Before you read

Answer first — the lesson below explains it.

A car's fuel efficiency is tracked. At $2$ gallons used, the car has traveled $5$ miles. At $6$ gallons used, it has traveled $13$ miles. What is the rate of miles per gallon (slope of the line through $(2, 5)$ and $(6, 13)$ )?

A
$\frac{1}{2}$
B
$2$
C
$4$
D
$8$

Check your answer

Answer:

Before you read

Answer first — the lesson below explains it.

A company's revenue (in thousands) was $$7$ k in month $- 3$ and dropped to $- $1$ k by month $1$ . What is the monthly rate of change (slope through $(- 3, 7)$ and $(1, - 1)$ )?

A
$- 2$
B
$2$
C
$- \frac{1}{2}$
D
$\frac{3}{2}$

Check your answer

Answer:

Theory

What Is Slope?

Slope measures how steep a line is — it tells you how much $y$ changes for every 1 unit increase in $x$ .

$\text{slope} = m = \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

Types of slope:
- Positive slope ( $m > 0$ ): line goes up from left to right ↗
- Negative slope ( $m < 0$ ): line goes down from left to right ↘
- Zero slope ( $m = 0$ ): horizontal line $\to$
- Undefined slope: vertical line ↑ (division by zero)

Line with slope 2 and y-intercept -1

Line with negative slope

Example 1

Find the slope of the line through $(1, 3)$ and $(4, 9)$ .

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{9 - 3}{4 - 1} = \frac{6}{3} = 2$

The slope is $2$ : for every 1 unit right, the line goes up 2 units.

m = 2

Example 2

Find the slope of the line through $(-2, 5)$ and $(3, -5)$ .

$m = \frac{-5 - 5}{3 - (-2)} = \frac{-10}{5} = -2$

Negative slope: the line goes downward from left to right.

m = -2

Quick check

You just read it. Can you apply it?

The temperature was measured at $3°$ F at hour $4$ and at $9°$ F at hour $4$ (same time, different sensors). What is the slope of the line through $(4, 3)$ and $(4, 9)$ ?

A
$0$
B
$6$
C
$1$
D
Undefined

Check your answer

Answer:

$m = \frac{9 - 3}{4 - 4} = \frac{6}{0}$
Division by zero $\to$ the slope is undefined.
This is a vertical line ( $x = 4$ ).

Theory

What Is the Y-Intercept?

The y-intercept is the point where the line crosses the y-axis. At this point, $x = 0$ .

We write it as the point $(0, b)$ or just the value $b$ .

To find the y-intercept from an equation: set $x = 0$ and solve for $y$ .
To find it from a graph: look where the line crosses the y-axis.

Example 1

Find the y-intercept of $y = 3x + 7$ .

Set $x = 0$ : $y = 3(0) + 7 = 7$

The y-intercept is $(0, 7)$ or simply $b = 7$ .

b = 7

Example 2

Find the y-intercept of $2x + 3y = 12$ .

Set $x = 0$ : $2(0) + 3y = 12$

$3y = 12$

$y = 4$

The y-intercept is $(0, 4)$ .

b = 4

Quick check

You just read it. Can you apply it?

A business analyst converts the revenue equation $4 x + 2 y = 10$ to find the y-intercept. What is the y-intercept of this line?

A
$10$
B
$5$
C
$4$
D
$\frac{5}{2}$

Check your answer

Answer:

Set $x = 0$ : $4 (0) + 2 y = 10$ $\to$ $2 y = 10$ $\to$ $y = 5$ .
The y-intercept is $(0, 5)$ , so $b = 5$ .

Theory

Interpreting Slope and Y-Intercept in Context

On the SAT, slope and y-intercept often represent real-world quantities:

- Slope = rate of change (cost per item, speed, growth rate)
- Y-intercept = starting value (initial cost, initial amount, base fee)

Example: In $C = 15t + 50$ , where $C$ is total cost and $t$ is hours:
- Slope = $15$ $\to$ the cost increases by $\$15$ per hour
- Y-intercept = $50$ → there's a $\$50$ base fee (the cost when $t = 0$ )

Example 1

A pool is being filled. The depth (in inches) after $t$ minutes is modeled by $d = 0.5t + 4$ . What do the slope and y-intercept represent?

