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Question
y=-x^{2}+4x+12
Function
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Find the vertex
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Find the axis of symmetry
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Rewrite in vertex form
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Evaluate the derivative
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Find the domain
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\text{Find the }x\text{-intercept/zero}
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Find the y-intercept
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Find the critical numbers
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Find the local extrema
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Find the increasing or decreasing interval
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Find the range
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Find the vertical asymptotes
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Find the horizontal asymptotes
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Find the oblique asymptotes
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Determine if even, odd or neither
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Find the stationary points
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Find the inflection points
More methods
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\left(2,16\right)
Evaluate
y=-x^{2}+4x+12
\text{Find the }x\text{-coordinate of the vertex by substituting a=}-1\text{ and b=}4\text{ into }x\text{ = }-\frac{b}{2a}
x=-\frac{4}{2\left(-1\right)}
\text{Solve the equation for }x
x=2
\text{Find the y-coordinate of the vertex by evaluating the function for }x\text{=}2
y=-2^{2}+4\times 2+12
Calculate
More Steps
Evaluate
-2^{2}+4\times 2+12
Multiply the numbers
-2^{2}+8+12
Evaluate the power
-4+8+12
Add the numbers
16
y=16
Solution
\left(2,16\right)
Show Solutions
Testing for symmetry
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Testing for symmetry about the origin
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Testing for symmetry about the x-axis
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Testing for symmetry about the y-axis
\textrm{Not symmetry with respect to the origin}
Evaluate
y=-x^{2}+4x+12
\text{To test if the graph of }y=-x^{2}+4x+12\text{ is symmetry with respect to the origin,substitute -x for x and -y for y}
-y=-\left(-x\right)^{2}+4\left(-x\right)+12
Simplify
More Steps
Evaluate
-\left(-x\right)^{2}+4\left(-x\right)+12
Multiply the numbers
-\left(-x\right)^{2}-4x+12
Rewrite the expression
-x^{2}-4x+12
-y=-x^{2}-4x+12
Change the signs both sides
y=x^{2}+4x-12
Solution
\textrm{Not symmetry with respect to the origin}
Show Solutions
Identify the conic
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Find the standard equation of the parabola
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Find the vertex of the parabola
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Find the focus of the parabola
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Find the directrix of the parabola
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\left(x-2\right)^{2}=-\left(y-16\right)
Evaluate
y=-x^{2}+4x+12
Swap the sides of the equation
-x^{2}+4x+12=y
Move the constant to the right-hand side and change its sign
-x^{2}+4x=y-12
\text{Multiply both sides of the equation by }-1
\left(-x^{2}+4x\right)\left(-1\right)=\left(y-12\right)\left(-1\right)
Multiply the terms
More Steps
Evaluate
\left(-x^{2}+4x\right)\left(-1\right)
Use the the distributive property to expand the expression
-x^{2}\left(-1\right)+4x\left(-1\right)
Multiplying or dividing an even number of negative terms equals a positive
x^{2}+4x\left(-1\right)
Multiply the numbers
x^{2}-4x
x^{2}-4x=\left(y-12\right)\left(-1\right)
Multiply the terms
More Steps
Evaluate
\left(y-12\right)\left(-1\right)
Apply the distributive property
y\left(-1\right)-12\left(-1\right)
Multiplying or dividing an odd number of negative terms equals a negative
-y-12\left(-1\right)
Simplify
-y+12
x^{2}-4x=-y+12
To complete the square, the same value needs to be added to both sides
