Answer:
Part a) The function’s equation written in vertex form is
[tex]f(x)=2(x-2)^2-8[/tex]
Part b) The function’s equation written in factored form is equal to
[tex]f(x)=2x(x-4)[/tex]
Step-by-step explanation:
Part a) What is the function’s equation written in vertex form?
we know that
The equation of a vertical parabola written in vertex form is equal to
[tex]f(x)=a(x-h)^2+k[/tex]
where
a is a coefficient
(h,k) is the vertex
Looking at the graph
The vertex is the point (2,-8)
substitute
[tex]f(x)=a(x-2)^2-8[/tex]
Find the value of the coefficient a
take one point from the graph
(0,0)
substitute in the equation
[tex]0=a(0-2)^2-8\\0=4a-8\\4a=8\\a=2[/tex]
therefore
The function’s equation written in vertex form is
[tex]f(x)=2(x-2)^2-8[/tex]
Part b) What is the function’s equation written in factored form?
we know that
The equation of a vertical parabola written in factored form is equal to
[tex]f(x)=a(x-x_1)(x-x_2)[/tex]
where
a is a coefficient
x_1 and x_2 are the zeros or x-intercepts of the function
Remember that the x-intercept is the value of x when the value of the function is equal to zero
Looking at the graph
The zeros or x-intercepts of the function are
x=0 and x=4
so
[tex]f(x)=a(x-0)(x-4)[/tex]
[tex]f(x)=ax(x-4)[/tex]
Find the value of the coefficient a
take one point from the graph
(2,-8)
substitute
[tex]-8=a(2)(2-4)\\-8=-4a\\a=2[/tex]
therefore
The function’s equation written in factored form is equal to
[tex]f(x)=2x(x-4)[/tex]
