Find slope and rate of change

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Lesson progressPractice problems 0/4

Difficulty: Beginner
Estimated time: 31 minutes
Domains: Algebra
Techniques: SlopeConsistent-subtractionScaled-axesRate-interpretation

What you’ll learn

Find slope from two points, a table, a graph or an equation.
Subtract in the same order on top and bottom.
Say what a positive, negative or zero slope means, with the right units.
Read graphs with scaled axes, and turn a change over several units into a rate.

Prerequisites

Solve linear equations0%
Use equivalent steps to isolate the output variable when a line is not already written in slope-intercept form.

Why this matters on the SAT

Read a line as a rate

On the SAT, a line can show up as an equation, two points, a table or a graph. Whatever form it takes, its slope answers one question: when the input changes, how much does the output change?

You'll need to subtract negative numbers and simplify fractions. When an equation doesn't show its slope yet, you'll also solve for the output variable, the skill from Solve linear equations.

Solution to the example

The slope is the number multiplying $t$ :

-1.6.

The minus sign tells you the output goes down as the input goes up. The output is temperature in degrees Celsius and the input is time in minutes, so the slope is in degrees Celsius per minute. The temperature drops $1.6^\circ\text{C}$ every minute.

That's D. Choice B is tempting because $84$ is right there in the equation, but it's the temperature at $t=0$ . That's where the sample starts, not how fast it changes.

SAT example

The function $T$ models the temperature, in degrees Celsius, of a sample $t$ minutes after cooling begins:

T(t)=84-1.6t.

Which choice best interprets the slope of the graph of $y=T(t)$ ?

A
The sample begins at $1.6^\circ\text{C}$ .
B
The sample begins at $84^\circ\text{C}$ .
C
The temperature increases by $1.6^\circ\text{C}$ each minute.
D
The temperature decreases by $1.6^\circ\text{C}$ each minute.

Spot the rate the question wants

On a line, every equal step in the input changes the output by the same amount. In the SAT example, each extra minute drops the temperature by the same $1.6^\circ\text{C}$ . That steady comparison is the slope:

\text{slope} = \frac{\text{change in output}}{\text{change in input}}.

With $x$ as the input and $y$ as the output, you write it as

m=\frac{\Delta y}{\Delta x}.

The symbol $\Delta$ is read "change in." It means the second value you pick minus the first. So slope tells you how much $y$ changes each time $x$ goes up by $1$ .

Questions name slope in several ways. Watch for "slope," "rate of change," "the change in $y$ with respect to $x$ ," an amount per hour, per mile or per item, or how much something goes up or down for each additional unit.

The order of the units matters too. A rate in dollars per hour puts dollars on top and hours on the bottom. Whatever comes before "per" goes on top.

Check your understanding:

A graph has time in seconds on the horizontal axis and distance in meters on the vertical axis. What units does its slope have, and which change goes on top?

Sometimes slope is only part of the job. If a question asks for an intercept, a whole equation or a prediction, you'll find the slope along the way, but it isn't your final answer.

Subtract in the same order

For two points $(x_1,y_1)$ and $(x_2,y_2)$ on a line,

m=\frac{y_2-y_1}{x_2-x_1}.

You can start with either point. The one rule is same order, top and bottom: if the top is the second point minus the first, the bottom has to be too.

Let's try it with $(-2,7)$ and $(4,-5)$ . Take the second point minus the first, in both places:

\begin{aligned} m &=\frac{-5-7}{4-(-2)}\\[1.4em] &=\frac{-12}{6}\\[1.4em] &=-2. \end{aligned}

Does a negative slope make sense? As $x$ goes from $-2$ to $4$ , $y$ drops from $7$ to $-5$ . The line falls, so yes.

Flip both subtractions and you get the same answer:

\frac{7-(-5)}{-2-4} = \frac{12}{-6} =-2.

Common mistake:

If you got a positive answer here, you probably subtracted the $y$ -values in one direction and the $x$ -values in the other. That flips only one sign. Write one point above the other so each $y$ -value stays with its own $x$ -value, then subtract the same way on top and bottom. Last, check the direction: a line that falls as you move right has a negative slope.

Check your understanding:

A line passes through $(1,-3)$ and $(7,9)$ . What is its slope? Then use the points to check its sign.

Slope means the same thing wherever a line shows up. Only your first move changes:

One slope idea, four starting points

What you’re given	Your first move
Two points	Use $\dfrac{y_2-y_1}{x_2-x_1}$ .
A table	Treat two rows as points, then compare the output change with the input change.
A graph	Pick two points you can read exactly, and take their values from the axis labels.
An equation	Solve for the output variable. The number multiplying the input is the slope.

Read slope from a table

Each row of a two-column table is one point: an input and its output. The rows don't have to be next to each other, so pick two with friendly numbers.

$x$	$f(x)$
$-1$	$10$
$2$	$4$
$6$	$-4$

Using the first and second rows,

m=\frac{4-10}{2-(-1)}=\frac{-6}{3}=-2.

The second and third rows give the same rate, as they should on a line:

m=\frac{-4-4}{6-2}=\frac{-8}{4}=-2.

For a short table of exact values like this one, subtracting by hand (the difference quotient) is usually faster than typing in a regression. If the points don't all sit exactly on one line, or the question asks for a fitted model or a prediction, use Desmos regression from a supplied table instead.

