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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. A y-intercept happens when x = 0.
  4. Substitute x = 0 to get y = -2.
  5. The y-intercept value 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. A y-intercept happens when x = 0.
  4. Substitute x = 0 to get y = -2.
  5. The y-intercept value 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 y-intercept of y = 4x - 1.

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

Answer: -1

3. Find the positive x-intercept of y = x^2 - 25.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the positive x-intercept of y = x^2 - 25.
  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 - 25 = 0 gives x^2 = 25.
  5. The positive x-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

4. Find the positive x-intercept of y = x^2 - 4.

Show solution
  1. Core Practice: First identify exactly what the question is asking: Find the positive x-intercept of y = x^2 - 4.
  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 - 4 = 0 gives x^2 = 4.
  5. The positive x-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

5. A local maximum is a point where the graph:

Choices: changes from increasing to decreasing nearby · crosses the y-axis · has no domain · must be below the x-axis

Show solution
  1. A local maximum is a nearby high point.
  2. The graph rises into it and falls after it.
  3. It does not have to be the highest point forever.

Answer: changes from increasing to decreasing nearby

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