Answer:
Step-by-step explanation:
The x-intercepts are the points with y- coordinate of 0
a) x-intercepts are
b) midpoint of x-intercepts is
c) The extheme value has x- coordinate of - 2.5 and y- coordinate
d) f(- 2.5) = - 30.25
Answer:
[tex]\sf a) \quad x = -8 \:\:and \:\:x = 3[/tex]
[tex]\sf b) \quad \left(-\dfrac{5}{2},0\right)[/tex]
[tex]\sf c) \quad \left(-\dfrac{5}{2},-\dfrac{121}{4}\right)[/tex]
[tex]\sf d) \quad f(x)=\left(x+\dfrac{5}{2}\right)^2-\dfrac{121}{4}[/tex]
Step-by-step explanation:
Part (a)
The x-intercepts are the x-values when the curve crosses the x-axis, so when the function is equal to zero:
[tex]\implies f(x)=0[/tex]
[tex]\implies (x-3)(x+8)=0[/tex]
Therefore:
[tex](x-3)=0 \implies x=3[/tex]
[tex](x+8)=0 \implies x=-8[/tex]
Therefore, the x-intercepts are x = -8 and x = 3
Part (b)
[tex]\textsf{Midpoint}=\left(\dfrac{x_2+x_1}{2},\dfrac{y_2+y_1}{2}\right)\quad \textsf{where}\:(x_1,y_1)\:\textsf{and}\:(x_2,y_2)\:\textsf{are the endpoints}}\right)[/tex]
[tex]\sf \implies Midpoint=\left(\dfrac{-8+3}{2},\dfrac{0+0}{2}\right)=\left(-\dfrac{5}{2},0\right)[/tex]
Therefore, the midpoint of the x-intercepts is (-5/2, 0)
Part (c)
The extreme value is the vertex. The vertex is:
The x-coordinate of the vertex is the x-value of the midpoint of the x-intercepts. Therefore, the x-value of the extreme value is x = -5/2.
To find the y-coordinate, substitute this into the given equation:
[tex]\implies \sf y=\left(-\dfrac{5}{2}-3\right)\left(-\dfrac{5}{2}+8\right)=-\dfrac{121}{4}[/tex]
Therefore, the extreme value is:
[tex]\sf \left(-\dfrac{5}{2},-\dfrac{121}{4}\right)[/tex]
Part (d)
The function in vertex form is:
[tex]\sf f(x)=\left(x+\dfrac{5}{2}\right)^2-\dfrac{121}{4}[/tex]
[tex]\begin{aligned}\implies \sf f\left(-\dfrac{5}{2}\right)& = \sf \left(-\dfrac{5}{2}+\dfrac{5}{2}\right)^2-\dfrac{121}{4}\\\\ & = \sf 0 -\dfrac{121}{4}\\\\ & = \sf -\dfrac{121}{4} \end{aligned}[/tex]
