Linear functions and graphs — Digital SAT Math practice
Slope, intercepts, parallel and perpendicular lines, function notation.
Key terms
- Slope: The constant rate of change in y for every one unit increase in x.
- Y-intercept: The point where a graph crosses the vertical axis, occurring when x equals zero.
- X-intercept: The point where a graph crosses the horizontal axis, found by setting y to zero.
- Parallel Lines: Two distinct lines that never intersect and share identical slopes.
- Function Notation: A compact way to write equations using f(x), where x is input and f(x) is output.
Common traps
- Confusing slope formula order by subtracting coordinates in different directions for x and y.
- Assuming parallel lines have equal y-intercepts instead of only matching slopes.
- Forgetting that perpendicular slopes multiply to negative one rather than just being opposite signs.
Key ideas to know
Calculating slope from two points
Find the slope of the line passing through (2, 5) and (6, 13). Show your work using the slope formula.
Use m = (y2 - y1)/(x2 - x1). Plug in: (13 - 5)/(6 - 2) = 8/4 = 2. The slope is 2.
Watch out: Swapping x and y values or subtracting in the wrong order will flip the sign or give a wrong fraction. Always keep numerator as change in y and denominator as change in x.
Parallel vs perpendicular
Line L1 has equation y = 3x + 2. Which line is parallel to L1 and which is perpendicular? A) y = −3x + 5 B) y = 3x − 4 C) y = −(1/3)x + 1
B is parallel because it has the same slope, 3. C is perpendicular because −1/3 is the negative reciprocal of 3.
Watch out: Choosing −3 for the perpendicular slope. Flip the fraction and change the sign: 3 becomes −1/3.
Evaluating and solving linear functions
Given f(x) = 4x - 7, find f(3) and solve for x when f(x) = 9.
To find f(3), substitute 3 for x: f(3) = 4(3) - 7 = 5. To solve f(x) = 9, set 4x - 7 = 9, add 7 to both sides to get 4x = 16, so x = 4.
Watch out: Treating f(3) as multiplication instead of substitution, or mixing up input and output when solving f(x) = k. Always replace the variable inside the parentheses with the given number.
Converting standard form correctly
A student converts 2x + 5y = 10 to slope-intercept form and writes y = (2/5)x + 2. What is wrong, and what is the correct equation?
The student flipped the sign of the slope. To isolate y, subtract 2x from both sides to get 5y = -2x + 10, then divide by 5. The correct form is y = (-2/5)x + 2.
Watch out: Dropping the negative sign when moving the x-term across the equals sign or dividing by a negative coefficient for y. Always track signs carefully during division.
Finding x and y intercepts
For the line y = -2x + 8, find both the x-intercept and the y-intercept as coordinate points.
The y-intercept occurs when x = 0: y = -2(0) + 8 = 8, so (0, 8). The x-intercept occurs when y = 0: 0 = -2x + 8 -> 2x = 8 -> x = 4, so (4, 0).
Watch out: Confusing which variable to set to zero. Always set x = 0 to find the y-intercept, and set y = 0 to find the x-intercept.
Sample questions
Question 1
If f(x) = 3x^2 - 1, what is f(2)?
- 5
- 35
- 11
- 17
Show answer and explanation
Answer: 11
f(2) = 3(4) - 1 = 11.
Question 2
If f(x) = 2x - 3 and g(x) = x^2, what is the value of f(g(2))?
- 1
- 11
- 8
- 5
Show answer and explanation
Answer: 5
g(2) = 4, then f(4) = 2(4) - 3 = 5.
Question 3
Line ℓ has equation y = 1/3x + 2. Line n is perpendicular to ℓ and passes through (−6, 3). What is the y-coordinate of the y-intercept of line n?
- −15
- 15
- −14
- 3
Show answer and explanation
Answer: −15
The slope of ℓ is 1/3, so the perpendicular slope is −3. Through (−6, 3): b = 3 − (−3)(−6) = −15.