Answer:
The vertex of the function is the point [tex](-3,-13)[/tex]
The graph increase over the interval--------> (-3,∞)
Step-by-step explanation:
we have
[tex]f(x)=6x-4+x^{2}[/tex]
1) Convert to vertex form
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]f(x)+4=x^{2}+6x[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]f(x)+4+9=x^{2}+6x+9[/tex]
[tex]f(x)+13=x^{2}+6x+9[/tex]
Rewrite as perfect squares
[tex]f(x)+13=(x+3)^{2}[/tex]
[tex]f(x)=(x+3)^{2}-13[/tex] -----> function in vertex form
2) Find the vertex
The vertex of the function is the point [tex](-3,-13)[/tex]
3) Find the axis of symmetry
we know that
In a vertical parabola, the axis of symmetry is equal to the x-coordinate of the vertex
The x-coordinate of the vertex in this problem is equal to [tex]x=-3[/tex]
therefore
the equation of the axis of symmetry is [tex]x=-3[/tex]
4) Find the increase-decrease intervals
The graph increase over the interval--------> (-3,∞)
The graph decrease over the interval--------> (-∞,-3)
see the attached figure to better understand the problem
5) Find the x-intercepts of the function
we know that
the x-intercepts are the values of x when the value of the function is equal to zero
In this problem the x-intercepts are
[tex](-6.61,0)[/tex] and [tex](0.61,0)[/tex]
so
The function cross the x-axis twice
see the attached figure
