Graph f(x)=18x-9x^2

Math
f(x)=18x-9×2
Find the properties of the given parabola.
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Rewrite the equation in vertex form.
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Reorder 18x and -9×2.
y=-9×2+18x
Complete the square for -9×2+18x.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=-9,b=18,c=0
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=182(-9)
Simplify the right side.
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Cancel the common factor of 18 and 2.
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Factor 2 out of 18.
d=2⋅92⋅-9
Cancel the common factors.
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Factor 2 out of 2⋅-9.
d=2⋅92(-9)
Cancel the common factor.
d=2⋅92⋅-9
Rewrite the expression.
d=9-9
d=9-9
d=9-9
Cancel the common factor of 9 and -9.
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Factor 9 out of 9.
d=9(1)-9
Move the negative one from the denominator of 1-1.
d=-1⋅1
d=-1⋅1
Multiply -1 by 1.
d=-1
d=-1
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raise 18 to the power of 2.
e=0-3244⋅-9
Multiply 4 by -9.
e=0-324-36
Divide 324 by -36.
e=0+9
Multiply -1 by -9.
e=0+9
e=0+9
Add 0 and 9.
e=9
e=9
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-9(x-1)2+9
-9(x-1)2+9
Set y equal to the new right side.
y=-9(x-1)2+9
y=-9(x-1)2+9
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-9
h=1
k=9
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(1,9)
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⋅-9
Simplify.
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Multiply 4 by -9.
1-36
Move the negative in front of the fraction.
-136
-136
-136
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.
(1,32336)
(1,32336)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=1
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=32536
y=32536
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (1,9)
Focus: (1,32336)
Axis of Symmetry: x=1
Directrix: y=32536
Direction: Opens Down
Vertex: (1,9)
Focus: (1,32336)
Axis of Symmetry: x=1
Directrix: y=32536
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 0 in the expression.
f(0)=-9(0)2+18(0)
Simplify the result.
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Simplify each term.
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Raising 0 to any positive power yields 0.
f(0)=-9⋅0+18(0)
Multiply -9 by 0.
f(0)=0+18(0)
Multiply 18 by 0.
f(0)=0+0
f(0)=0+0
Add 0 and 0.
f(0)=0
The final answer is 0.
0
0
The y value at x=0 is 0.
y=0
Replace the variable x with 2 in the expression.
f(2)=-9(2)2+18(2)
Simplify the result.
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Simplify each term.
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Raise 2 to the power of 2.
f(2)=-9⋅4+18(2)
Multiply -9 by 4.
f(2)=-36+18(2)
Multiply 18 by 2.
f(2)=-36+36
f(2)=-36+36
Add -36 and 36.
f(2)=0
The final answer is 0.
0
0
The y value at x=2 is 0.
y=0
Graph the parabola using its properties and the selected points.
xy001920
xy001920
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (1,9)
Focus: (1,32336)
Axis of Symmetry: x=1
Directrix: y=32536
xy001920
Graph f(x)=18x-9x^2

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