Question
Function
Find the vertex
Find the axis of symmetry
Rewrite in vertex form
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![arrow-down](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTAiIGhlaWdodD0iMTAiIHZpZXdCb3g9IjAgMCAxMCAxMCIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPGcgb3BhY2l0eT0iMC4zNSI+CjxwYXRoIGQ9Ik0yLjUgMy43NUw1IDYuMjVMNy41IDMuNzUiIHN0cm9rZT0iIzIzMjAyRSIgc3Ryb2tlLXdpZHRoPSIxLjUiIHN0cm9rZS1saW5lY2FwPSJyb3VuZCIgc3Ryb2tlLWxpbmVqb2luPSJyb3VuZCIvPgo8L2c+Cjwvc3ZnPgo=)
(−31,32)
Evaluate
y=3x2+2x+1
Find the x-coordinate of the vertex by substituting a=3 and b=2 into x = −2ab
x=−2×32
Solve the equation for x
x=−31
Find the y-coordinate of the vertex by evaluating the function for x=−31
y=3(−31)2+2(−31)+1
Calculate
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
3(−31)2+2(−31)+1
Multiply the terms
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
3(−31)2
Evaluate the power
3×91
Multiply the numbers
31
31+2(−31)+1
Multiply the numbers
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
2(−31)
Multiplying or dividing an odd number of negative terms equals a negative
−2×31
Multiply the numbers
−32
31−32+1
Reduce fractions to a common denominator
31−32+33
Add or subtract the terms
31−2+3
Calculate
32
y=32
Solution
(−31,32)
Show Solution
![solution-arrow](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTciIGhlaWdodD0iMTYiIHZpZXdCb3g9IjAgMCAxNyAxNiIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPHBhdGggZD0iTTguNSAyLjY2Nzk3TDguNSAxMi42NjgiIHN0cm9rZT0id2hpdGUiIHN0cm9rZS13aWR0aD0iMS41IiBzdHJva2UtbGluZWNhcD0icm91bmQiIHN0cm9rZS1saW5lam9pbj0icm91bmQiLz4KPHBhdGggZD0iTTEyLjUgOC42Njc5N0w4LjUgMTIuNjY4TDQuNSA4LjY2Nzk3IiBzdHJva2U9IndoaXRlIiBzdHJva2Utd2lkdGg9IjEuNSIgc3Ryb2tlLWxpbmVjYXA9InJvdW5kIiBzdHJva2UtbGluZWpvaW49InJvdW5kIi8+Cjwvc3ZnPgo=)
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=3x2+2x+1
To test if the graph of y=3x2+2x+1 is symmetry with respect to the origin,substitute -x for x and -y for y
−y=3(−x)2+2(−x)+1
Simplify
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
3(−x)2+2(−x)+1
Multiply the terms
3x2+2(−x)+1
Multiply the numbers
3x2−2x+1
−y=3x2−2x+1
Change the signs both sides
y=−3x2+2x−1
Solution
Not symmetry with respect to the origin
Show Solution
![solution-arrow](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTciIGhlaWdodD0iMTYiIHZpZXdCb3g9IjAgMCAxNyAxNiIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPHBhdGggZD0iTTguNSAyLjY2Nzk3TDguNSAxMi42NjgiIHN0cm9rZT0id2hpdGUiIHN0cm9rZS13aWR0aD0iMS41IiBzdHJva2UtbGluZWNhcD0icm91bmQiIHN0cm9rZS1saW5lam9pbj0icm91bmQiLz4KPHBhdGggZD0iTTEyLjUgOC42Njc5N0w4LjUgMTIuNjY4TDQuNSA4LjY2Nzk3IiBzdHJva2U9IndoaXRlIiBzdHJva2Utd2lkdGg9IjEuNSIgc3Ryb2tlLWxpbmVjYXA9InJvdW5kIiBzdHJva2UtbGluZWpvaW49InJvdW5kIi8+Cjwvc3ZnPgo=)
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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![arrow-down](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTAiIGhlaWdodD0iMTAiIHZpZXdCb3g9IjAgMCAxMCAxMCIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPGcgb3BhY2l0eT0iMC4zNSI+CjxwYXRoIGQ9Ik0yLjUgMy43NUw1IDYuMjVMNy41IDMuNzUiIHN0cm9rZT0iIzIzMjAyRSIgc3Ryb2tlLXdpZHRoPSIxLjUiIHN0cm9rZS1saW5lY2FwPSJyb3VuZCIgc3Ryb2tlLWxpbmVqb2luPSJyb3VuZCIvPgo8L2c+Cjwvc3ZnPgo=)
