Answer:
Part 1) The smallest x-intercept is x=-1
Part 2) The largest x-intercept is x=6
Part 3) The y-intercept is y=-6
Part 4) The vertex is the point (2.5,-12.25)
Part 5) The equation of the line of symmetry is x=2.5
Step-by-step explanation:
we have
[tex]f(x)=x^{2}-5x-6[/tex]
step 1
Find the x-intercepts
we know that
The x-intercept is the value of x when the value of the function is equal to zero
so
equate the function to zero
[tex]x^{2}-5x-6=0[/tex]
The formula to solve a quadratic equation of the form [tex]ax^{2} +bx+c=0[/tex] is equal to
[tex]x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}[/tex]
in this problem we have
[tex]x^{2}-5x-6=0[/tex]
so
[tex]a=1\\b=-5\\c=-6[/tex]
substitute in the formula
[tex]x=\frac{-(-5)(+/-)\sqrt{-5^{2}-4(1)(-6)}} {2(1)}[/tex]
[tex]x=\frac{5(+/-)\sqrt{49}} {2}[/tex]
[tex]x=\frac{5(+/-)7} {2}[/tex]
[tex]x=\frac{5(+)7} {2}=6[/tex]
[tex]x=\frac{5(-)7} {2}=-1[/tex]
therefore
The x-intercepts are
x=-1 and x=6
The smallest x-intercept is x=-1
The largest x-intercept is x=6
step 2
Find the y-intercept
we know that
The y-intercept is the value of y when the value of x is equal to zero
so
For x=0
[tex]f(0)=(0)^{2}-5(0)-6[/tex]
[tex]f(0)=-6[/tex]
therefore
The y-intercept is y=-6
step 3
Find the vertex
we know that
The equation of a vertical parabola into vertex form is equal to
[tex]f(x)=a(x-h)^{2}+k[/tex]
where
(h,k) is the vertex
Convert the function into vertex form
[tex]f(x)=x^{2}-5x-6[/tex]
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]f(x)+6=(x^{2}-5x)[/tex]
Complete the square, Remember to balance the equation by adding the same constants to each side
[tex]f(x)+6+2.5^{2}=(x^{2}-5x+2.5^{2})[/tex]
[tex]f(x)+12.25=(x^{2}-5x+6.25)[/tex]
Rewrite as perfect squares
[tex]f(x)+12.25=(x-2.5)^{2}[/tex]
[tex]f(x)=(x-2.5)^{2}-12.25[/tex]
The vertex is the point (2.5,-12.25)
step 4
Find the equation of the line of symmetry
we know that
In a vertical parabola the equation of the line of symmetry is equal to the x-coordinate of the vertex
we have
vertex (2.5,-12.25)
The x-coordinate of the vertex is 2.5
therefore
The equation of the line of symmetry is x=2.5
