Algebra
Calculus
Trigonometry
Matrix
Question
Function
Find the vertex
Find the axis of symmetry
Rewrite in vertex form
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(71,−71)
Evaluate
y=7x2−2x×1
Simplify
y=7x2−2x
Find the x-coordinate of the vertex by substituting a=7 and b=−2 into x = −2ab
x=−2×7−2
Solve the equation for x
x=71
Find the y-coordinate of the vertex by evaluating the function for x=71
y=7(71)2−2×71
Calculate
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Evaluate
7(71)2−2×71
Multiply the terms
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Evaluate
7(71)2
Rewrite the expression
(71)−1(71)2
Rewrite the expression
(71)−1+2
Calculate
(71)1
Calculate
71
71−2×71
Multiply the numbers
71−72
Write all numerators above the common denominator
71−2
Subtract the numbers
7−1
Use b−a=−ba=−ba to rewrite the fraction
−71
y=−71
Solution
(71,−71)
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=7x2−2x1
Simplify the expression
y=7x2−2x
To test if the graph of y=7x2−2x is symmetry with respect to the origin,substitute -x for x and -y for y
−y=7(−x)2−2(−x)
Simplify
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Evaluate
7(−x)2−2(−x)
Multiply the terms
7x2−2(−x)
Multiply the numbers
7x2−(−2x)
Rewrite the expression
7x2+2x
−y=7x2+2x
Change the signs both sides
y=−7x2−2x
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−71)2=71(y+71)
Evaluate
y=7x2−2x×1
Simplify
y=7x2−2x
Swap the sides of the equation
7x2−2x=y
Multiply both sides of the equation by 71
(7x2−2x)×71=y×71
Multiply the terms
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Evaluate
(7x2−2x)×71
Use the the distributive property to expand the expression
7x2×71−2x×71
Multiply the numbers
x2−2x×71
Multiply the numbers
x2−72x
x2−72x=y×71
Use the commutative property to reorder the terms
x2−72x=71y
To complete the square, the same value needs to be added to both sides
x2−72x+491=71y+491
Use a2−2ab+b2=(a−b)2 to factor the expression
(x−71)2=71y+491
Solution
(x−71)2=71(y+71)
Show Solution
Solve the equation
Solve for x
Solve for y
x=71+1+7yx=71−1+7y
Evaluate
y=7x2−2x×1
Simplify
y=7x2−2x
Swap the sides of the equation
7x2−2x=y
Move the expression to the left side
7x2−2x−y=0
Substitute a=7,b=−2 and c=−y into the quadratic formula x=2a−b±b2−4ac
x=2×72±(−2)2−4×7(−y)
Simplify the expression
x=142±(−2)2−4×7(−y)
Simplify the expression
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Evaluate
(−2)2−4×7(−y)
Multiply
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Multiply the terms
4×7(−y)
Rewrite the expression
−4×7y
Multiply the terms
−28y
(−2)2−(−28y)
Rewrite the expression
22−(−28y)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
22+28y
Evaluate the power
4+28y
x=142±4+28y
Simplify the radical expression
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Evaluate
4+28y
Factor the expression
4(1+7y)
The root of a product is equal to the product of the roots of each factor
4×1+7y
Evaluate the root
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Evaluate
4
Write the number in exponential form with the base of 2
22
Reduce the index of the radical and exponent with 2
2
21+7y
x=142±21+7y
Separate the equation into 2 possible cases
x=142+21+7yx=142−21+7y
Simplify the expression
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Evaluate
x=142+21+7y
Divide the terms
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Evaluate
142+21+7y
Rewrite the expression
142(1+1+7y)
Cancel out the common factor 2
71+1+7y
x=71+1+7y
x=71+1+7yx=142−21+7y
Solution
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Evaluate
x=142−21+7y
Divide the terms
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Evaluate
142−21+7y
Rewrite the expression
142(1−1+7y)
Cancel out the common factor 2
71−1+7y
x=71−1+7y
x=71+1+7yx=71−1+7y
Show Solution
Rewrite the equation
r=0r=7cos2(θ)sin(θ)+2cos(θ)
Evaluate
y=7x2−2x×1
Simplify
y=7x2−2x
Move the expression to the left side
y−7x2+2x=0
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
sin(θ)×r−7(cos(θ)×r)2+2cos(θ)×r=0
Factor the expression
−7cos2(θ)×r2+(sin(θ)+2cos(θ))r=0
Factor the expression
r(−7cos2(θ)×r+sin(θ)+2cos(θ))=0
When the product of factors equals 0,at least one factor is 0
r=0−7cos2(θ)×r+sin(θ)+2cos(θ)=0
Solution
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Factor the expression
−7cos2(θ)×r+sin(θ)+2cos(θ)=0
Subtract the terms
−7cos2(θ)×r+sin(θ)+2cos(θ)−(sin(θ)+2cos(θ))=0−(sin(θ)+2cos(θ))
Evaluate
−7cos2(θ)×r=−sin(θ)−2cos(θ)
Divide the terms
r=7cos2(θ)sin(θ)+2cos(θ)
r=0r=7cos2(θ)sin(θ)+2cos(θ)
Show Solution
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