Question

Which statements are true about the graph of the function f(x) = x2 – 8x + 5? Check all that apply.

a. The function in vertex form is f(x) = (x – 4)2 – 11.
b. The vertex of the function is (–8, 5).
c. The axis of symmetry is x = 5.
d. The y-intercept of the function is (0, 5).
e. The function crosses the x-axis twice.

Answer 1

d. The y-intercept of the function is (0, 5) and e. The function crosses the x-axis twice are true

Answer 2

Option a,d and e are true statements.

Explanation:

Given : Function
`f(x)=x^2-8x+5`

To find : Which statements are true about the graph of the function, Check all that apply?

Solution :

The quadratic function is in the form,
`y=ax^2+bx+c`

The vertex form is
`y=a(x-h)^2+k`

To find vertex form we apply completing the square,

`f(x)=x^2-8x+5+4^2-4^2`

`f(x)=x^2-8x+(4)^2+5-16`

`f(x)=(x-4)^2-11`

The vertex form of the function is
`f(x)=(x-4)^2-11`

The 'a' statement is true.

The vertex of the function is (h,k).

On comparing with vertex form, h=4 and k=11

Vertex of the function is (4,11)

The 'b' statement is not true.

The x-coordinate of the vertex i.e.
`x=-(b)/(2a)` is the axis of symmetry,

So,
`x=-(-8)/(2(1))`

`x=4`

The axis of symmetry is x=4.

The 'c' statement is not true.

The y-intercept of the function is at x=0

So, Put x=0 in the equation
`f(x)=x^2-8x+5`

`f(0)=0^2-8(0)+5`

`y=5`

The y-intercept of the function is (0,5).

The 'd' statement is true.

To find the function crosses the x-axis twice refer the attached figure below.

Yes the function crosses the x-axis twice.

The 'e' statement is true.