(x+31)2=31(y−32)
Evaluate
y=3x2+2x+1
Swap the sides of the equation
3x2+2x+1=y
Move the constant to the right-hand side and change its sign
3x2+2x=y−1
Multiply both sides of the equation by 31
(3x2+2x)×31=(y−1)×31
Multiply the terms
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
(3x2+2x)×31
Use the the distributive property to expand the expression
3x2×31+2x×31
Multiply the numbers
x2+2x×31
Multiply the numbers
x2+32x
x2+32x=(y−1)×31
Multiply the terms
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
(y−1)×31
Apply the distributive property
y×31−31
Use the commutative property to reorder the terms
31y−31
x2+32x=31y−31
To complete the square, the same value needs to be added to both sides
x2+32x+91=31y−31+91
Use a2+2ab+b2=(a+b)2 to factor the expression
(x+31)2=31y−31+91
Add the terms
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
−31+91
Reduce fractions to a common denominator
−3×33+91
Calculate
−93+91
Add or subtract the terms
9−3+1
Calculate
9−2
Calculate
−92
(x+31)2=31y−92
Solution
(x+31)2=31(y−32)
Show Solution
![solution-arrow](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTciIGhlaWdodD0iMTYiIHZpZXdCb3g9IjAgMCAxNyAxNiIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPHBhdGggZD0iTTguNSAyLjY2Nzk3TDguNSAxMi42NjgiIHN0cm9rZT0id2hpdGUiIHN0cm9rZS13aWR0aD0iMS41IiBzdHJva2UtbGluZWNhcD0icm91bmQiIHN0cm9rZS1saW5lam9pbj0icm91bmQiLz4KPHBhdGggZD0iTTEyLjUgOC42Njc5N0w4LjUgMTIuNjY4TDQuNSA4LjY2Nzk3IiBzdHJva2U9IndoaXRlIiBzdHJva2Utd2lkdGg9IjEuNSIgc3Ryb2tlLWxpbmVjYXA9InJvdW5kIiBzdHJva2UtbGluZWpvaW49InJvdW5kIi8+Cjwvc3ZnPgo=)
Solve the equation
x=3−1+−2+3yx=−31+−2+3y
Evaluate
y=3x2+2x+1
Swap the sides of the equation
3x2+2x+1=y
Move the expression to the left side
3x2+2x+1−y=0
Substitute a=3,b=2 and c=1−y into the quadratic formula x=2a−b±b2−4ac
x=2×3−2±22−4×3(1−y)
Simplify the expression
x=6−2±22−4×3(1−y)
Simplify the expression
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
22−4×3(1−y)
Multiply the terms
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Multiply the terms
4×3(1−y)
Multiply the terms
12(1−y)
Apply the distributive property
12−12y
22−(12−12y)
If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses
22−12+12y
Evaluate the power
4−12+12y
Subtract the terms
−8+12y
x=6−2±−8+12y
Simplify the radical expression
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
−8+12y
Factor the expression
4(−2+3y)
The root of a product is equal to the product of the roots of each factor
4×−2+3y
Evaluate the root
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
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
2−2+3y
x=6−2±2−2+3y
Separate the equation into 2 possible cases
x=6−2+2−2+3yx=6−2−2−2+3y
Simplify the expression
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
x=6−2+2−2+3y
Divide the terms
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
