Algebra
Calculus
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Matrix
Question
Function
Find the vertex
Find the axis of symmetry
Rewrite in vertex form
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(45,−8121)
Evaluate
y=2x2−5x−12
Find the x-coordinate of the vertex by substituting a=2 and b=−5 into x = −2ab
x=−2×2−5
Solve the equation for x
x=45
Find the y-coordinate of the vertex by evaluating the function for x=45
y=2(45)2−5×45−12
Calculate
More Steps
Evaluate
2(45)2−5×45−12
Multiply the terms
More Steps
Evaluate
2(45)2
Evaluate the power
2×1625
Multiply the numbers
825
825−5×45−12
Multiply the numbers
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Evaluate
−5×45
Multiply the numbers
−45×5
Multiply the numbers
−425
825−425−12
Reduce fractions to a common denominator
825−4×225×2−4×212×4×2
Multiply the numbers
825−825×2−4×212×4×2
Multiply the numbers
825−825×2−812×4×2
Write all numerators above the common denominator
825−25×2−12×4×2
Multiply the numbers
825−50−12×4×2
Multiply the terms
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Evaluate
12×4×2
Multiply the terms
48×2
Multiply the numbers
96
825−50−96
Subtract the numbers
8−121
Use b−a=−ba=−ba to rewrite the fraction
−8121
y=−8121
Solution
(45,−8121)
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=2x2−5x−12
To test if the graph of y=2x2−5x−12 is symmetry with respect to the origin,substitute -x for x and -y for y
−y=2(−x)2−5(−x)−12
Simplify
More Steps
Evaluate
2(−x)2−5(−x)−12
Multiply the terms
2x2−5(−x)−12
Multiply the numbers
2x2+5x−12
−y=2x2+5x−12
Change the signs both sides
y=−2x2−5x+12
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−45)2=21(y+8121)
Evaluate
y=2x2−5x−12
Swap the sides of the equation
2x2−5x−12=y
Move the constant to the right-hand side and change its sign
2x2−5x=y−(−12)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
2x2−5x=y+12
Multiply both sides of the equation by 21
(2x2−5x)×21=(y+12)×21
Multiply the terms
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Evaluate
(2x2−5x)×21
Use the the distributive property to expand the expression
2x2×21−5x×21
Multiply the numbers
x2−5x×21
Multiply the numbers
x2−25x
x2−25x=(y+12)×21
Multiply the terms
More Steps
Evaluate
(y+12)×21
Apply the distributive property
y×21+12×21
Use the commutative property to reorder the terms
21y+12×21
Multiply the numbers
21y+6
x2−25x=21y+6
To complete the square, the same value needs to be added to both sides
x2−25x+1625=21y+6+1625
Use a2−2ab+b2=(a−b)2 to factor the expression
(x−45)2=21y+6+1625
Add the numbers
More Steps
Evaluate
6+1625
Reduce fractions to a common denominator
166×16+1625
Write all numerators above the common denominator
166×16+25
Multiply the numbers
1696+25
Add the numbers
16121
(x−45)2=21y+16121
Solution
(x−45)2=21(y+8121)
Show Solution
Solve the equation
x=45+121+8yx=45−121+8y
Evaluate
y=2x2−5x−12
Swap the sides of the equation
2x2−5x−12=y
Move the expression to the left side
2x2−5x−12−y=0
Substitute a=2,b=−5 and c=−12−y into the quadratic formula x=2a−b±b2−4ac
x=2×25±(−5)2−4×2(−12−y)
Simplify the expression
x=45±(−5)2−4×2(−12−y)
Simplify the expression
More Steps
Evaluate
(−5)2−4×2(−12−y)
Multiply the terms
More Steps
Multiply the terms
4×2(−12−y)
Multiply the terms
8(−12−y)
Apply the distributive property
−8×12−8y
Multiply the numbers
−96−8y
(−5)2−(−96−8y)
Rewrite the expression
52−(−96−8y)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
52+96+8y
Evaluate the power
25+96+8y
Add the numbers
121+8y
x=45±121+8y
Solution
x=45+121+8yx=45−121+8y
Show Solution
Rewrite the equation
r=4cos2(θ)sin(θ)+5cos(θ)−1+120cos2(θ)+5sin(2θ)r=4cos2(θ)sin(θ)+5cos(θ)+1+120cos2(θ)+5sin(2θ)
Evaluate
y=2x2−5x−12
Move the expression to the left side
y−2x2+5x=−12
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+5cos(θ)×r=−12
Factor the expression
−2cos2(θ)×r2+(sin(θ)+5cos(θ))r=−12
Subtract the terms
−2cos2(θ)×r2+(sin(θ)+5cos(θ))r−(−12)=−12−(−12)
Evaluate
−2cos2(θ)×r2+(sin(θ)+5cos(θ))r+12=0
Solve using the quadratic formula
r=−4cos2(θ)−sin(θ)−5cos(θ)±(sin(θ)+5cos(θ))2−4(−2cos2(θ))×12
Simplify
r=−4cos2(θ)−sin(θ)−5cos(θ)±1+120cos2(θ)+5sin(2θ)
Separate the equation into 2 possible cases
r=−4cos2(θ)−sin(θ)−5cos(θ)+1+120cos2(θ)+5sin(2θ)r=−4cos2(θ)−sin(θ)−5cos(θ)−1+120cos2(θ)+5sin(2θ)
Use b−a=−ba=−ba to rewrite the fraction
r=4cos2(θ)sin(θ)+5cos(θ)−1+120cos2(θ)+5sin(2θ)r=−4cos2(θ)−sin(θ)−5cos(θ)−1+120cos2(θ)+5sin(2θ)
Solution
r=4cos2(θ)sin(θ)+5cos(θ)−1+120cos2(θ)+5sin(2θ)r=4cos2(θ)sin(θ)+5cos(θ)+1+120cos2(θ)+5sin(2θ)
Show Solution
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