Answer:
a. [tex]f(x)=x^2-5x-24[/tex]
b. (2.5,-30.25) is the vertex
c. The x-intercepts are (-3,0) and (8,0)
d. See graph
Step-by-step explanation:
a. The given function is;
[tex]f(x)=x^2-5x-24[/tex]
We complete the square to obtain;
[tex]f(x)=x^2-5x+(-\frac{5}{2})^2 -(-\frac{5}{2})^2 -24[/tex]
Observe that the first three term is a perfect square trinomial.
[tex]f(x)=(x-\frac{5}{2})^2 -30.25[/tex]
This is the same as
[tex]f(x)=(x-2.5)^2 -30.25[/tex]
b.
The function is now of the form;
[tex]f(x)=a(x-h)^2+k[/tex]
Where (h,k)=(2.5,-30.25) is the vertex.
c.
At x-intercept, y=0
This implies that;
[tex](x-2.5)^2 -30.25=0[/tex]
[tex](x-2.5)^2 =30.25[/tex]
[tex]x-2.5=\pm \sqrt{30.25}[/tex]
[tex]x=2.5\pm 5.5[/tex]
[tex]x=2.5\pm 5.5[/tex]
[tex]x=-3[/tex] or [tex]x=8[/tex]
The x-intercepts are (-3,0) and (8,0)
At y-intercept x=0,
This implies that;
[tex]f(0)=0^2-5(0)-24=-24[/tex]
The y-intercept is (0,-24)
d) We now plot the vertex (2.5,-30.25).
The a=1, this means the graph opens upwards.
We also plot the intercepts and graph the function to obtain the graph shown in the attachment.
