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
