Math Problem Statement

The range of the parabola shown in the graph above is y ≥ -4. If the equation y = ax^2 + bx + c is used to represent the graph, what is the value of a?

Solution

The graph represents a parabola, and the equation given is y=ax2+bx+cy = ax^2 + bx + c, with the range of the parabola being y4y \geq -4. The vertex is at the point (6,4)(6, -4), and one of the points on the parabola is (12,8)(12, 8).

We are tasked with finding the value of aa.

Step-by-step approach:

  1. Vertex form of the parabola: Since the vertex is given at (6,4)(6, -4), we can start by writing the equation in vertex form: y=a(xh)2+ky = a(x - h)^2 + k where (h,k)=(6,4)(h, k) = (6, -4). This gives us: y=a(x6)24y = a(x - 6)^2 - 4

  2. Using the point (12,8)(12, 8): Now, substitute x=12x = 12 and y=8y = 8 into the equation to find aa: 8=a(126)248 = a(12 - 6)^2 - 4 Simplify the equation: 8=a(6)248 = a(6)^2 - 4 8=36a48 = 36a - 4 Solve for aa: 8+4=36a8 + 4 = 36a 12=36a12 = 36a a=1236=13a = \frac{12}{36} = \frac{1}{3}

Thus, the value of aa is 13\frac{1}{3}.

Answer: The correct option is A) 13\frac{1}{3}.

Would you like a more detailed explanation or help with another question?

Here are some related questions:

  1. How do you find the vertex of a parabola from its equation in standard form?
  2. What is the significance of the value of aa in a parabola's equation?
  3. How do you convert from vertex form to standard form for a parabola equation?
  4. What does the range of a parabola tell you about its graph?
  5. How do you determine the direction a parabola opens based on its equation?

Tip: Always double-check the vertex and points on the graph to ensure accuracy in solving for the constants in the equation.

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Quadratic Equations
Vertex Form of a Parabola
Coordinate Geometry

Formulas

Vertex form of a parabola: y = a(x - h)^2 + k

Theorems

The vertex form of a parabola represents its minimum or maximum point.

Suitable Grade Level

Grades 9-11