Slope = 0.5: The water depth increases by $0.5$ inches per minute (the fill rate).

Y-intercept = 4: The pool already had $4$ inches of water when filling began ( $t = 0$ ).

Slope = fill rate (0.5 in/min), y-intercept = initial water depth (4 in)

Quick check

You just read it. Can you apply it?

A population study records data points at $(0, - 3)$ and $(2, 1)$ . What are the slope and y-intercept of the line through these points?

A
Slope $= 2$ , y-intercept $= - 3$
B
Slope $= - 2$ , y-intercept $= - 3$
C
Slope $= 2$ , y-intercept $= 1$
D
Slope $= \frac{1}{2}$ , y-intercept $= - 3$

Check your answer

Answer:

Slope: $m = \frac{1 - ( - 3 )}{2 - 0} = \frac{4}{2} = 2$
Y-intercept: the line passes through $(0, - 3)$ , so $b = - 3$ .

Explore

Explore: How Slope Affects a Line

Drag the slider to change $m$ . Watch how the steepness and direction of the line change.

$y = mx + 1$

Quick check

You just read it. Can you apply it?

The equation $C = 0.12 m + 35$ models the monthly cost $C$ (in dollars) of a cell phone plan, where $m$ is the number of minutes used. What does the number $0.12$ represent?

A
The base monthly cost
B
The cost per minute of usage
C
The total cost for 1 minute
D
The number of free minutes

Check your answer

Answer:

In $C = 0.12 m + 35$ , the coefficient of $m$ is the slope.
Slope represents the rate of change: how much $C$ increases per 1-unit increase in $m$ .
So $0.12$ = the cost per minute ( $$0.12$ per minute).
The $35$ is the y-intercept = base monthly cost.

Common Mistakes

Common Mistakes to Avoid

Swapping the

x

and

y

coordinates in the slope formula

Using $\frac{x_2 - x_1}{y_2 - y_1}$ instead of $\frac{y_2 - y_1}{x_2 - x_1}$

Remember: slope = rise over run =

\frac{\Delta y}{\Delta x}

. The

y

's go on top.

Inconsistent point order

Using $\frac{y_2 - y_1}{x_1 - x_2}$ (mixing up which point is 1 and 2)

It doesn't matter which point you call

(x_1, y_1)

, but be consistent: if you subtract point 1 from point 2 on top, do the same on the bottom.

Quick check

You just read it. Can you apply it?

A scientist measures the temperature of a cooling liquid. The equation $T = - 2.5 t + 80$ models the temperature $T$ (in $^{\circ}$ F) after $t$ minutes. What does $- 2.5$ represent in this context?

A
The initial temperature
B
The temperature drops $2.5$ $^{\circ}$ F per minute
C
The final temperature
D
The time it takes to cool $2.5$ degrees

Check your answer

Answer:

The slope is $- 2.5$ . In context, this means the temperature decreases by $2.5$ $^{\circ}$ F for every additional minute. The negative sign indicates the temperature is going down (cooling).

Strategy

SAT Strategy Tip

Slope questions appear on every SAT. The fastest way to find slope from a graph: pick two points where the line clearly crosses grid intersections, then count the rise and run. For interpretation questions, always translate slope as 'the change in $y$ for each 1-unit increase in $x$ '.

After reading

Same question as before. What do you say now?

A
$\frac{1}{2}$
B
$2$
C
$4$
D
$8$

Check your answer

Answer:

$m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2$

After reading

Same question as before. What do you say now?

A company's revenue (in thousands) was $$7$ k in month $- 3$ and dropped to $- $1$ k by month $1$ . What is the monthly rate of change (slope through $(- 3, 7)$ and $(1, - 1)$ )?

A
$- 2$
B
$2$
C
$- \frac{1}{2}$
D
$\frac{3}{2}$

Check your answer

Answer:

$m = \frac{- 1 - 7}{1 - ( - 3 )} = \frac{- 8}{4} = - 2$

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