Shown below
Explanation:
We have the following function:
[tex]g(x)= -2(x-1)(x+5)[/tex]
We can get some key features of the graph:
1. y-intercept:
By setting [tex]x=0[/tex] we get the y-intercept:
[tex]y=-2(0-1)(0+5) \\ \\ y=-2(-1)(5) \\ \\ y=10[/tex]
2. x-intercepts:
This is a parabola that cuts the x-axis two times and we can find those values by setting [tex]y=0[/tex]:
[tex]-2(x-1)(x+5)=0 \\ \\ \\ So: \\ \\ x=1 \\ \\ x=-5[/tex]
So x = 1 and x = -5 are the roots of the polynomial function
3. Vertex
Applying distributive property:
[tex]g(x)= -2(x-1)(x+5) \\ \\ g(x)=-2(x^2+5x-x-5) \\ \\ g(x)=-2(x^2+4x-5) \\ \\ g(x)=-2x^2-8x+10[/tex]
So the vertex can be found as:
[tex]x=-\frac{b}{2a} \\ \\ \\ a=-2 \\ \\ b=-8 \\ \\ 10 \\ \\ \\ x=-\frac{-8}{2(-2)} \\ \\ x=-2 \\ \\ y=g(-2)=-2(-2)^2-8(-2)+10=-8+16+10=18 \\ \\ \\ So \ vertex: \\ \\ (h,k)=(-2,18)[/tex]
So the graph is shown below with these key features.
Learn more:
Parabola: https://brainly.com/question/10525169
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