6−2+2−2+3y
Rewrite the expression
62(−1+−2+3y)
Reduce the fraction
3−1+−2+3y
x=3−1+−2+3y
x=3−1+−2+3yx=6−2−2−2+3y
Solution
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
x=6−2−2−2+3y
Divide the terms
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![solution-arrow](../../wwwstatic/images/solution-arrow-down-2.svg)
Evaluate
6−2−2−2+3y
Rewrite the expression
62(−1−−2+3y)
Reduce the fraction
3−1−−2+3y
Use b−a=−ba=−ba to rewrite the fraction
−31+−2+3y
x=−31+−2+3y
x=3−1+−2+3yx=−31+−2+3y
Show Solution
![solution-arrow](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTciIGhlaWdodD0iMTYiIHZpZXdCb3g9IjAgMCAxNyAxNiIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPHBhdGggZD0iTTguNSAyLjY2Nzk3TDguNSAxMi42NjgiIHN0cm9rZT0id2hpdGUiIHN0cm9rZS13aWR0aD0iMS41IiBzdHJva2UtbGluZWNhcD0icm91bmQiIHN0cm9rZS1saW5lam9pbj0icm91bmQiLz4KPHBhdGggZD0iTTEyLjUgOC42Njc5N0w4LjUgMTIuNjY4TDQuNSA4LjY2Nzk3IiBzdHJva2U9IndoaXRlIiBzdHJva2Utd2lkdGg9IjEuNSIgc3Ryb2tlLWxpbmVjYXA9InJvdW5kIiBzdHJva2UtbGluZWpvaW49InJvdW5kIi8+Cjwvc3ZnPgo=)
Rewrite the equation
r=6cos2(θ)sin(θ)−2cos(θ)−1−9cos2(θ)−2sin(2θ)r=6cos2(θ)sin(θ)−2cos(θ)+1−9cos2(θ)−2sin(2θ)
Evaluate
y=3x2+2x+1
Move the expression to the left side
y−3x2−2x=1
To convert the equation to polar coordinates,substitute x for rcos(θ) and y for rsin(θ)
sin(θ)×r−3(cos(θ)×r)2−2cos(θ)×r=1
Factor the expression
−3cos2(θ)×r2+(sin(θ)−2cos(θ))r=1
Subtract the terms
−3cos2(θ)×r2+(sin(θ)−2cos(θ))r−1=1−1
Evaluate
−3cos2(θ)×r2+(sin(θ)−2cos(θ))r−1=0
Solve using the quadratic formula
r=−6cos2(θ)−sin(θ)+2cos(θ)±(sin(θ)−2cos(θ))2−4(−3cos2(θ))(−1)
Simplify
r=−6cos2(θ)−sin(θ)+2cos(θ)±1−9cos2(θ)−2sin(2θ)
Separate the equation into 2 possible cases
r=−6cos2(θ)−sin(θ)+2cos(θ)+1−9cos2(θ)−2sin(2θ)r=−6cos2(θ)−sin(θ)+2cos(θ)−1−9cos2(θ)−2sin(2θ)
Evaluate
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
−6cos2(θ)−sin(θ)+2cos(θ)+1−9cos2(θ)−2sin(2θ)
Rewrite the expression
−6cos2(θ)−sin(θ)+2cos(θ)+1−9cos2(θ)−2sin(2θ)
Rewrite the expression
6cos2(θ)sin(θ)−2cos(θ)−1−9cos2(θ)−2sin(2θ)
r=6cos2(θ)sin(θ)−2cos(θ)−1−9cos2(θ)−2sin(2θ)r=−6cos2(θ)−sin(θ)+2cos(θ)−1−9cos2(θ)−2sin(2θ)
Solution
More Steps
![solution-arrow](../../wwwstatic/images/solution-arrow-down-1.svg)
Evaluate
−6cos2(θ)−sin(θ)+2cos(θ)−1−9cos2(θ)−2sin(2θ)
Rewrite the expression
−6cos2(θ)−sin(θ)+2cos(θ)−1−9cos2(θ)−2sin(2θ)
Rewrite the expression
6cos2(θ)sin(θ)−2cos(θ)+1−9cos2(θ)−2sin(2θ)
r=6cos2(θ)sin(θ)−2cos(θ)−1−9cos2(θ)−2sin(2θ)r=6cos2(θ)sin(θ)−2cos(θ)+1−9cos2(θ)−2sin(2θ)
Show Solution
![solution-arrow](data:image/svg+xml;base64,PHN2ZyB3aWR0aD0iMTciIGhlaWdodD0iMTYiIHZpZXdCb3g9IjAgMCAxNyAxNiIgZmlsbD0ibm9uZSIgeG1sbnM9Imh0dHA6Ly93d3cudzMub3JnLzIwMDAvc3ZnIj4KPHBhdGggZD0iTTguNSAyLjY2Nzk3TDguNSAxMi42NjgiIHN0cm9rZT0id2hpdGUiIHN0cm9rZS13aWR0aD0iMS41IiBzdHJva2UtbGluZWNhcD0icm91bmQiIHN0cm9rZS1saW5lam9pbj0icm91bmQiLz4KPHBhdGggZD0iTTEyLjUgOC42Njc5N0w4LjUgMTIuNjY4TDQuNSA4LjY2Nzk3IiBzdHJva2U9IndoaXRlIiBzdHJva2Utd2lkdGg9IjEuNSIgc3Ryb2tlLWxpbmVjYXA9InJvdW5kIiBzdHJva2UtbGluZWpvaW49InJvdW5kIi8+Cjwvc3ZnPgo=)
Graph