Let's solve the problem step-by-step.
(a) Express [tex]\( f \)[/tex] in vertex form.
Given the quadratic function:
[tex]\[ f(x) = -x^2 + 16x \][/tex]
To express this in vertex form, we need to rewrite it in the form:
[tex]\[ f(x) = a(x-h)^2 + k \][/tex]
Completing the square:
1. Factor out the coefficient of [tex]\( x^2 \)[/tex] from the quadratic and linear term:
[tex]\[ f(x) = -\left(x^2 - 16x\right) \][/tex]
2. Take half of the coefficient of [tex]\( x \)[/tex], which is 16, divide it by 2 to get 8, and then square it to get [tex]\( 64 \)[/tex]:
[tex]\[ \left(\frac{16}{2}\right)^2 = 8^2 = 64 \][/tex]
3. Add and subtract this square inside the parentheses:
[tex]\[ f(x) = -\left(x^2 - 16x + 64 - 64\right) \][/tex]
4. Write the quadratic expression as a square and combine the constants:
[tex]\[ f(x) = -\left((x - 8)^2 - 64\right) \][/tex]
5. Distribute the negative sign:
[tex]\[ f(x) = -(x - 8)^2 + 64 \][/tex]
So the vertex form of the quadratic function is:
[tex]\[ f(x) = -(x - 8)^2 + 64 \][/tex]
(b) Find the vertex, [tex]\( x \)[/tex]-intercepts, and [tex]\( y \)[/tex]-intercept of [tex]\( f \)[/tex].
(i) Vertex:
The vertex form of the function is:
[tex]\[ f(x) = -(x - 8)^2 + 64 \][/tex]
From this, we can see that the vertex [tex]\((h, k)\)[/tex] is:
[tex]\[ (x, y) = (8, 64) \][/tex]
(ii) [tex]\( x \)[/tex]-intercepts:
The [tex]\( x \)[/tex]-intercepts occur where [tex]\( f(x) = 0 \)[/tex]. Therefore, we solve for [tex]\( x \)[/tex] in the equation:
[tex]\[ -x^2 + 16x = 0 \][/tex]
Factor out [tex]\( x \)[/tex]:
[tex]\[ x(-x + 16) = 0 \][/tex]
Setting each factor to zero gives:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = 16 \][/tex]
So the [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = 16 \][/tex]
(iii) [tex]\( y \)[/tex]-intercept:
The [tex]\( y \)[/tex]-intercept occurs where [tex]\( x = 0 \)[/tex]. Substitute [tex]\( x = 0 \)[/tex] into the function:
[tex]\[ f(0) = -(0)^2 + 16(0) = 0 \][/tex]
So the [tex]\( y \)[/tex]-intercept is:
[tex]\[ y = 0 \][/tex]
In summary:
- The vertex form is:
[tex]\[ f(x) = -(x - 8)^2 + 64 \][/tex]
- The vertex is:
[tex]\[ (x, y) = (8, 64) \][/tex]
- The [tex]\( x \)[/tex]-intercepts are:
[tex]\[ x = 0 \][/tex]
[tex]\[ x = 16 \][/tex]
- The [tex]\( y \)[/tex]-intercept is:
[tex]\[ y = 0 \][/tex]
