Recalling the General Form of a Quadratic Function

Oct 13, 2025 by talynt.org 140 views

Understanding the Problem

The problem presents a quadratic function with zeros at -8 and 4 and a maximum at (-2,18). We are asked to find the value of 'a' in the function's equation. To solve this problem, we need to use the information given about the zeros and the maximum point to determine the equation of the quadratic function.

Recalling the General Form of a Quadratic Function

A quadratic function can be written in the general form:

f(x) = a(x - r)(x - s)

where 'a' is the coefficient, and 'r' and 's' are the zeros of the function.

Using the Zeros to Write the Function

Given that the zeros are -8 and 4, we can write the function as:

f(x) = a(x + 8)(x - 4)

Finding the Value of a

To find the value of 'a', we can use the information about the maximum point. The maximum point is given as (-2,18), which means that when x = -2, f(x) = 18.

Substituting the Values into the Function

Substituting x = -2 and f(x) = 18 into the function, we get:

18 = a(-2 + 8)(-2 - 4)

Simplifying the Equation

Simplifying the equation, we get:

18 = a(6)(-6)

Solving for a

Solving for 'a', we get:

a = 18 / (-36)

a = -18 / 36

a = -1/2

However, we can also express this as a positive fraction by flipping the numerator and denominator:

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Understanding the Problem

Recalling the General Form of a Quadratic Function

A quadratic function can be written in the general form:

f(x) = a(x - r)(x - s)

where 'a' is the coefficient, and 'r' and 's' are the zeros of the function.

Using the Zeros to Write the Function

Given that the zeros are -8 and 4, we can write the function as:

f(x) = a(x + 8)(x - 4)

Finding the Value of a

To find the value of 'a', we can use the information about the maximum point. The maximum point is given as (-2,18), which means that when x = -2, f(x) = 18.

Substituting the Values into the Function

Substituting x = -2 and f(x) = 18 into the function, we get:

18 = a(-2 + 8)(-2 - 4)

Simplifying the Equation

Simplifying the equation, we get:

18 = a(6)(-6)

Solving for a

Solving for 'a', we get:

a = 18 / (-36)

a = -18 / 36

a = -1/2

However, we can also express this as a positive fraction by flipping the numerator and denominator: