Linear Functions
SAT Math Mastery

Master slope, intercepts, and equations of lines: the foundation of SAT algebra and one of the most tested topics on the exam.

SAT Math

What Is a Linear Function?

A linear function is any function whose graph is a straight line. It has a constant rate of change: for every equal increase in $ x $, the value of $ y $ increases (or decreases) by the same amount.

Slope: The Rate of Change

The slope $ m $ tells you how steep a line is and in which direction it tilts. It is the ratio of vertical change to horizontal change between any two points on the line.

Slope Formula

$ m = \dfrac{y_2 - y_1}{x_2 - x_1} $

Positive Slope

Line rises from left to right. Example: $ m = 2 $

Negative Slope

Line falls from left to right. Example: $ m = -3 $

The y-Intercept

The y-intercept is the point where the line crosses the y-axis. At this point, $ x = 0 $. In the equation $ y = mx + b $, the value $ b $ is the y-intercept.

Core Forms of Linear Equations

The SAT tests three main forms. You must be able to recognize, convert between, and extract information from each one.

Slope-Intercept

$ y = mx + b $
Slope $ m $, y-intercept $ b $ are immediately visible.

Point-Slope

$ y - y_1 = m(x - x_1) $
Uses a known point $ (x_1, y_1) $ and slope $ m $.

Standard Form

$ Ax + By = C $
Useful for intercepts: set $ x=0 $ or $ y=0 $.

Converting Between Forms

On the SAT, you will often need to rewrite one form into another. The key moves are distributing, isolating $ y $, and rearranging terms.

1

Standard to Slope-Intercept: Solve $ Ax + By = C $ for $ y $. Subtract $ Ax $ from both sides, then divide by $ B $.

2

Point-Slope to Slope-Intercept: Distribute $ m $ on the right side, then add $ y_1 $ to both sides.

3

Slope-Intercept to Standard: Move all variable terms to one side. Multiply through if needed to clear fractions.

Parallel vs Perpendicular Lines

The SAT frequently asks about the relationship between two lines. The answer always comes from comparing their slopes.

	Parallel Lines	Perpendicular Lines
Slopes	Equal: $ m_1 = m_2 $	Negative reciprocals: $ m_1 \cdot m_2 = -1 $
Visual	Lines never intersect	Lines cross at 90°
Example	$ y = 2x + 1 $ and $ y = 2x - 5 $	$ y = 2x + 1 $ and $ y = -\frac{1}{2}x + 3 $

How to Solve Linear Function Problems

1

Identify what is given. Do you have two points? A point and a slope? An equation? A graph?

2

Find the slope. Use $ m = \frac{y_2 - y_1}{x_2 - x_1} $ if you have two points, or read $ m $ directly from the equation.

3

Find the y-intercept. Plug a known point and the slope into $ y = mx + b $ and solve for $ b $.

4

Write the equation. Put $ m $ and $ b $ into the form the question asks for (slope-intercept, standard, or point-slope).

5

Check your answer. Plug in the original points to verify the equation holds.

Example 1: Two Points

Problem: Find the equation of the line through $ (1, 3) $ and $ (4, 9) $.

Example 2: Parallel Line Through a Point

Problem: A line is parallel to $ y = -3x + 7 $ and passes through $ (2, 1) $. Find its equation.

1

Identify the slope. Parallel lines share the same slope: $ m = -3 $.

2

Use point-slope form: $ y - 1 = -3(x - 2) $.

3

Simplify: $ y - 1 = -3x + 6 $, so $ y = -3x + 7 $.

Example 3: Perpendicular Line

Problem: Write the equation of a line perpendicular to $ y = \frac{2}{5}x - 4 $ that passes through $ (0, 3) $.

1

Flip and negate the slope: The given slope is $ \frac{2}{5} $. The perpendicular slope is $ -\frac{5}{2} $.

2

The point is the y-intercept: Since $ x = 0 $, we know $ b = 3 $.

3

Write the equation: $ y = -\frac{5}{2}x + 3 $.

Interpreting Slope in Context

Many SAT questions present a linear model in a real-world scenario. You must connect $ m $ and $ b $ to their practical meanings.

Slope: $ 50 $

The plumber charges $50 per hour. This is the rate of change: for each additional hour, the cost increases by $50.

y-intercept: $ 80 $

The plumber charges an $80 service fee regardless of hours worked. This is the initial value when $ h = 0 $.

Systems of Linear Equations

When two linear equations are graphed together, they can intersect at one point, no points, or infinitely many points. The SAT tests all three cases.

One Solution

Different slopes. Lines cross at exactly one point.