x^{2}-4x+4=-y+12+4
\text{Use }a^2-2ab+b^2=(a-b)^2\text{ to factor the expression}
\left(x-2\right)^{2}=-y+12+4
Add the numbers
\left(x-2\right)^{2}=-y+16
Solution
\left(x-2\right)^{2}=-\left(y-16\right)
Show Solutions
Solve the equation
\begin{align}&x=2+\sqrt{16-y}\\&x=2-\sqrt{16-y}\end{align}
Evaluate
y=-x^{2}+4x+12
Swap the sides of the equation
-x^{2}+4x+12=y
Move the expression to the left side
-x^{2}+4x+12-y=0
Multiply both sides
x^{2}-4x-12+y=0
\text{Substitute a=}1\text{,b=}-4\text{ and c=}-12+y\text{ into the quadratic formula }x\text{=}\frac{-b\pm\sqrt{b^2-4ac}}{2a}
x=\frac{4\pm \sqrt{\left(-4\right)^{2}-4\left(-12+y\right)}}{2}
Simplify the expression
More Steps
Evaluate
\left(-4\right)^{2}-4\left(-12+y\right)
Multiply the terms
More Steps
Evaluate
4\left(-12+y\right)
Apply the distributive property
-4\times 12+4y
Multiply the numbers
-48+4y
\left(-4\right)^{2}-\left(-48+4y\right)
Rewrite the expression
4^{2}-\left(-48+4y\right)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
4^{2}+48-4y
Evaluate the power
16+48-4y
Add the numbers
64-4y
x=\frac{4\pm \sqrt{64-4y}}{2}
Simplify the radical expression
More Steps
Evaluate
\sqrt{64-4y}
Factor the expression
\sqrt{4\left(16-y\right)}
The root of a product is equal to the product of the roots of each factor
\sqrt{4}\times \sqrt{16-y}
Evaluate the root
More Steps
Evaluate
\sqrt{4}
\text{Write the number in exponential form with the base of }2
\sqrt{2^{2}}
\text{Reduce the index of the radical and exponent with }2
2
2\sqrt{16-y}
x=\frac{4\pm 2\sqrt{16-y}}{2}
\text{Separate the equation into }2\text{ possible cases}
\begin{align}&x=\frac{4+2\sqrt{16-y}}{2}\\&x=\frac{4-2\sqrt{16-y}}{2}\end{align}
Simplify the expression
More Steps
Evaluate
x=\frac{4+2\sqrt{16-y}}{2}
Divide the terms
More Steps
Evaluate
\frac{4+2\sqrt{16-y}}{2}
Rewrite the expression
\frac{2\left(2+\sqrt{16-y}\right)}{2}
Reduce the fraction
2+\sqrt{16-y}
x=2+\sqrt{16-y}
\begin{align}&x=2+\sqrt{16-y}\\&x=\frac{4-2\sqrt{16-y}}{2}\end{align}
Solution
More Steps
Evaluate
x=\frac{4-2\sqrt{16-y}}{2}
Divide the terms
More Steps
Evaluate
\frac{4-2\sqrt{16-y}}{2}
Rewrite the expression
\frac{2\left(2-\sqrt{16-y}\right)}{2}
Reduce the fraction
2-\sqrt{16-y}
x=2-\sqrt{16-y}
\begin{align}&x=2+\sqrt{16-y}\\&x=2-\sqrt{16-y}\end{align}
Show Solutions
Rewrite the equation
\begin{align}&r=\frac{-\sin\left(\theta \right)+4\cos\left(\theta \right)+\sqrt{1+63\cos^{2}\left(\theta \right)-4\sin\left(2\theta \right)}}{2\cos^{2}\left(\theta \right)}\\&r=\frac{-\sin\left(\theta \right)+4\cos\left(\theta \right)-\sqrt{1+63\cos^{2}\left(\theta \right)-4\sin\left(2\theta \right)}}{2\cos^{2}\left(\theta \right)}\end{align}
Evaluate
y=-x^{2}+4x+12
Move the expression to the left side
y+x^{2}-4x=12
\text{To convert the equation to polar coordinates,substitute }r\cos\left(\theta \right)\text{ for }x\text{ and }r\sin\left(\theta \right)\text{ for }y
\sin\left(\theta \right)\times r+\left(\cos\left(\theta \right)\times r\right)^{2}-4\cos\left(\theta \right)\times r=12
Factor the expression
\cos^{2}\left(\theta \right)\times r^{2}+\left(\sin\left(\theta \right)-4\cos\left(\theta \right)\right)r=12
Subtract the terms
\cos^{2}\left(\theta \right)\times r^{2}+\left(\sin\left(\theta \right)-4\cos\left(\theta \right)\right)r-12=12-12
Evaluate
\cos^{2}\left(\theta \right)\times r^{2}+\left(\sin\left(\theta \right)-4\cos\left(\theta \right)\right)r-12=0
Solve using the quadratic formula
r=\frac{-\sin\left(\theta \right)+4\cos\left(\theta \right)\pm \sqrt{\left(\sin\left(\theta \right)-4\cos\left(\theta \right)\right)^{2}-4\cos^{2}\left(\theta \right)\left(-12\right)}}{2\cos^{2}\left(\theta \right)}
Simplify
r=\frac{-\sin\left(\theta \right)+4\cos\left(\theta \right)\pm \sqrt{1+63\cos^{2}\left(\theta \right)-4\sin\left(2\theta \right)}}{2\cos^{2}\left(\theta \right)}
Solution
\begin{align}&r=\frac{-\sin\left(\theta \right)+4\cos\left(\theta \right)+\sqrt{1+63\cos^{2}\left(\theta \right)-4\sin\left(2\theta \right)}}{2\cos^{2}\left(\theta \right)}\\&r=\frac{-\sin\left(\theta \right)+4\cos\left(\theta \right)-\sqrt{1+63\cos^{2}\left(\theta \right)-4\sin\left(2\theta \right)}}{2\cos^{2}\left(\theta \right)}\end{align}
Show Solutions
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