Solve y=3x^2+2x+1

Question

Function

Find the vertex

Find the axis of symmetry

Rewrite in vertex form

(- \frac{1}{3}, \frac{2}{3})

Evaluate

y = 3 x^{2} + 2 x + 1

Find the x -coordinate of the vertex by substituting a= 3 and b= 2 into x = - \frac{b}{2 a}

x = - \frac{2}{2 \times 3}

Solve the equation for x

x = - \frac{1}{3}

Find the y-coordinate of the vertex by evaluating the function for x = - \frac{1}{3}

y = 3 (- \frac{1}{3})^{2} + 2 (- \frac{1}{3}) + 1

Calculate

More Steps

Evaluate

3 (- \frac{1}{3})^{2} + 2 (- \frac{1}{3}) + 1

Multiply the terms

More Steps

Evaluate

3 (- \frac{1}{3})^{2}

Evaluate the power

3 \times \frac{1}{9}

Multiply the numbers

\frac{1}{3}

\frac{1}{3} + 2 (- \frac{1}{3}) + 1

Multiply the numbers

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Evaluate

2 (- \frac{1}{3})

Multiplying or dividing an odd number of negative terms equals a negative

- 2 \times \frac{1}{3}

Multiply the numbers

- \frac{2}{3}

\frac{1}{3} - \frac{2}{3} + 1

Reduce fractions to a common denominator

\frac{1}{3} - \frac{2}{3} + \frac{3}{3}

Write all numerators above the common denominator

\frac{1 - 2 + 3}{3}

Calculate the sum or difference

\frac{2}{3}

y = \frac{2}{3}

Solution

(- \frac{1}{3}, \frac{2}{3})

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 = 3 x^{2} + 2 x + 1

To test if the graph of y = 3 x^{2} + 2 x + 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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Evaluate

3 (- x)^{2} + 2 (- x) + 1

Multiply the terms

3 x^{2} + 2 (- x) + 1

Multiply the numbers

3 x^{2} - 2 x + 1

- y = 3 x^{2} - 2 x + 1

Change the signs both sides

y = - 3 x^{2} + 2 x - 1

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

(x + \frac{1}{3})^{2} = \frac{1}{3} (y - \frac{2}{3})

Evaluate

y = 3 x^{2} + 2 x + 1

Swap the sides of the equation

3 x^{2} + 2 x + 1 = y

Move the constant to the right-hand side and change its sign

3 x^{2} + 2 x = y - 1

Multiply both sides of the equation by \frac{1}{3}

(3 x^{2} + 2 x) \times \frac{1}{3} = (y - 1) \times \frac{1}{3}

Multiply the terms

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Evaluate

(3 x^{2} + 2 x) \times \frac{1}{3}

Use the the distributive property to expand the expression

3 x^{2} \times \frac{1}{3} + 2 x \times \frac{1}{3}

Multiply the numbers

x^{2} + 2 x \times \frac{1}{3}

Multiply the numbers

x^{2} + \frac{2}{3} x

x^{2} + \frac{2}{3} x = (y - 1) \times \frac{1}{3}

Multiply the terms

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Evaluate

(y - 1) \times \frac{1}{3}

Apply the distributive property

y \times \frac{1}{3} - \frac{1}{3}

Use the commutative property to reorder the terms

\frac{1}{3} y - \frac{1}{3}

x^{2} + \frac{2}{3} x = \frac{1}{3} y - \frac{1}{3}

To complete the square, the same value needs to be added to both sides

x^{2} + \frac{2}{3} x + \frac{1}{9} = \frac{1}{3} y - \frac{1}{3} + \frac{1}{9}

Use a^{2} + 2 ab + b^{2} = (a + b)^{2} to factor the expression

(x + \frac{1}{3})^{2} = \frac{1}{3} y - \frac{1}{3} + \frac{1}{9}

Add the numbers

More Steps

Evaluate

- \frac{1}{3} + \frac{1}{9}

Reduce fractions to a common denominator

- \frac{3}{3 \times 3} + \frac{1}{9}

Multiply the numbers

- \frac{3}{9} + \frac{1}{9}

Write all numerators above the common denominator

\frac{- 3 + 1}{9}

Add the numbers

\frac{- 2}{9}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{2}{9}

(x + \frac{1}{3})^{2} = \frac{1}{3} y - \frac{2}{9}

Solution

(x + \frac{1}{3})^{2} = \frac{1}{3} (y - \frac{2}{3})

Show Solution

Solve the equation

x = \frac{- 2 + 3 y - 1}{3} x = - \frac{- 2 + 3 y + 1}{3}

Evaluate

y = 3 x^{2} + 2 x + 1

Swap the sides of the equation

3 x^{2} + 2 x + 1 = y

Move the expression to the left side

3 x^{2} + 2 x + 1 - y = 0

Move the constant to the right side

3 x^{2} + 2 x = 0 - (1 - y)

