ANSWER
a)The x-intercepts are:
(-4,0) and (2,0)
b) The axis of symmetry is x=-1
c) The vertex of this function is (-1,-9)
d)Domain: { [tex]x|x \in \: R[/tex]}
e) Range : {[tex]y |y \geqslant - 9[/tex]}
EXPLANATION
The given function is:
[tex]f(x) = {x}^{2} + 2x - 8[/tex]
We complete the square to write this function in the form:
[tex]f(x)=a{(x - h)}^{2} + k[/tex]
We add and subtract the square of half the coefficient of x.
[tex]f(x) = {x}^{2} + 2x + {1}^{2} - {1}^{2} - 8[/tex]
[tex]f(x) = {(x + 1)}^{2} - 9[/tex]
The vertex of this function is (h,k) which is (-1,-9)
The equation of axis of symmetry is x=h
But h=-1, hence the axis of symmetry isx=-1
To find the x-intercepts, we put f(x)=0
[tex]{(x + 1)}^{2} - 9 = 0[/tex]
[tex]{(x + 1)}^{2} = 9[/tex]
[tex]x + 1= \pm \sqrt{9} [/tex]
[tex]x= - 1 \pm3[/tex]
x=-4, 2
The x-intercepts are:
(-4,0) and (2,0)
The given function is a polynomial function, the domain is all real numbers.
[tex]x|x \in \: R[/tex]
e) The function has a minimum value of y=-9.
Therefore the range is
[tex]y |y \geqslant - 9[/tex]
Using the intercepts and vertex we can now draw this graph easily.
The graph of this function is shown in the attachment.
