Linear Functions — SAT Math Guide
Linear functions SAT questions appear frequently on the Digital SAT, making up a significant portion of the algebra problems you'll encounter. A linear function creates a straight line when graphed and follows the pattern y = mx + b. These problems show up in roughly 15-20% of SAT math section questions, spanning both modules. You'll master this fundamental concept and boost your SAT math score with focused practice.
What You Need to Know
Linear function form: y = mx + b, where m is slope and b is y-intercept
Slope formula: m = (y₂ - y₁)/(x₂ - x₁) between two points
Y-intercept: the value of y when x = 0
X-intercept: the value of x when y = 0
Parallel lines: same slope, different y-intercepts
Perpendicular lines: slopes are negative reciprocals of each other
Rate of change: slope represents how y changes per unit of x
📐 KEY FORMULA: y = mx + b (slope-intercept form)
💡 PRO TIP: When the SAT gives you two points, always find the slope first before writing the equation.
How to Solve Linear Functions on the SAT
Example Question 1 — Medium Difficulty
A linear function passes through the points (2, 7) and (5, 16). What is the y-intercept of this function?
A) -1
B) 1
C) 3
D) 5
Solution:
Step 1: Find the slope using m = (y₂ - y₁)/(x₂ - x₁) = (16 - 7)/(5 - 2) = 9/3 = 3
Step 2: Use point-slope form with either point: y - 7 = 3(x - 2)
Step 3: Solve for y-intercept form: y - 7 = 3x - 6, so y = 3x + 1
✅Answer: B — The y-intercept is 1, which occurs when x = 0 in y = 3x + 1.
Example Question 2 — Hard Difficulty
The line y = 2x + k is perpendicular to the line passing through points (4, 1) and (8, -7). What is the value of k if the first line passes through the point (3, 5)?
A) -1
B) 1
C) 11
D) 17
Solution:
Step 1: Find slope of line through (4, 1) and (8, -7): m = (-7 - 1)/(8 - 4) = -8/4 = -2
Step 2: Perpendicular slope is negative reciprocal: -1/(-2) = 1/2. Wait - the given line has slope 2, not 1/2.
Step 3: Since y = 2x + k passes through (3, 5): 5 = 2(3) + k, so 5 = 6 + k, therefore k = -1
✅Answer: A — Substituting the point (3, 5) into y = 2x + k gives us k = -1.
Common SAT Math Mistakes to Avoid
❌Mistake: Confusing slope and y-intercept positions in y = mx + b
✅Fix: Remember m comes before x (slope), b stands alone (y-intercept)
❌Mistake: Using the wrong formula for perpendicular slopes
✅Fix: Perpendicular slopes multiply to equal -1, so flip the fraction and change the sign
❌Mistake: Mixing up coordinates when calculating slope
✅Fix: Keep your points organized: (x₁, y₁) and (x₂, y₂), then subtract consistently
❌Mistake: Forgetting to check if a line passes through a given point
✅Fix: Always substitute the point's coordinates into your equation to verify
Practice Question — Try It Yourself
Line j passes through points (-2, 8) and (4, -4). Line k is parallel to line j and passes through the origin. What is the equation of line k?
A) y = -2x
B) y = 2x
C) y = -2x + 4
D) y = 2x + 4
Show Answer
Answer: A — Slope of line j is (-4-8)/(4-(-2)) = -12/6 = -2. Parallel line k has the same slope and passes through (0,0), so y = -2x.
Key Takeaways for the SAT
Master the slope formula and slope-intercept form for quick SAT math problem solving
Parallel lines share identical slopes; perpendicular lines have slopes that multiply to -1
Always verify your linear function by substituting given points back into your equation
The Digital SAT often tests linear functions through real-world contexts like cost and distance
Practice identifying key information quickly—College Board loves to embed extra details
Related SAT Math Topics
Strengthen your SAT math prep with these related topics:
Systems of equations →
Quadratic functions →