q(x)=-(x-5)2-6

Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.

a=-1

h=5

k=-6

Since the value of a is negative, the parabola opens down.

Opens Down

Find the vertex (h,k).

(5,-6)

Find p, the distance from the vertex to the focus.

Find the distance from the vertex to a focus of the parabola by using the following formula.

14a

Substitute the value of a into the formula.

14⋅-1

Cancel the common factor of 1 and -1.

Rewrite 1 as -1(-1).

-1(-1)4⋅-1

Move the negative in front of the fraction.

-14

-14

-14

Find the focus.

The focus of a parabola can be found by adding p to the y-coordinate k if the parabola opens up or down.

(h,k+p)

Substitute the known values of h, p, and k into the formula and simplify.

(5,-254)

(5,-254)

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

x=5

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting p from the y-coordinate k of the vertex if the parabola opens up or down.

y=k-p

Substitute the known values of p and k into the formula and simplify.

y=-234

y=-234

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Down

Vertex: (5,-6)

Focus: (5,-254)

Axis of Symmetry: x=5

Directrix: y=-234

Direction: Opens Down

Vertex: (5,-6)

Focus: (5,-254)

Axis of Symmetry: x=5

Directrix: y=-234

Replace the variable x with 4 in the expression.

f(4)=-(4)2+10(4)-31

Simplify the result.

Simplify each term.

Raise 4 to the power of 2.

f(4)=-1⋅16+10(4)-31

Multiply -1 by 16.

f(4)=-16+10(4)-31

Multiply 10 by 4.

f(4)=-16+40-31

f(4)=-16+40-31

Simplify by adding and subtracting.

Add -16 and 40.

f(4)=24-31

Subtract 31 from 24.

f(4)=-7

f(4)=-7

The final answer is -7.

-7

-7

The y value at x=4 is -7.

y=-7

Replace the variable x with 3 in the expression.

f(3)=-(3)2+10(3)-31

Simplify the result.

Simplify each term.

Raise 3 to the power of 2.

f(3)=-1⋅9+10(3)-31

Multiply -1 by 9.

f(3)=-9+10(3)-31

Multiply 10 by 3.

f(3)=-9+30-31

f(3)=-9+30-31

Simplify by adding and subtracting.

Add -9 and 30.

f(3)=21-31

Subtract 31 from 21.

f(3)=-10

f(3)=-10

The final answer is -10.

-10

-10

The y value at x=3 is -10.

y=-10

Replace the variable x with 6 in the expression.

f(6)=-(6)2+10(6)-31

Simplify the result.

Simplify each term.

Raise 6 to the power of 2.

f(6)=-1⋅36+10(6)-31

Multiply -1 by 36.

f(6)=-36+10(6)-31

Multiply 10 by 6.

f(6)=-36+60-31

f(6)=-36+60-31

Simplify by adding and subtracting.

Add -36 and 60.

f(6)=24-31

Subtract 31 from 24.

f(6)=-7

f(6)=-7

The final answer is -7.

-7

-7

The y value at x=6 is -7.

y=-7

Replace the variable x with 7 in the expression.

f(7)=-(7)2+10(7)-31

Simplify the result.

Simplify each term.

Raise 7 to the power of 2.

f(7)=-1⋅49+10(7)-31

Multiply -1 by 49.

f(7)=-49+10(7)-31

Multiply 10 by 7.

f(7)=-49+70-31

f(7)=-49+70-31

Simplify by adding and subtracting.

Add -49 and 70.

f(7)=21-31

Subtract 31 from 21.

f(7)=-10

f(7)=-10

The final answer is -10.

-10

-10

The y value at x=7 is -10.

y=-10

Graph the parabola using its properties and the selected points.

xy3-104-75-66-77-10

xy3-104-75-66-77-10

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (5,-6)

Focus: (5,-254)

Axis of Symmetry: x=5

Directrix: y=-234

xy3-104-75-66-77-10

Graph q(x)=-(x-5)^2-6