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Answer :
Sure! Let's go through each part of the question step by step:
Given the quadratic function [tex]\( f(x) = -2x^2 + 8x + 24 \)[/tex].
### a. Rewrite [tex]\( f(x) \)[/tex] in factored form.
To factor the quadratic function, you can use the quadratic formula to find its roots:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the quadratic [tex]\( x^2 - 4x - 12 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = -12 \)[/tex]
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-12) = 16 + 48 = 64 \][/tex]
Find the square root of the discriminant:
[tex]\[ \sqrt{64} = 8 \][/tex]
Now, calculate the roots:
[tex]\[ x_1 = \frac{-(-4) + 8}{2 \times 1} = \frac{4 + 8}{2} = 6 \][/tex]
[tex]\[ x_2 = \frac{-(-4) - 8}{2 \times 1} = \frac{4 - 8}{2} = -2 \][/tex]
So, the factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = -2(x - 6)(x + 2) \][/tex]
### b. Find all zeros of [tex]\( f \)[/tex].
The zeros of the function, or the x-values where the function equals zero, are the roots we just calculated:
[tex]\[ x = 6 \][/tex] and [tex]\[ x = -2 \][/tex]
### c. Find the axis of symmetry.
The axis of symmetry of a quadratic function can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function:
[tex]\[ a = -2, b = 8 \][/tex]
So, the axis of symmetry is:
[tex]\[ x = -\frac{8}{2 \times -2} = \frac{8}{4} = 2 \][/tex]
### d. Find the coordinates of the vertex of the graph of [tex]\( f \)[/tex].
The vertex of a quadratic function can be found at the axis of symmetry. To find the y-coordinate of the vertex, substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(x) = -2x^2 + 8x + 24 \][/tex]
[tex]\[ f(2) = -2(2)^2 + 8 \times 2 + 24 \][/tex]
[tex]\[ = -2(4) + 16 + 24 \][/tex]
[tex]\[ = -8 + 16 + 24 = 32 \][/tex]
Thus, the vertex of the graph is at:
[tex]\[ (2, 32) \][/tex]
These steps provide a detailed breakdown of each part of the problem.
Given the quadratic function [tex]\( f(x) = -2x^2 + 8x + 24 \)[/tex].
### a. Rewrite [tex]\( f(x) \)[/tex] in factored form.
To factor the quadratic function, you can use the quadratic formula to find its roots:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the quadratic [tex]\( x^2 - 4x - 12 \)[/tex], we have:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -4 \)[/tex]
- [tex]\( c = -12 \)[/tex]
Calculate the discriminant:
[tex]\[ b^2 - 4ac = (-4)^2 - 4 \times 1 \times (-12) = 16 + 48 = 64 \][/tex]
Find the square root of the discriminant:
[tex]\[ \sqrt{64} = 8 \][/tex]
Now, calculate the roots:
[tex]\[ x_1 = \frac{-(-4) + 8}{2 \times 1} = \frac{4 + 8}{2} = 6 \][/tex]
[tex]\[ x_2 = \frac{-(-4) - 8}{2 \times 1} = \frac{4 - 8}{2} = -2 \][/tex]
So, the factored form of [tex]\( f(x) \)[/tex] is:
[tex]\[ f(x) = -2(x - 6)(x + 2) \][/tex]
### b. Find all zeros of [tex]\( f \)[/tex].
The zeros of the function, or the x-values where the function equals zero, are the roots we just calculated:
[tex]\[ x = 6 \][/tex] and [tex]\[ x = -2 \][/tex]
### c. Find the axis of symmetry.
The axis of symmetry of a quadratic function can be found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For our function:
[tex]\[ a = -2, b = 8 \][/tex]
So, the axis of symmetry is:
[tex]\[ x = -\frac{8}{2 \times -2} = \frac{8}{4} = 2 \][/tex]
### d. Find the coordinates of the vertex of the graph of [tex]\( f \)[/tex].
The vertex of a quadratic function can be found at the axis of symmetry. To find the y-coordinate of the vertex, substitute [tex]\( x = 2 \)[/tex] into the function:
[tex]\[ f(x) = -2x^2 + 8x + 24 \][/tex]
[tex]\[ f(2) = -2(2)^2 + 8 \times 2 + 24 \][/tex]
[tex]\[ = -2(4) + 16 + 24 \][/tex]
[tex]\[ = -8 + 16 + 24 = 32 \][/tex]
Thus, the vertex of the graph is at:
[tex]\[ (2, 32) \][/tex]
These steps provide a detailed breakdown of each part of the problem.
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