Question
Function
Find the vertex
Find the axis of symmetry
Rewrite in vertex form
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(−2,−1)
Evaluate
y=2x2+8x+7
Find the x-coordinate of the vertex by substituting a=2 and b=8 into x = −2ab
x=−2×28
Solve the equation for x
x=−2
Find the y-coordinate of the vertex by evaluating the function for x=−2
y=2(−2)2+8(−2)+7
Calculate
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Evaluate
2(−2)2+8(−2)+7
Multiply the terms
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Evaluate
2(−2)2
Calculate the product
−(−2)3
A negative base raised to an odd power equals a negative
23
23+8(−2)+7
Multiply the numbers
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Evaluate
8(−2)
Multiplying or dividing an odd number of negative terms equals a negative
−8×2
Multiply the numbers
−16
23−16+7
Evaluate the power
8−16+7
Calculate the sum or difference
−1
y=−1
Solution
(−2,−1)
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Testing for symmetry
Testing for symmetry about the origin
Testing for symmetry about the x-axis
Testing for symmetry about the y-axis
Not symmetry with respect to the origin
Evaluate
y=2x2+8x+7
To test if the graph of y=2x2+8x+7 is symmetry with respect to the origin,substitute -x for x and -y for y
−y=2(−x)2+8(−x)+7
Simplify
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Evaluate
2(−x)2+8(−x)+7
Multiply the terms
2x2+8(−x)+7
Multiply the numbers
2x2−8x+7
−y=2x2−8x+7
Change the signs both sides
y=−2x2+8x−7
Solution
Not symmetry with respect to the origin
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Identify the conic
Find the standard equation of the parabola
Find the vertex of the parabola
Find the focus of the parabola
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(x+2)2=21(y+1)
Evaluate
y=2x2+8x+7
Swap the sides of the equation
2x2+8x+7=y
Move the constant to the right-hand side and change its sign
2x2+8x=y−7
Multiply both sides of the equation by 21
(2x2+8x)×21=(y−7)×21
Multiply the terms
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Evaluate
(2x2+8x)×21
Use the the distributive property to expand the expression
2x2×21+8x×21
Multiply the numbers
x2+8x×21
Multiply the numbers
x2+4x
x2+4x=(y−7)×21
Multiply the terms
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Evaluate
(y−7)×21
Apply the distributive property
y×21−7×21
Use the commutative property to reorder the terms
21y−7×21
Multiply the numbers
21y−27
x2+4x=21y−27
To complete the square, the same value needs to be added to both sides
x2+4x+4=21y−27+4
Use a2+2ab+b2=(a+b)2 to factor the expression
(x+2)2=21y−27+4
Add the numbers
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Evaluate
−27+4
Reduce fractions to a common denominator
−27+24×2
Write all numerators above the common denominator
2−7+4×2
Multiply the numbers
2−7+8
Add the numbers
21
(x+2)2=21y+21
Solution
(x+2)2=21(y+1)
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Solve the equation
Solve for x
x=22+2y−4x=−22+2y+4
Evaluate
y=2x2+8x+7
Swap the sides of the equation
2x2+8x+7=y
Move the expression to the left side
2x2+8x+7−y=0
Move the constant to the right side
2x2+8x=0−(7−y)
Add the terms
2x2+8x=−7+y
Evaluate
x2+4x=2−7+y
Add the same value to both sides
x2+4x+4=2−7+y+4
Evaluate
x2+4x+4=21+y
Evaluate
(x+2)2=21+y
Take the root of both sides of the equation and remember to use both positive and negative roots
x+2=±21+y
Simplify the expression
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Evaluate
21+y
To take a root of a fraction,take the root of the numerator and denominator separately
21+y
Multiply by the Conjugate
2×21+y×2
Calculate
21+y×2
Calculate
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Evaluate
1+y×2
The product of roots with the same index is equal to the root of the product
(1+y)×2
Calculate the product
2+2y
22+2y
x+2=±22+2y
Separate the equation into 2 possible cases
x+2=22+2yx+2=−22+2y
Calculate
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Evaluate
x+2=22+2y
Move the constant to the right-hand side and change its sign
x=22+2y−2
Subtract the terms
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Evaluate
22+2y−2
Reduce fractions to a common denominator
22+2y−22×2
Write all numerators above the common denominator
22+2y−2×2
Multiply the numbers
22+2y−4
x=22+2y−4
x=22+2y−4x+2=−22+2y
Solution
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Evaluate
x+2=−22+2y
Move the constant to the right-hand side and change its sign
x=−22+2y−2
Subtract the terms
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Evaluate
−22+2y−2
Reduce fractions to a common denominator
−22+2y−22×2
Write all numerators above the common denominator
2−2+2y−2×2
Multiply the numbers
2−2+2y−4
Use b−a=−ba=−ba to rewrite the fraction
−22+2y+4
x=−22+2y+4
x=22+2y−4x=−22+2y+4
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Rewrite the equation
Rewrite in polar form
r=4cos2(θ)sin(θ)−8cos(θ)−1+7cos2(θ)−8sin(2θ)r=4cos2(θ)sin(θ)−8cos(θ)+1+7cos2(θ)−8sin(2θ)
Evaluate
y=2x2+8x+7
Move the expression to the left side
y−2x2−8x=7
To convert the equation to polar coordinates,substitute rcos(θ) for x and rsin(θ) for y
sin(θ)×r−2(cos(θ)×r)2−8cos(θ)×r=7
Factor the expression
−2cos2(θ)×r2+(sin(θ)−8cos(θ))r=7
Subtract the terms
−2cos2(θ)×r2+(sin(θ)−8cos(θ))r−7=7−7
Evaluate
−2cos2(θ)×r2+(sin(θ)−8cos(θ))r−7=0
Solve using the quadratic formula
r=−4cos2(θ)−sin(θ)+8cos(θ)±(sin(θ)−8cos(θ))2−4(−2cos2(θ))(−7)
Simplify
r=−4cos2(θ)−sin(θ)+8cos(θ)±1+7cos2(θ)−8sin(2θ)
Separate the equation into 2 possible cases
r=−4cos2(θ)−sin(θ)+8cos(θ)+1+7cos2(θ)−8sin(2θ)r=−4cos2(θ)−sin(θ)+8cos(θ)−1+7cos2(θ)−8sin(2θ)
Use b−a=−ba=−ba to rewrite the fraction
r=4cos2(θ)sin(θ)−8cos(θ)−1+7cos2(θ)−8sin(2θ)r=−4cos2(θ)−sin(θ)+8cos(θ)−1+7cos2(θ)−8sin(2θ)
Solution
r=4cos2(θ)sin(θ)−8cos(θ)−1+7cos2(θ)−8sin(2θ)r=4cos2(θ)sin(θ)−8cos(θ)+1+7cos2(θ)−8sin(2θ)
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