No Solution

Same slope, different intercepts. Lines are parallel and never meet.

Infinite Solutions

Same slope, same intercept. The equations describe the same line.

Common Mistakes to Avoid

✕ Flipping Rise and Run

Slope is $ \frac{\text{rise}}{\text{run}} = \frac{\Delta y}{\Delta x} $, not the other way around. Putting $ x $ on top gives the reciprocal of the slope.

✕ Sign Errors in Subtraction

When computing $ y_2 - y_1 $ and $ x_2 - x_1 $, keep the same order for both. Swapping order in only one changes the sign of $ m $.

✕ Confusing Parallel and Perpendicular

Parallel: same slope. Perpendicular: negative reciprocal. Students often forget to negate the reciprocal.

Summary of Key Concepts

Concept	Formula / Rule	Key Insight
Slope	$ m = \frac{y_2 - y_1}{x_2 - x_1} $	Rate of change
Slope-Intercept	$ y = mx + b $	Read $ m $ and $ b $ directly
Point-Slope	$ y - y_1 = m(x - x_1) $	Best when given a point + slope
Standard Form	$ Ax + By = C $	Good for finding intercepts
Parallel	$ m_1 = m_2 $	Same slope, different $ b $
Perpendicular	$ m_1 \cdot m_2 = -1 $	Flip fraction, change sign

Question 1 of 5

What is the slope of the line that passes through the points $ (2, 5) $ and $ (6, 13) $?

The slope formula gives $ m = \frac{13 - 5}{6 - 2} = \frac{8}{4} = 2 $. Be careful to subtract y-values in the numerator and x-values in the denominator.

Question 2 of 5

Which equation represents a line perpendicular to $ y = 3x - 2 $?

The perpendicular slope is the negative reciprocal of $ 3 $, which is $ -\frac{1}{3} $. Choice (a) is parallel, (b) is the reciprocal without negation, and (d) is just the negation without reciprocal.

Question 3 of 5

A gym membership costs a one-time fee of $50 plus $30 per month. If $ C $ represents the total cost after $ m $ months, what does the slope represent?

The equation is $ C = 30m + 50 $. The slope is $ 30 $, representing the rate of change: the cost per additional month. The y-intercept $ 50 $ is the one-time fee.

Question 4 of 5

The equation $ 4x - 2y = 10 $ is rewritten in slope-intercept form. What are the slope and y-intercept?

Solve for $ y $: subtract $ 4x $ from both sides to get $ -2y = -4x + 10 $, then divide by $ -2 $: $ y = 2x - 5 $. So $ m = 2 $ and $ b = -5 $. Watch the sign when dividing by a negative!

Question 5 of 5

The system of equations $ y = 5x + 3 $ and $ y = 5x - 2 $ has:

Both lines have slope $ m = 5 $ but different y-intercepts ($ 3 $ and $ -2 $). Same slope, different intercept means the lines are parallel and never intersect: no solution. Two distinct lines can never intersect at exactly two points.

	Parallel Lines	Perpendicular Lines
Slopes	Equal: \( m_1 = m_2 \)	Negative reciprocals: \( m_1 \cdot m_2 = -1 \)
Visual	Lines never intersect	Lines cross at 90°
Example	\( y = 2x + 1 \) and \( y = 2x - 5 \)	\( y = 2x + 1 \) and \( y = -\frac{1}{2}x + 3 \)

Concept	Formula / Rule	Key Insight
Slope	\( m = \frac{y_2 - y_1}{x_2 - x_1} \)	Rate of change
Slope-Intercept	\( y = mx + b \)	Read \( m \) and \( b \) directly
Point-Slope	\( y - y_1 = m(x - x_1) \)	Best when given a point + slope
Standard Form	\( Ax + By = C \)	Good for finding intercepts
Parallel	\( m_1 = m_2 \)	Same slope, different \( b \)
Perpendicular	\( m_1 \cdot m_2 = -1 \)	Flip fraction, change sign

Linear FunctionsSAT Math Mastery

What Is a Linear Function?

Slope: The Rate of Change

The y-Intercept

Core Forms of Linear Equations

Converting Between Forms

Parallel vs Perpendicular Lines

How to Solve Linear Function Problems

Example 1: Two Points

Example 2: Parallel Line Through a Point

Example 3: Perpendicular Line

Interpreting Slope in Context

Systems of Linear Equations

Common Mistakes to Avoid

Summary of Key Concepts

Question 1 of 5

Question 2 of 5

Question 3 of 5

Question 4 of 5

Question 5 of 5

Linear Functions
SAT Math Mastery