Question

Which statements are true about the graph of the function f(x) = 6x – 4 + x2?

The vertex form of the function is f(x) = (x – 2)2 + 2.
The vertex of the function is (–3, –13).
The axis of symmetry for the function is x = 3.
The graph increases over the interval (–3, ).
The function does not cross the x-axis.

Answer 1

The vertex of the function is (–3, –13).

Explanation:

The graph is shown below. From the graph you can see, the vertex is (-3,-13).

Answer 2

Option 2,4 are true statement.

Explanation:

Given : Function
`f(x)=6x-4+x^2`

To find : Which statements are true about the graph of the function?

Solution :

First we have to convert the quadratic function
`y=ax^2+bx+c` into vertex form
`y=a(x-h)^2+k` where (h,k) are the vertex.

Function
`f(x)=x^2+6x-4`

Applying completing the square i.e. add and subtract half square of b,

`f(x)=x^2+6x+3^2-3^2-4`

`f(x)=(x+3)^2-9-4`

`f(x)=(x+3)^2-13`

Option 1 is incorrect.

On comparing with vertex form,

h=-3 and k=-13

So, The vertex of the function is (h,k)=(-3,-13).

Option 2 is correct.

Axis of symmetry is
`x=-(b)/(2a)`

Substitute a=1 and b=6

`x=-(6)/(2(1))`

`x=-3`

Option 3 is incorrect.

Now, We plot the graph of the function.

Refer the attached figure below.

The graph increases over the interval (–3,-13).

Option 4 is correct.

From the graph we see that it crosses x-axis.

Option 5 is incorrect.

Therefore, Option 2,4 are true statement.