Linear functions are the bread and butter of SAT Math. They show up in pure algebra questions, word problems, and data analysis. If you master this topic, you're picking up easy points across multiple sections.
The Slope-Intercept Form
The most common form you'll see:
- = slope (rate of change)
- = y-intercept (starting value)
Example 1: In , the slope is 3 (y increases by 3 for every 1-unit increase in x) and the y-intercept is 5 (the line crosses the y-axis at ).
Calculating Slope
Given two points and :
Example 2: Find the slope of the line through and .
Interpreting Slope and Intercept in Context
The SAT frequently asks what the slope and intercept mean in a real-world scenario.
Example 3: A plumber charges according to , where is the total cost and is the number of hours worked.
- The slope (75) means the plumber charges $75 per hour
- The y-intercept (50) means there's a $50 service fee regardless of hours worked
Finding the Equation of a Line
Given a point and the slope, use point-slope form:
Example 4: Write the equation of a line with slope passing through .
Parallel and Perpendicular Lines
- Parallel lines have the same slope:
- Perpendicular lines have negative reciprocal slopes:
Example 5: A line has slope . A perpendicular line has slope .
Word Problems with Linear Functions
Example 6: A lake has 200 fish. Each month, 15 new fish are added. Write a function for the fish population after months.
How many fish after 8 months? fish.
Practice Problems
Problem 1: Find the equation of the line through and .
Solution
Slope:
Using point-slope:
Problem 2: A taxi charges 1.75 per mile. What is the cost for a 12-mile ride?
Solution
23.50$
Problem 3: Line has equation . Line is perpendicular to and passes through . What is the equation of Line ?
Solution
Perpendicular slope:
Key Takeaways
- Know cold — slope is the rate, intercept is the starting value
- For context questions, translate slope and intercept into plain English
- Parallel = same slope; perpendicular = negative reciprocal slopes
- Point-slope form is your friend when given a point and slope
- Most SAT linear function questions are testing interpretation, not just calculation
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