Algebra
Calculus
Trigonometry
Matrix
Question
Function
Find the vertex
Find the axis of symmetry
Rewrite in vertex form
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(−7,7)
Evaluate
y=−2x2−28x−91
Find the x-coordinate of the vertex by substituting a=−2 and b=−28 into x = −2ab
x=−2(−2)−28
Solve the equation for x
x=−7
Find the y-coordinate of the vertex by evaluating the function for x=−7
y=−2(−7)2−28(−7)−91
Calculate
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Evaluate
−2(−7)2−28(−7)−91
Multiply the terms
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Evaluate
−2(−7)2
Evaluate the power
−2×49
Multiply the numbers
−98
−98−28(−7)−91
Multiply the numbers
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Evaluate
−28(−7)
Multiplying or dividing an even number of negative terms equals a positive
28×7
Multiply the numbers
196
−98+196−91
Calculate the sum or difference
7
y=7
Solution
(−7,7)
Show Solution
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=−2x2−28x−91
To test if the graph of y=−2x2−28x−91 is symmetry with respect to the origin,substitute -x for x and -y for y
−y=−2(−x)2−28(−x)−91
Simplify
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Evaluate
−2(−x)2−28(−x)−91
Multiply the terms
−2x2−28(−x)−91
Multiply the numbers
−2x2+28x−91
−y=−2x2+28x−91
Change the signs both sides
y=2x2−28x+91
Solution
Not symmetry with respect to the origin
Show Solution
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+7)2=−21(y−7)
Evaluate
y=−2x2−28x−91
Swap the sides of the equation
−2x2−28x−91=y
Move the constant to the right-hand side and change its sign
−2x2−28x=y−(−91)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
−2x2−28x=y+91
Multiply both sides of the equation by −21
(−2x2−28x)(−21)=(y+91)(−21)
Multiply the terms
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Evaluate
(−2x2−28x)(−21)
Use the the distributive property to expand the expression
−2x2(−21)−28x(−21)
Multiply the numbers
x2−28x(−21)
Multiply the numbers
x2+14x
x2+14x=(y+91)(−21)
Multiply the terms
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Evaluate
(y+91)(−21)
Apply the distributive property
y(−21)+91(−21)
Use the commutative property to reorder the terms
−21y+91(−21)
Multiply the numbers
−21y−291
x2+14x=−21y−291
To complete the square, the same value needs to be added to both sides
x2+14x+49=−21y−291+49
Use a2+2ab+b2=(a+b)2 to factor the expression
(x+7)2=−21y−291+49
Add the numbers
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Evaluate
−291+49
Reduce fractions to a common denominator
−291+249×2
Write all numerators above the common denominator
2−91+49×2
Multiply the numbers
2−91+98
Add the numbers
27
(x+7)2=−21y+27
Solution
(x+7)2=−21(y−7)
Show Solution
Solve the equation
x=2−14+14−2yx=−214+14−2y
Evaluate
y=−2x2−28x−91
Swap the sides of the equation
−2x2−28x−91=y
Move the expression to the left side
−2x2−28x−91−y=0
Multiply both sides
2x2+28x+91+y=0
Substitute a=2,b=28 and c=91+y into the quadratic formula x=2a−b±b2−4ac
x=2×2−28±282−4×2(91+y)
Simplify the expression
x=4−28±282−4×2(91+y)
Simplify the expression
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Evaluate
282−4×2(91+y)
Multiply the terms
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Multiply the terms
4×2(91+y)
Multiply the terms
8(91+y)
Apply the distributive property
8×91+8y
Multiply the numbers
728+8y
282−(728+8y)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
282−728−8y
Evaluate the power
784−728−8y
Subtract the numbers
56−8y
x=4−28±56−8y
Simplify the radical expression
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Evaluate
56−8y
Factor the expression
8(7−y)
The root of a product is equal to the product of the roots of each factor
8×7−y
Evaluate the root
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Evaluate
8
Write the expression as a product where the root of one of the factors can be evaluated
4×2
Write the number in exponential form with the base of 2
22×2
The root of a product is equal to the product of the roots of each factor
22×2
Reduce the index of the radical and exponent with 2
22
22×7−y
Calculate the product
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Evaluate
2×7−y
The product of roots with the same index is equal to the root of the product
2(7−y)
Calculate the product
14−2y
214−2y
x=4−28±214−2y
Separate the equation into 2 possible cases
x=4−28+214−2yx=4−28−214−2y
Simplify the expression
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Evaluate
x=4−28+214−2y
Divide the terms
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Evaluate
4−28+214−2y
Rewrite the expression
42(−14+14−2y)
Cancel out the common factor 2
2−14+14−2y
x=2−14+14−2y
x=2−14+14−2yx=4−28−214−2y
Solution
More Steps
Evaluate
x=4−28−214−2y
Divide the terms
More Steps
Evaluate
4−28−214−2y
Rewrite the expression
42(−14−14−2y)
Cancel out the common factor 2
2−14−14−2y
Use b−a=−ba=−ba to rewrite the fraction
−214+14−2y
x=−214+14−2y
x=2−14+14−2yx=−214+14−2y
Show Solution
Rewrite the equation
r=4cos2(θ)−sin(θ)−28cos(θ)+1+55cos2(θ)+28sin(2θ)r=−4cos2(θ)sin(θ)+28cos(θ)+1+55cos2(θ)+28sin(2θ)
Evaluate
y=−2x2−28x−91
Move the expression to the left side
y+2x2+28x=−91
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
sin(θ)×r+2(cos(θ)×r)2+28cos(θ)×r=−91
Factor the expression
2cos2(θ)×r2+(sin(θ)+28cos(θ))r=−91
Subtract the terms
2cos2(θ)×r2+(sin(θ)+28cos(θ))r−(−91)=−91−(−91)
Evaluate
2cos2(θ)×r2+(sin(θ)+28cos(θ))r+91=0
Solve using the quadratic formula
r=4cos2(θ)−sin(θ)−28cos(θ)±(sin(θ)+28cos(θ))2−4×2cos2(θ)×91
Simplify
r=4cos2(θ)−sin(θ)−28cos(θ)±1+55cos2(θ)+28sin(2θ)
Separate the equation into 2 possible cases
r=4cos2(θ)−sin(θ)−28cos(θ)+1+55cos2(θ)+28sin(2θ)r=4cos2(θ)−sin(θ)−28cos(θ)−1+55cos2(θ)+28sin(2θ)
Solution
r=4cos2(θ)−sin(θ)−28cos(θ)+1+55cos2(θ)+28sin(2θ)r=−4cos2(θ)sin(θ)+28cos(θ)+1+55cos2(θ)+28sin(2θ)
Show Solution
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