Answer:
Part 1) The vertex is the point (1,-9)
Part 2) The equation of the axis of symmetry is x=1
Part 3) The y-intercept is the point (0,-8)
Part 4) The x-intercepts are (-2,0) and (4,0)
Part 5) The graph in the attached figure
Step-by-step explanation:
we have
![h(x)=(x-1)^2-9](https://tex.z-dn.net/?f=h%28x%29%3D%28x-1%29%5E2-9)
This is a vertical parabola open upward
The vertex represent a minimum
Part 1) Find the vertex
The quadratic equation is written in vertex form
![y=a(x-h)^2+k](https://tex.z-dn.net/?f=y%3Da%28x-h%29%5E2%2Bk)
where
(h,k) is the vertex of the parabola
so
The vertex is the point (1,-9)
Part 2) Find the axis of symmetry
The equation of the axis of symmetry of a vertical parabola is equal the the x-coordinate of the vertex
so
The equation of the axis of symmetry is
![x=1](https://tex.z-dn.net/?f=x%3D1)
Part 3) Find the y-intercept
we know that
The y-intercept is the value of the function when the value of x is equal to zero
so
For x=0
![h(x)=(0-1)^2-9](https://tex.z-dn.net/?f=h%28x%29%3D%280-1%29%5E2-9)
![h(0)=-8](https://tex.z-dn.net/?f=h%280%29%3D-8)
therefore
The y-intercept is the point (0,-8)
Part 4) Find the x-intercepts
we know that
The x-intercepts are the values of x when the value of the function is equal to zero
so
For h(x)=0
![(x-1)^2-9=0](https://tex.z-dn.net/?f=%28x-1%29%5E2-9%3D0)
solve for x
![(x-1)^2=9](https://tex.z-dn.net/?f=%28x-1%29%5E2%3D9)
square root both sides
![x-1=\pm3](https://tex.z-dn.net/?f=x-1%3D%5Cpm3)
![x=1\pm3](https://tex.z-dn.net/?f=x%3D1%5Cpm3)
![x=1+3=4\\x=1-3=-2](https://tex.z-dn.net/?f=x%3D1%2B3%3D4%5C%5Cx%3D1-3%3D-2)
therefore
The x-intercepts are (-2,0) and (4,0)
Part 5) Plot the graph of the quadratic function
Plot the following points to graph the function
vertex (1,-9)
axis of symmetry x=1
y-intercept (0,-8)
x-intercepts (-2,0) and (4,0)
The graph in the attached figure