Question

Please help ASAP !

Answer 1

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

$f(x)=x^{2}-5x-6$

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

$x^{2}-5x-6=0$

The formula to solve a quadratic equation of the form $ax^{2} +bx+c=0$ is equal to

$x=\frac{-b(+/-)\sqrt{b^{2}-4ac}} {2a}$

in this problem we have

$x^{2}-5x-6=0$

so

$a=1\\b=-5\\c=-6$

substitute in the formula

$x=\frac{-(-5)(+/-)\sqrt{-5^{2}-4(1)(-6)}} {2(1)}$

$x=\frac{5(+/-)\sqrt{49}} {2}$

$x=\frac{5(+/-)7} {2}$

$x=\frac{5(+)7} {2}=6$

$x=\frac{5(-)7} {2}=-1$

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

$f(0)=(0)^{2}-5(0)-6$

$f(0)=-6$

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

$f(x)=a(x-h)^{2}+k$

where

(h,k) is the vertex

Convert the function into vertex form

$f(x)=x^{2}-5x-6$

Group terms that contain the same variable, and move the constant to the opposite side of the equation

$f(x)+6=(x^{2}-5x)$

Complete the square, Remember to balance the equation by adding the same constants to each side

$f(x)+6+2.5^{2}=(x^{2}-5x+2.5^{2})$

$f(x)+12.25=(x^{2}-5x+6.25)$

Rewrite as perfect squares

$f(x)+12.25=(x-2.5)^{2}$

$f(x)=(x-2.5)^{2}-12.25$

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