Answer :
To find the axis of symmetry for the quadratic function given in the factored form \( f(x) = -(x+9)(x-21) \), we need to use the following approach.
### Step-by-Step Solution:
1. Identify the roots of the function:
- The given function is in the form \( f(x) = -(x + 9)(x - 21) \).
- The roots of the quadratic function are found by setting each factor equal to zero.
[tex]\[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \][/tex]
[tex]\[ x - 21 = 0 \quad \Rightarrow \quad x = 21 \][/tex]
Therefore, the roots are \( x = -9 \) and \( x = 21 \).
2. Find the axis of symmetry:
- The axis of symmetry of a parabola in the factored form is the vertical line that passes through the midpoint of the roots.
- To find the midpoint of the roots \( x = -9 \) and \( x = 21 \), we use the formula for the midpoint of two points \( \frac{x_1 + x_2}{2} \).
[tex]\[ \text{Axis of symmetry} \quad x = \frac{-9 + 21}{2} \][/tex]
3. Calculate the midpoint:
- Add the roots \( -9 \) and \( 21 \):
[tex]\[ -9 + 21 = 12 \][/tex]
- Divide the sum by 2 to find the midpoint:
[tex]\[ \frac{12}{2} = 6 \][/tex]
Therefore, the axis of symmetry for the function \( f(x) = -(x+9)(x-21) \) is:
[tex]\[ x = 6 \][/tex]
Hence, the correct answer is:
[tex]\[ x = 6 \][/tex]
### Step-by-Step Solution:
1. Identify the roots of the function:
- The given function is in the form \( f(x) = -(x + 9)(x - 21) \).
- The roots of the quadratic function are found by setting each factor equal to zero.
[tex]\[ x + 9 = 0 \quad \Rightarrow \quad x = -9 \][/tex]
[tex]\[ x - 21 = 0 \quad \Rightarrow \quad x = 21 \][/tex]
Therefore, the roots are \( x = -9 \) and \( x = 21 \).
2. Find the axis of symmetry:
- The axis of symmetry of a parabola in the factored form is the vertical line that passes through the midpoint of the roots.
- To find the midpoint of the roots \( x = -9 \) and \( x = 21 \), we use the formula for the midpoint of two points \( \frac{x_1 + x_2}{2} \).
[tex]\[ \text{Axis of symmetry} \quad x = \frac{-9 + 21}{2} \][/tex]
3. Calculate the midpoint:
- Add the roots \( -9 \) and \( 21 \):
[tex]\[ -9 + 21 = 12 \][/tex]
- Divide the sum by 2 to find the midpoint:
[tex]\[ \frac{12}{2} = 6 \][/tex]
Therefore, the axis of symmetry for the function \( f(x) = -(x+9)(x-21) \) is:
[tex]\[ x = 6 \][/tex]
Hence, the correct answer is:
[tex]\[ x = 6 \][/tex]
