Graph f(x)=x^2+8x+9

Math
f(x)=x2+8x+9
Find the properties of the given parabola.
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Rewrite the equation in vertex form.
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Complete the square for x2+8x+9.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=1,b=8,c=9
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=82(1)
Cancel the common factor of 8 and 2.
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Factor 2 out of 8.
d=2⋅42⋅1
Cancel the common factors.
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Factor 2 out of 2⋅1.
d=2⋅42(1)
Cancel the common factor.
d=2⋅42⋅1
Rewrite the expression.
d=41
Divide 4 by 1.
d=4
d=4
d=4
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise 8 to the power of 2.
e=9-644⋅1
Multiply 4 by 1.
e=9-644
Divide 64 by 4.
e=9-1⋅16
Multiply -1 by 16.
e=9-16
e=9-16
Subtract 16 from 9.
e=-7
e=-7
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
(x+4)2-7
(x+4)2-7
Set y equal to the new right side.
y=(x+4)2-7
y=(x+4)2-7
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=1
h=-4
k=-7
Since the value of a is positive, the parabola opens up.
Opens Up
Find the vertex (h,k).
(-4,-7)
Find p, the distance from the vertex to the focus.
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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.
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Cancel the common factor.
14⋅1
Rewrite the expression.
14
14
14
Find the focus.
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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.
(-4,-274)
(-4,-274)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=-4
Find the directrix.
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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=-294
y=-294
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Up
Vertex: (-4,-7)
Focus: (-4,-274)
Axis of Symmetry: x=-4
Directrix: y=-294
Direction: Opens Up
Vertex: (-4,-7)
Focus: (-4,-274)
Axis of Symmetry: x=-4
Directrix: y=-294
Select a few x values, and plug them into the equation to find the corresponding y values. The x values should be selected around the vertex.
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Replace the variable x with -5 in the expression.
f(-5)=(-5)2+8(-5)+9
Simplify the result.
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Simplify each term.
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Raise -5 to the power of 2.
f(-5)=25+8(-5)+9
Multiply 8 by -5.
f(-5)=25-40+9
f(-5)=25-40+9
Simplify by adding and subtracting.
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Subtract 40 from 25.
f(-5)=-15+9
Add -15 and 9.
f(-5)=-6
f(-5)=-6
The final answer is -6.
-6
-6
The y value at x=-5 is -6.
y=-6
Replace the variable x with -6 in the expression.
f(-6)=(-6)2+8(-6)+9
Simplify the result.
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Simplify each term.
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Raise -6 to the power of 2.
f(-6)=36+8(-6)+9
Multiply 8 by -6.
f(-6)=36-48+9
f(-6)=36-48+9
Simplify by adding and subtracting.
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Subtract 48 from 36.
f(-6)=-12+9
Add -12 and 9.
f(-6)=-3
f(-6)=-3
The final answer is -3.
-3
-3
The y value at x=-6 is -3.
y=-3
Replace the variable x with -3 in the expression.
f(-3)=(-3)2+8(-3)+9
Simplify the result.
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Simplify each term.
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Raise -3 to the power of 2.
f(-3)=9+8(-3)+9
Multiply 8 by -3.
f(-3)=9-24+9
f(-3)=9-24+9
Simplify by adding and subtracting.
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Subtract 24 from 9.
f(-3)=-15+9
Add -15 and 9.
f(-3)=-6
f(-3)=-6
The final answer is -6.
-6
-6
The y value at x=-3 is -6.
y=-6
Replace the variable x with -2 in the expression.
f(-2)=(-2)2+8(-2)+9
Simplify the result.
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Simplify each term.
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Raise -2 to the power of 2.
f(-2)=4+8(-2)+9
Multiply 8 by -2.
f(-2)=4-16+9
f(-2)=4-16+9
Simplify by adding and subtracting.
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Subtract 16 from 4.
f(-2)=-12+9
Add -12 and 9.
f(-2)=-3
f(-2)=-3
The final answer is -3.
-3
-3
The y value at x=-2 is -3.
y=-3
Graph the parabola using its properties and the selected points.
xy-6-3-5-6-4-7-3-6-2-3
xy-6-3-5-6-4-7-3-6-2-3
Graph the parabola using its properties and the selected points.
Direction: Opens Up
Vertex: (-4,-7)
Focus: (-4,-274)
Axis of Symmetry: x=-4
Directrix: y=-294
xy-6-3-5-6-4-7-3-6-2-3
Graph f(x)=x^2+8x+9

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