (x, 0), and that x-value is what you want.

Step 3

Two points? Get the slope

Type each point as (x, y) on its own line — say (0, 3) and (4, 11). Compute rise over run, or graph a line through them and read its coefficient of x.

Step 4

Assemble the equation

Drop the slope and y-intercept into y = mx + b. A quick glance at the graph confirms the line truly passes through the points or hits the intercepts the question gives.

Make the read instant.

Linear functions are everywhere in the Algebra domain. Practice graphing lines and pulling slope and intercepts on SAT-style questions with the built-in calculator.

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Worked example

From two points to the equation

A classic "find the equation of the line" question, handled with points and a quick read.

The line through (0, 3) and (4, 11)

Question (SAT-style). A line in the xy-plane passes through the points (0, 3) and (4, 11). Which equation defines the line?

Plot: (0, 3) and (4, 11)
Slope = rise/run = (11 − 3) / (4 − 0) = 8/4 = 2
The point (0, 3) sits on the y-axis, so the y-intercept is 3

Putting it together gives y = 2x + 3. To confirm, graph y = 2x + 3 on top of the two plotted points — the line runs cleanly through both, so the equation is right. (No second point on the y-axis? Read b by clicking where your line crosses it instead.)

Line y = 2x + 3: Make the mathematical object in this brief the hero: a rising straight line passing through a ringed point on the y-axis at (0, 3) and a ringed point at (4, 11), with a small step-triangle showing run 4 and rise 8 between them

Reading slope and intercepts from an equation

Question (SAT-style). Line k is defined by y = −¾x + 6. What is the x-intercept of line k?

Graph y = -3/4 x + 6. From the equation alone you can read the slope −¾ and the y-intercept 6. For the x-intercept, click where the line crosses the x-axis: Desmos shows (8, 0), so the x-intercept is 8. One line on the screen delivered slope, both intercepts, and the answer.

When to use it / pitfalls

Where graphing a line helps most

Use it

For intercepts and "which equation" items.

Reading an x-intercept off the graph, or graphing answer choices to see which line passes through given points, is faster and surer than algebra under time pressure.

Watch out

For slope sign and "per unit" wording.

A downward line has a negative slope — don't drop the sign. In word problems the slope is the rate "per" something; make sure rise and run line up with the quantities the question names.

Skip it

When the equation already shows it.

If a line is handed to you as y = mx + b and the question wants the slope or y-intercept, just read m or b — opening the graph adds nothing.

FAQ

Linear functions on Desmos, answered

How do you graph a line on Desmos?

Type the equation in slope-intercept form, such as y = 2x + 3, on its own line. Desmos draws the line immediately, with slope m and y-intercept b read straight from the equation. You can also enter a line in standard form and Desmos will graph it.

How do I find the slope of a line on Desmos?

If the line is in the form y = mx + b, the slope is m, the coefficient of x. To find the slope through two points, type both points and compute rise over run, or graph a line through them and read the coefficient of x in y = mx + b.

How do I read the x-intercept and y-intercept?

The y-intercept is the b in y = mx + b, where the line crosses the y-axis. For the x-intercept, click the point where the line crosses the x-axis; its x-value is the x-intercept.

How do I plot points in Desmos?

Type a point as (x, y) on its own line, for example (4, 11), and Desmos plots it. Enter several points on separate lines to see them together, which helps when you are finding a slope or checking that a line passes through given points.

Can Desmos give the equation of a line through two points?

You can find it quickly: plot both points, read the y-intercept where the line crosses the y-axis, and compute the slope as rise over run between the points. Put them together as y = mx + b. Desmos also supports a linear regression on two points, which returns the same line.

Turn Desmos into points.

Start free and practice SAT-style Math with the built-in calculator, until reading slope and intercepts off a line is automatic.

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