Math Problem Statement

Graph the equation. Select integers for x from -3 to 3, inclusive. y = x^2 - 4

Solution

The given equation is y=x24y = x^2 - 4, and you are asked to graph it for integer values of xx from 3-3 to 33, inclusive.

Let's first compute the corresponding yy-values for each xx-value in that range:

  • For x=3x = -3: y=(3)24=94=5y = (-3)^2 - 4 = 9 - 4 = 5
  • For x=2x = -2: y=(2)24=44=0y = (-2)^2 - 4 = 4 - 4 = 0
  • For x=1x = -1: y=(1)24=14=3y = (-1)^2 - 4 = 1 - 4 = -3
  • For x=0x = 0: y=024=04=4y = 0^2 - 4 = 0 - 4 = -4
  • For x=1x = 1: y=124=14=3y = 1^2 - 4 = 1 - 4 = -3
  • For x=2x = 2: y=224=44=0y = 2^2 - 4 = 4 - 4 = 0
  • For x=3x = 3: y=324=94=5y = 3^2 - 4 = 9 - 4 = 5

Now that we have the points (3,5)(-3, 5), (2,0)(-2, 0), (1,3)(-1, -3), (0,4)(0, -4), (1,3)(1, -3), (2,0)(2, 0), and (3,5)(3, 5), we can plot these on a graph. The shape of the graph will be a parabola, as this is a quadratic equation.

Would you like me to provide the graph as well?

Expanding the concept:

  1. What is the general shape of a quadratic function like y=x24y = x^2 - 4?
  2. How would the graph change if the equation were y=x24y = -x^2 - 4?
  3. What is the vertex of the parabola for the equation y=x24y = x^2 - 4?
  4. How does the value of 4-4 in x24x^2 - 4 affect the graph's position?
  5. What happens to the graph if we change the equation to y=(x1)24y = (x - 1)^2 - 4?

Tip:

In a quadratic function y=ax2+bx+cy = ax^2 + bx + c, the coefficient aa controls the direction and width of the parabola, while cc controls the vertical shift.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Equations
Graphing

Formulas

Quadratic equation in the form y = ax^2 + bx + c

Theorems

Properties of quadratic functions
Graph of a parabola

Suitable Grade Level

Grades 8-10