Read slope from a graph

On a graph, pick two points whose coordinates you can read exactly, then read their values off the axes. Don't count grid boxes unless each box is one unit on both axes. One box might mean $1$ hour across but $3$ centimeters up. Read the labels, not the boxes.

The graph below shows how a line's direction matches the sign of its slope:

$y=2x+1$ rises as $x$ increases, so its slope is positive.
$y=-2x+1$ falls as $x$ increases, so its slope is negative.
$y=3$ is flat. The output never changes, so its slope is $0$ .
$x=3$ is vertical. The horizontal change is $0$ , and you can't divide by $0$ , so its slope is undefined.

A vertical line isn't the graph of a function of $x$ , but it can still show up in a question about lines in the coordinate plane.

Check your understanding:

A sample stays at $18^\circ\text{C}$ from minute $2$ through minute $7$ . What is its slope in degrees Celsius per minute, and what does that slope tell you?

Calculator loads as you approach

Turn each line off and on to compare directions. Change the number in front of

x

in either slanted equation and watch the slope change. Reset brings back all four lines.

Read slope from an equation

When an equation looks like

y=mx+b,

the number $m$ in front of $x$ is the slope. If $y$ isn't by itself yet, solve for it first.

Take

6x+3y=12.

Subtract $6x$ from both sides, then divide by $3$ :

\begin{aligned} 3y&=-6x+12\\[1.4em] y&=-2x+4. \end{aligned}

The slope is $-2$ . It's tempting to grab the $6$ , since it sits right next to $x$ . But the $6$ isn't the slope, because $y$ wasn't by itself yet.

Check your understanding:

What is the slope of the line $4x-2y=7$ ? Why isn’t it $4$ ?

Example: Read a scaled graph without counting boxes

Worked example

Read coordinate values from the axes before calculating rise over run.

The graph shows the height of a candle as it burns. The horizontal axis gives time in hours, and the vertical axis gives candle height in centimeters.

Which choice gives the slope of the line?

A
$-12$ centimeters per hour
B
$-3$ centimeters per hour
C
$3$ centimeters per hour
D
$12$ centimeters per hour

Step 1

Read the labels, not the boxes

Follow each dot straight down to the time axis and straight across to the height axis. The marked points are $(0,15)$ and $(4,3)$ . From the first point to the second, time goes up while height goes down, so expect a negative slope.

Here's the trap: each box up is $3$ centimeters, but each box across is only $1$ hour. Counting boxes gives $4$ down and $4$ across, a slope of $-1$ , which isn't even one of the choices.

Step 2

Divide the height change by the time change

Height change goes on top and time change on the bottom:

\begin{aligned} m &=\frac{3-15}{4-0}\\[1.4em] &=\frac{-12}{4}\\[1.4em] &=-3. \end{aligned}

That's $-12$ centimeters of height over $4$ hours of time.

Step 3

Say what the rate means

The slope is

-3\text{ centimeters per hour}.

The answer is B. The candle gets $3$ centimeters shorter each hour.

Common mistake:

An answer of $12$ uses the total height change and forgets that it happened over $4$ hours. An answer of $3$ has the right size but has lost its sign. Divide the signed height change by the time change, and let the shrinking candle remind you the slope is negative.

Practice problems

Your turn. You'll meet a line as a table, an equation, a graph and two points.

Find slope from a table

Practice problem

The table shows values of a linear function $g$ .

$x$	$g(x)$
$-2$	$9$
$1$	$3$
$5$	$-5$

Which choice is the slope of the graph of $y=g(x)$ ?

Calculator loads as you approach

Two rows by hand is the quick way. Use the graph only if you want to check.

Interpret slope from a two-variable equation

Practice problem

A student council spends its entire $120 supply budget on notebooks and poster boards. Each notebook costs $6, and each poster board costs $4. If $n$ is the number of notebooks and $p$ is the number of poster boards, then

6n+4p=120.

The equation is graphed with $n$ on the horizontal axis and $p$ on the vertical axis. Which choice best interprets the slope?

Calculator loads as you approach

Solve for

p

and read the slope by hand. Use the graph only to check.

Read and convert a scaled-graph rate

Practice problem

Choose two exact points from the line and read their values from the labeled axes.

The graph shows the amount of water in a tank after filling begins.

According to the graph, at what rate is the amount of water increasing, in liters per hour?

Calculator loads as you approach

Read the graph and do the math by hand. Use this graph only to check.

Find slope from two points

Practice problem

A line in the $xy$ -plane passes through $(-3,7)$ and $(5,1)$ . What is the slope of the line?

Your answer

Calculator loads as you approach

Plot the two points here if you want to see which way the line runs.

Finish the lesson

4 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

Slope is the signed change in output divided by the change in input: $m=\frac{\Delta y}{\Delta x}$ .
Same order, top and bottom.
Read the labels, not the boxes.
In a table, rows are points. In an equation, solve for the output and read the number in front of the input.
Units are output per input, and the sign tells you whether the output goes up, goes down or stays the same.

Next lesson

Understand intercepts and starting values

Identify where a line crosses an axis and explain what that starting or boundary value means.

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Practice

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1,416 SAT questions use what this lesson teaches. Practice a few in a study session at the difficulty you choose.

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