Graph f(x)=-9/2x^2+1/3+6

Math
f(x)=-92×2+13+6
Find the properties of the given parabola.
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Rewrite the equation in vertex form.
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Simplify -92×2+13+6.
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Simplify each term.
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Combine x2 and 92.
y=-x2⋅92+13+6
Move 9 to the left of x2.
y=-9×22+13+6
y=-9×22+13+6
To write 6 as a fraction with a common denominator, multiply by 33.
y=-9×22+13+6⋅33
Combine 6 and 33.
y=-9×22+13+6⋅33
Combine the numerators over the common denominator.
y=-9×22+1+6⋅33
Simplify the numerator.
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Multiply 6 by 3.
y=-9×22+1+183
Add 1 and 18.
y=-9×22+193
y=-9×22+193
y=-9×22+193
Complete the square for -9×22+193.
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Use the form ax2+bx+c, to find the values of a, b, and c.
a=-92,b=0,c=193
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=02(-92)
Simplify the right side.
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Cancel the common factor of 0 and 2.
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Factor 2 out of 0.
d=2(0)2(-92)
Cancel the common factors.
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Cancel the common factor.
d=2⋅02(-92)
Rewrite the expression.
d=0-92
d=0-92
d=0-92
Multiply the numerator by the reciprocal of the denominator.
d=0(-29)
Multiply 0(-29).
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Multiply -1 by 0.
d=0(29)
Multiply 0 by 29.
d=0
d=0
d=0
Find the value of e using the formula e=c-b24a.
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Simplify each term.
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Raising 0 to any positive power yields 0.
e=193-04⋅(-1(92))
Simplify the denominator.
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Multiply 4 by -1.
e=193-0-4(92)
Combine -4 and 92.
e=193-0-4⋅92
e=193-0-4⋅92
Multiply -4 by 9.
e=193-0-362
Divide -36 by 2.
e=193-0-18
Divide 0 by -18.
e=193-0
Multiply -1 by 0.
e=193+0
e=193+0
Add 193 and 0.
e=193
e=193
Substitute the values of a, d, and e into the vertex form a(x+d)2+e.
-92⋅(x+0)2+193
-92⋅(x+0)2+193
Set y equal to the new right side.
y=-92⋅(x+0)2+193
y=-92⋅(x+0)2+193
Use the vertex form, y=a(x-h)2+k, to determine the values of a, h, and k.
a=-92
h=0
k=193
Since the value of a is negative, the parabola opens down.
Opens Down
Find the vertex (h,k).
(0,193)
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⋅(-92)
Simplify.
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Cancel the common factor of 1 and -1.
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Rewrite 1 as -1(-1).
-1(-1)4⋅(-92)
Move the negative in front of the fraction.
-14(92)
-14(92)
Combine 4 and 92.
-14⋅92
Simplify the expression.
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Multiply 4 by 9.
-1362
Divide 36 by 2.
-118
-118
-118
-118
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.
(0,11318)
(0,11318)
Find the axis of symmetry by finding the line that passes through the vertex and the focus.
x=0
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=11518
y=11518
Use the properties of the parabola to analyze and graph the parabola.
Direction: Opens Down
Vertex: (0,193)
Focus: (0,11318)
Axis of Symmetry: x=0
Directrix: y=11518
Direction: Opens Down
Vertex: (0,193)
Focus: (0,11318)
Axis of Symmetry: x=0
Directrix: y=11518
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 -1 in the expression.
f(-1)=-9(-1)22+193
Simplify the result.
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Simplify each term.
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Raise -1 to the power of 2.
f(-1)=-9⋅12+193
Multiply 9 by 1.
f(-1)=-92+193
f(-1)=-92+193
To write -92 as a fraction with a common denominator, multiply by 33.
f(-1)=-92⋅33+193
To write 193 as a fraction with a common denominator, multiply by 22.
f(-1)=-92⋅33+193⋅22
Write each expression with a common denominator of 6, by multiplying each by an appropriate factor of 1.
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Multiply 92 and 33.
f(-1)=-9⋅32⋅3+193⋅22
Multiply 2 by 3.
f(-1)=-9⋅36+193⋅22
Multiply 193 and 22.
f(-1)=-9⋅36+19⋅23⋅2
Multiply 3 by 2.
f(-1)=-9⋅36+19⋅26
f(-1)=-9⋅36+19⋅26
Combine the numerators over the common denominator.
f(-1)=-9⋅3+19⋅26
Simplify the numerator.
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Multiply -9 by 3.
f(-1)=-27+19⋅26
Multiply 19 by 2.
f(-1)=-27+386
Add -27 and 38.
f(-1)=116
f(-1)=116
The final answer is 116.
