4×2+8x-3-5×2+4x-9

Subtract 5×2 from 4×2.

y=-x2+8x-3+4x-9

Add 8x and 4x.

y=-x2+12x-3-9

Subtract 9 from -3.

y=-x2+12x-12

y=-x2+12x-12

Rewrite the equation in vertex form.

Complete the square for -x2+12x-12.

Use the form ax2+bx+c, to find the values of a, b, and c.

a=-1,b=12,c=-12

Consider the vertex form of a parabola.

a(x+d)2+e

Substitute the values of a and b into the formula d=b2a.

d=122(-1)

Simplify the right side.

Cancel the common factor of 12 and 2.

Factor 2 out of 12.

d=2⋅62⋅-1

Move the negative one from the denominator of 6-1.

d=-1⋅6

d=-1⋅6

Multiply -1 by 6.

d=-6

d=-6

Find the value of e using the formula e=c-b24a.

Simplify each term.

Raise 12 to the power of 2.

e=-12-1444⋅-1

Multiply 4 by -1.

e=-12-144-4

Divide 144 by -4.

e=-12+36

Multiply -1 by -36.

e=-12+36

e=-12+36

Add -12 and 36.

e=24

e=24

Substitute the values of a, d, and e into the vertex form a(x+d)2+e.

-(x-6)2+24

-(x-6)2+24

Set y equal to the new right side.

y=-(x-6)2+24

y=-(x-6)2+24

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

a=-1

h=6

k=24

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

Opens Down

Find the vertex (h,k).

(6,24)

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.

(6,954)

(6,954)

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

x=6

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=974

y=974

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

Direction: Opens Down

Vertex: (6,24)

Focus: (6,954)

Axis of Symmetry: x=6

Directrix: y=974

Direction: Opens Down

Vertex: (6,24)

Focus: (6,954)

Axis of Symmetry: x=6

Directrix: y=974

Replace the variable x with 5 in the expression.

f(5)=-(5)2+12(5)-12

Simplify the result.

Simplify each term.

Raise 5 to the power of 2.

f(5)=-1⋅25+12(5)-12

Multiply -1 by 25.

f(5)=-25+12(5)-12

Multiply 12 by 5.

f(5)=-25+60-12

f(5)=-25+60-12

Simplify by adding and subtracting.

Add -25 and 60.

f(5)=35-12

Subtract 12 from 35.

f(5)=23

f(5)=23

The final answer is 23.

23

23

The y value at x=5 is 23.

y=23

Replace the variable x with 4 in the expression.

f(4)=-(4)2+12(4)-12

Simplify the result.

Simplify each term.

Raise 4 to the power of 2.

f(4)=-1⋅16+12(4)-12

Multiply -1 by 16.

f(4)=-16+12(4)-12

Multiply 12 by 4.

f(4)=-16+48-12

f(4)=-16+48-12

Simplify by adding and subtracting.

Add -16 and 48.

f(4)=32-12

Subtract 12 from 32.

f(4)=20

f(4)=20

The final answer is 20.

20

20

The y value at x=4 is 20.

y=20

Replace the variable x with 7 in the expression.

f(7)=-(7)2+12(7)-12

Simplify the result.

Simplify each term.

Raise 7 to the power of 2.

f(7)=-1⋅49+12(7)-12

Multiply -1 by 49.

f(7)=-49+12(7)-12

Multiply 12 by 7.

f(7)=-49+84-12

f(7)=-49+84-12

Simplify by adding and subtracting.

Add -49 and 84.

f(7)=35-12

Subtract 12 from 35.

f(7)=23

f(7)=23

The final answer is 23.

23

23

The y value at x=7 is 23.

y=23

Replace the variable x with 8 in the expression.

f(8)=-(8)2+12(8)-12

Simplify the result.

Simplify each term.

Raise 8 to the power of 2.

f(8)=-1⋅64+12(8)-12

Multiply -1 by 64.

f(8)=-64+12(8)-12

Multiply 12 by 8.

f(8)=-64+96-12

f(8)=-64+96-12

Simplify by adding and subtracting.

Add -64 and 96.

f(8)=32-12

Subtract 12 from 32.

f(8)=20

f(8)=20

The final answer is 20.

20

20

The y value at x=8 is 20.

y=20

Graph the parabola using its properties and the selected points.

xy420523624723820

xy420523624723820

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: (6,24)

Focus: (6,954)

Axis of Symmetry: x=6

Directrix: y=974

xy420523624723820

Graph 4x^2+8x-3-5x^2+4x-9