CMClearMathAcademy

Intercepts and Key Features

A free Precalculus lesson from the “Function Foundations and Behavior” unit, with a worked example and practice problems including step-by-step solutions.

Key features are the landmarks of a graph: intercepts, turning points, zeros, and boundary behavior. This lesson is part of Precalculus: Advanced Functions, so the emphasis is on interpreting behavior, choosing the right representation, and explaining the result clearly rather than memorizing isolated algebra moves.

What you'll learn

Why it matters: Function behavior is the language behind Calculus, physics graphs, economic models, and computer-generated curves.

Worked example

Problem. Find the y-intercept of y = 3x + 2.

  1. Worked Example: First identify exactly what the question is asking: Find the y-intercept of y = 3x + 2.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. The y-intercept is the output when x = 0.
  4. 3(0) + 2 = 2.
  5. So the y-intercept is 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

Practice problems

1. Find the y-intercept of y = 3x + 2.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the y-intercept of y = 3x + 2.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. The y-intercept is the output when x = 0.
  4. 3(0) + 2 = 2.
  5. So the y-intercept is 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

2. Find the x-intercept of y = 4x - 12.

Show solution
  1. Warm-up: First identify exactly what the question is asking: Find the x-intercept of y = 4x - 12.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. The x-intercept is where y = 0.
  4. 4x - 12 = 0 gives 4x = 12, so x = 3.
  5. So the x-intercept is x = 3.
  6. Check the result by substituting or estimating: the response should match 3 and make sense in the original problem.

Answer: 3

3. Find the positive x-intercept of y = x^2 - 16.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the positive x-intercept of y = x^2 - 16.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Set y = 0.
  4. x^2 - 16 = 0 gives x^2 = 16.
  5. The positive root is x = 4.
  6. Check the result by substituting or estimating: the response should match 4 and make sense in the original problem.

Answer: 4

4. Find the y-intercept of y = 3x^2 + 5.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the y-intercept of y = 3x^2 + 5.
  2. For intercepts, remember that an x-intercept has y = 0 and a y-intercept has x = 0.
  3. Substitute x = 0.
  4. 3(0)^2 + 5 = 5.
  5. So the y-intercept is 5.
  6. Check the result by substituting or estimating: the response should match 5 and make sense in the original problem.

Answer: 5

5. The graph of y = (x - 2)^2 has its vertex at what x-value?

Show solution
  1. Core Practice: First identify exactly what the question is asking: The graph of y = (x - 2)^2 has its vertex at what x-value?
  2. For quadratics, track the zeros, vertex, or coefficients so the algebra matches the graph feature being asked about.
  3. A squared term is smallest when its inside is 0.
  4. x - 2 = 0 gives x = 2.
  5. So the vertex is at x = 2.
  6. Check the result by substituting or estimating: the response should match 2 and make sense in the original problem.

Answer: 2

Practice this interactively with instant feedback and an AI tutor.

Practice Intercepts and Key Features Take the free placement check

More Precalculus lessons