116
116
The y value at x=-1 is 116.
y=116
Replace the variable x with -2 in the expression.
f(-2)=-9(-2)22+193
Simplify the result.
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Simplify each term.
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Cancel the common factor of (-2)2 and 2.
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Rewrite -2 as -1(2).
f(-2)=-9(-1⋅2)22+193
Apply the product rule to -1(2).
f(-2)=-9((-1)2⋅22)2+193
Raise -1 to the power of 2.
f(-2)=-9(1⋅22)2+193
Multiply 22 by 1.
f(-2)=-9⋅222+193
Factor 2 out of 9⋅22.
f(-2)=-2(9⋅2)2+193
Cancel the common factors.
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Factor 2 out of 2.
f(-2)=-2(9⋅2)2(1)+193
Cancel the common factor.
f(-2)=-2(9⋅2)2⋅1+193
Rewrite the expression.
f(-2)=-9⋅21+193
Divide 9⋅2 by 1.
f(-2)=-(9⋅2)+193
f(-2)=-(9⋅2)+193
f(-2)=-(9⋅2)+193
Multiply 9 by 2.
f(-2)=-1⋅18+193
Multiply -1 by 18.
f(-2)=-18+193
f(-2)=-18+193
To write -18 as a fraction with a common denominator, multiply by 33.
f(-2)=-18⋅33+193
Combine -18 and 33.
f(-2)=-18⋅33+193
Combine the numerators over the common denominator.
f(-2)=-18⋅3+193
Simplify the numerator.
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Multiply -18 by 3.
f(-2)=-54+193
Add -54 and 19.
f(-2)=-353
f(-2)=-353
Move the negative in front of the fraction.
f(-2)=-353
The final answer is -353.
-353
-353
The y value at x=-2 is -353.
y=-353
Replace the variable x with 1 in the expression.
f(1)=-9(1)22+193
Simplify the result.
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Simplify each term.
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One to any power is one.
f(1)=-9⋅12+193
Multiply 9 by 1.
f(1)=-92+193
f(1)=-92+193
To write -92 as a fraction with a common denominator, multiply by 33.
f(1)=-92⋅33+193
To write 193 as a fraction with a common denominator, multiply by 22.
f(1)=-92⋅33+193⋅22
Write each expression with a common denominator of 6, by multiplying each by an appropriate factor of 1.
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Multiply 92 and 33.
f(1)=-9⋅32⋅3+193⋅22
Multiply 2 by 3.
f(1)=-9⋅36+193⋅22
Multiply 193 and 22.
f(1)=-9⋅36+19⋅23⋅2
Multiply 3 by 2.
f(1)=-9⋅36+19⋅26
f(1)=-9⋅36+19⋅26
Combine the numerators over the common denominator.
f(1)=-9⋅3+19⋅26
Simplify the numerator.
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Multiply -9 by 3.
f(1)=-27+19⋅26
Multiply 19 by 2.
f(1)=-27+386
Add -27 and 38.
f(1)=116
f(1)=116
The final answer is 116.
116
116
The y value at x=1 is 116.
y=116
Replace the variable x with 2 in the expression.
f(2)=-9(2)22+193
Simplify the result.
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Simplify each term.
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Cancel the common factor of (2)2 and 2.
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Factor 2 out of 9(2)2.
f(2)=-2(9⋅2)2+193
Cancel the common factors.
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Factor 2 out of 2.
f(2)=-2(9⋅2)2(1)+193
Cancel the common factor.
f(2)=-2(9⋅2)2⋅1+193
Rewrite the expression.
f(2)=-9⋅21+193
Divide 9⋅2 by 1.
f(2)=-(9⋅2)+193
f(2)=-(9⋅2)+193
f(2)=-(9⋅2)+193
Multiply 9 by 2.
f(2)=-1⋅18+193
Multiply -1 by 18.
f(2)=-18+193
f(2)=-18+193
To write -18 as a fraction with a common denominator, multiply by 33.
f(2)=-18⋅33+193
Combine -18 and 33.
f(2)=-18⋅33+193
Combine the numerators over the common denominator.
f(2)=-18⋅3+193
Simplify the numerator.
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Multiply -18 by 3.
f(2)=-54+193
Add -54 and 19.
f(2)=-353
f(2)=-353
Move the negative in front of the fraction.
f(2)=-353
The final answer is -353.
-353
-353
The y value at x=2 is -353.
y=-353
Graph the parabola using its properties and the selected points.
xy-2-353-1116019311162-353
xy-2-353-1116019311162-353
Graph the parabola using its properties and the selected points.
Direction: Opens Down
Vertex: (0,193)
Focus: (0,11318)
Axis of Symmetry: x=0
Directrix: y=11518
xy-2-353-1116019311162-353
Graph f(x)=-9/2x^2+1/3+6

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