Math Problem Statement

Find the intercepts and graph the equation by plotting points. yequalsx squared minus 4 Question content area bottom left Part 1 What are the​ intercepts? Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice. A. The​ intercept(s) is/are    enter your response here. ​(Use a comma to separate answers as needed. Type an ordered​ pair.)

Solution

To find the intercepts of the equation y=x24y = x^2 - 4, we need to determine where the graph crosses the x-axis and the y-axis.

X-intercepts:

X-intercepts occur where y=0y = 0. So, set y=0y = 0 and solve for xx:

0=x240 = x^2 - 4

This simplifies to:

x2=4x^2 = 4

Taking the square root of both sides:

x=±2x = \pm 2

Thus, the x-intercepts are at (2,0)(2, 0) and (2,0)(-2, 0).

Y-intercept:

The y-intercept occurs where x=0x = 0. Plug x=0x = 0 into the equation:

y=024=4y = 0^2 - 4 = -4

Thus, the y-intercept is at (0,4)(0, -4).

Intercepts:

  • X-intercepts: (2,0)(2, 0) and (2,0)(-2, 0)
  • Y-intercept: (0,4)(0, -4)

Would you like to see the graph, or do you have any questions about the solution?

Here are 5 questions to expand your understanding:

  1. How would the graph change if the equation was y=x2+4y = x^2 + 4?
  2. What would the intercepts be if the equation was y=x29y = x^2 - 9?
  3. What is the vertex of the parabola represented by y=x24y = x^2 - 4?
  4. How do the intercepts relate to the symmetry of the parabola?
  5. Can you find the axis of symmetry for the parabola?

Tip: When solving for x-intercepts, always remember to set y=0y = 0 and solve for xx.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Graphing

Formulas

Standard form of a quadratic equation: y = ax^2 + bx + c
Finding intercepts: Set y = 0 for x-intercepts and x = 0 for y-intercepts

Theorems

Properties of quadratic functions
Symmetry of parabolas

Suitable Grade Level

Grades 8-10