Answer:
Option 2,4 are true statement.
Step-by-step explanation:
Given : Function [tex]f(x)=6x-4+x^2[/tex]
To find : Which statements are true about the graph of the function?
Solution :
First we have to convert the quadratic function [tex]y=ax^2+bx+c[/tex] into vertex form [tex]y=a(x-h)^2+k[/tex] where (h,k) are the vertex.
Function [tex]f(x)=x^2+6x-4[/tex]
Applying completing the square i.e. add and subtract half square of b,
[tex]f(x)=x^2+6x+3^2-3^2-4[/tex]
[tex]f(x)=(x+3)^2-9-4[/tex]
[tex]f(x)=(x+3)^2-13[/tex]
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 [tex]x=-\frac{b}{2a}[/tex]
Substitute a=1 and b=6
[tex]x=-\frac{6}{2(1)}[/tex]
[tex]x=-3[/tex]
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.
