Linear Functions Test- Chapter Assessment with Answer Key

What You Need to Know About Linear Functions Before the Test

Linear functions are the foundation of algebra. If you can't solve these problems cold, you're going to struggle with everything that comes after. This isn't a "maybe you should review" situation. It's a "you must know this cold" situation.

The Core Formula

Every linear function follows this pattern:

f(x) = mx + b

Where:

• m = slope (rate of change)
• b = y-intercept (where the line crosses the y-axis)
• x = the input variable

That's it. Memorize it. Know what each piece represents. The test will throw different problems at you, but they all trace back to this formula.

Slope: The Most Important Concept

Slope measures how steep a line is. You calculate it as:

slope = (y₂ - y₁) / (x₂ - x₁)

Remember: rise over run. How much does y change? Divide that by how much x changes.

Slope can be:

• Positive — line goes up as you move right
• Negative — line goes down as you move right
• Zero — horizontal line
• Undefined — vertical line (you can't divide by zero)

Linear Functions Practice Test

This test covers the skills you'll encounter. No fluff, no trick questions. Just the math.

Section 1: Identifying Linear Functions

Question 1: Which of the following is a linear function?

• A) f(x) = 3x² + 2
• B) f(x) = 5x - 7
• C) f(x) = √x + 4
• D) f(x) = 2/x

Question 2: What is the slope of the line passing through (2, 5) and (6, 13)?

Question 3: Identify the slope and y-intercept of f(x) = -4x + 9.

Section 2: Writing Linear Equations

Question 4: Write the equation of a line with slope 3 that passes through the point (1, 4).

Question 5: Find the equation of the line passing through (-2, 3) and (4, -1).

Question 6: Write the equation in slope-intercept form for a line with slope -2 that passes through (0, 5).

Section 3: Graphing Linear Functions

Question 7: Sketch the graph of f(x) = 2x - 3. Identify the y-intercept and one other point.

Question 8: Two lines are parallel. One has equation y = 5x + 2. What is the slope of the second line?

Question 9: Are the lines y = 3x + 1 and y = -1/3x + 4 perpendicular? Show your reasoning.

Section 4: Applications

Question 10: A taxi charges $3.50 base fare plus $2.00 per mile. Write a linear function that models the cost, then calculate the cost for a 7-mile trip.

Question 11: A company has fixed costs of $500 and each unit costs $12 to produce. The revenue is $20 per unit. Find the break-even point.

Section 5: Solving Systems

Question 12: Solve the system by graphing:

• y = 2x + 1
• y = -x + 4

Question 13: Solve using substitution:

• 3x + y = 10
• x - 2y = -3

Answer Key

Check your work. If you got these wrong, go back and figure out why.

Section 1 Answers

Q1: B) f(x) = 5x - 7

This is the only option with x raised to the first power. Quadratics (x²), radicals (√x), and rational expressions (2/x) are all non-linear.

Q2: slope = 2

Calculate: (13 - 5) / (6 - 2) = 8/4 = 2

Q3: slope = -4, y-intercept = 9

Already in slope-intercept form. The coefficient of x is the slope. The constant term is the y-intercept.

Section 2 Answers

Q4: f(x) = 3x + 1

Use point-slope form: y - 4 = 3(x - 1). Simplify to get y = 3x + 1.

Q5: f(x) = (-2/3)x + 5/3

Slope = (-1 - 3) / (4 - (-2)) = -4/6 = -2/3. Then use point-slope with either point.

Q6: f(x) = -2x + 5

When x = 0, you're at the y-intercept. Slope -2 plus y-intercept 5 gives f(x) = -2x + 5.

Section 3 Answers

Q7: y-intercept at (0, -3)

Plug in x = 1: f(1) = 2(1) - 3 = -1. So another point is (1, -1). Plot both points and draw a line through them.

Q8: slope = 5

Parallel lines have identical slopes. The first line has slope 5, so the second line also has slope 5.

Q9: Yes, they are perpendicular.

Slopes: 3 and -1/3. Multiply them: 3 × (-1/3) = -1. Perpendicular lines have slopes that multiply to -1.

Section 4 Answers

Q10: f(x) = 2x + 3.50, Cost = $17.50

For 7 miles: f(7) = 2(7) + 3.50 = 14 + 3.50 = $17.50

Q11: 62.5 units

Cost function: C = 12x + 500. Revenue function: R = 20x. Set equal: 20x = 12x + 500. Solve: 8x = 500, x = 62.5. They must sell 63 units to break even (selling 62 units means a loss).

Section 5 Answers

Q12: (1, 3)

Graph both lines. They intersect at x = 1, y = 3.

Q13: x = 2.6, y = 2.2 (or x = 13/5, y = 11/5)

From first equation: y = 10 - 3x. Substitute into second: x - 2(10 - 3x) = -3. Solve: x - 20 + 6x = -3. 7x = 17. x = 17/7 ≈ 2.43. Then y = 10 - 3(17/7) = 31/7 ≈ 4.43.

Quick Reference Table

Common Mistakes to Avoid

• Getting slope sign wrong — watch your negatives. A common error is forgetting that (5 - 8) gives -3, not 3.
• Confusing x and y — slope always compares y-changes to x-changes, in that order.
• Forgetting to distribute negatives — when you see y - 5 = 2(x - 3), multiply the whole parenthesis by 2, including the sign.
• Saying "undefined" when you mean "zero"** — vertical lines have undefined slope. Horizontal lines have zero slope.
• Not checking your answer** — plug your solution back into the original equation. If it doesn't work, you made an error.