Answer :
To find the value of \( a \) in the equation of the quadratic function, follow these steps:
1. Identify the form of the quadratic function:
The zeros of the quadratic function are given as \( -8 \) and \( 4 \). We can start with the form:
[tex]\[ f(x) = a(x + 8)(x - 4) \][/tex]
where \( a \) is a constant that we need to determine.
2. Determine the vertex:
A vertex is a key feature of a parabola formed by a quadratic function. Given that the maximum point is \( (-2, 18) \), we know the vertex \( (-2, 18) \) lies on the graph of the quadratic function.
3. Substitute the vertex coordinates:
Substitute \( x = -2 \) and \( f(x) = 18 \) into the function to solve for \( a \).
[tex]\[ f(-2) = a((-2) + 8)((-2) - 4) = 18 \][/tex]
4. Simplify the equation:
Calculate the values inside the parentheses first:
[tex]\[ f(-2) = a(6)(-6) = 18 \][/tex]
Simplifying further,
[tex]\[ a \cdot 6 \cdot (-6) = 18 \][/tex]
[tex]\[ -36a = 18 \][/tex]
5. Solve for \( a \):
Isolate \( a \) by dividing both sides of the equation by \(-36\):
[tex]\[ a = \frac{18}{-36} = -\frac{1}{2} \][/tex]
Hence, the value of \( a \) in the quadratic function's equation is:
[tex]\[ \boxed{-\frac{1}{2}} \][/tex]
1. Identify the form of the quadratic function:
The zeros of the quadratic function are given as \( -8 \) and \( 4 \). We can start with the form:
[tex]\[ f(x) = a(x + 8)(x - 4) \][/tex]
where \( a \) is a constant that we need to determine.
2. Determine the vertex:
A vertex is a key feature of a parabola formed by a quadratic function. Given that the maximum point is \( (-2, 18) \), we know the vertex \( (-2, 18) \) lies on the graph of the quadratic function.
3. Substitute the vertex coordinates:
Substitute \( x = -2 \) and \( f(x) = 18 \) into the function to solve for \( a \).
[tex]\[ f(-2) = a((-2) + 8)((-2) - 4) = 18 \][/tex]
4. Simplify the equation:
Calculate the values inside the parentheses first:
[tex]\[ f(-2) = a(6)(-6) = 18 \][/tex]
Simplifying further,
[tex]\[ a \cdot 6 \cdot (-6) = 18 \][/tex]
[tex]\[ -36a = 18 \][/tex]
5. Solve for \( a \):
Isolate \( a \) by dividing both sides of the equation by \(-36\):
[tex]\[ a = \frac{18}{-36} = -\frac{1}{2} \][/tex]
Hence, the value of \( a \) in the quadratic function's equation is:
[tex]\[ \boxed{-\frac{1}{2}} \][/tex]