Add the terms

3 x^{2} + 2 x = - 1 + y

Evaluate

x^{2} + \frac{2}{3} x = \frac{- 1 + y}{3}

Add the same value to both sides

x^{2} + \frac{2}{3} x + \frac{1}{9} = \frac{- 1 + y}{3} + \frac{1}{9}

Evaluate

x^{2} + \frac{2}{3} x + \frac{1}{9} = \frac{- 2 + 3 y}{9}

Evaluate

(x + \frac{1}{3})^{2} = \frac{- 2 + 3 y}{9}

Take the root of both sides of the equation and remember to use both positive and negative roots

x + \frac{1}{3} = \pm \frac{- 2 + 3 y}{9}

Simplify the expression

More Steps

Evaluate

\frac{- 2 + 3 y}{9}

To take a root of a fraction,take the root of the numerator and denominator separately

\frac{- 2 + 3 y}{9}

Simplify the radical expression

More Steps

Evaluate

9

Write the number in exponential form with the base of 3

3^{2}

Reduce the index of the radical and exponent with 2

3

\frac{- 2 + 3 y}{3}

x + \frac{1}{3} = \pm \frac{- 2 + 3 y}{3}

Separate the equation into 2 possible cases

x + \frac{1}{3} = \frac{- 2 + 3 y}{3} x + \frac{1}{3} = - \frac{- 2 + 3 y}{3}

Calculate

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Evaluate

x + \frac{1}{3} = \frac{- 2 + 3 y}{3}

Move the constant to the right-hand side and change its sign

x = \frac{- 2 + 3 y}{3} - \frac{1}{3}

Write all numerators above the common denominator

x = \frac{- 2 + 3 y - 1}{3}

x = \frac{- 2 + 3 y - 1}{3} x + \frac{1}{3} = - \frac{- 2 + 3 y}{3}

Solution

More Steps

Evaluate

x + \frac{1}{3} = - \frac{- 2 + 3 y}{3}

Move the constant to the right-hand side and change its sign

x = - \frac{- 2 + 3 y}{3} - \frac{1}{3}

Subtract the terms

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Evaluate

- \frac{- 2 + 3 y}{3} - \frac{1}{3}

Write all numerators above the common denominator

\frac{- - 2 + 3 y - 1}{3}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

- \frac{- 2 + 3 y + 1}{3}

x = - \frac{- 2 + 3 y + 1}{3}

x = \frac{- 2 + 3 y - 1}{3} x = - \frac{- 2 + 3 y + 1}{3}

Show Solution

Rewrite the equation

r = \frac{sin ( θ ) - 2 cos ( θ ) - 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{6 cos ^{2} ( θ )} r = \frac{sin ( θ ) - 2 cos ( θ ) + 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{6 cos ^{2} ( θ )}

Evaluate

y = 3 x^{2} + 2 x + 1

Move the expression to the left side

y - 3 x^{2} - 2 x = 1

To convert the equation to polar coordinates,substitute r cos (θ) for x and r sin (θ) for y

sin (θ) \times r - 3 (cos (θ) \times r)^{2} - 2 cos (θ) \times r = 1

Factor the expression

- 3 cos^{2} (θ) \times r^{2} + (sin (θ) - 2 cos (θ)) r = 1

Subtract the terms

- 3 cos^{2} (θ) \times r^{2} + (sin (θ) - 2 cos (θ)) r - 1 = 1 - 1

Evaluate

- 3 cos^{2} (θ) \times r^{2} + (sin (θ) - 2 cos (θ)) r - 1 = 0

Solve using the quadratic formula

r = \frac{- sin ( θ ) + 2 cos ( θ ) \pm ( sin ( θ ) - 2 cos ( θ ) ) ^{2} - 4 ( - 3 cos ^{2} ( θ ) ) ( - 1 )}{- 6 cos ^{2} ( θ )}

Simplify

r = \frac{- sin ( θ ) + 2 cos ( θ ) \pm 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{- 6 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- sin ( θ ) + 2 cos ( θ ) + 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{- 6 cos ^{2} ( θ )} r = \frac{- sin ( θ ) + 2 cos ( θ ) - 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{- 6 cos ^{2} ( θ )}

Use \frac{- a}{b} = \frac{a}{- b} = - \frac{a}{b} to rewrite the fraction

r = \frac{sin ( θ ) - 2 cos ( θ ) - 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{6 cos ^{2} ( θ )} r = \frac{- sin ( θ ) + 2 cos ( θ ) - 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{- 6 cos ^{2} ( θ )}

Solution

r = \frac{sin ( θ ) - 2 cos ( θ ) - 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{6 cos ^{2} ( θ )} r = \frac{sin ( θ ) - 2 cos ( θ ) + 1 - 9 cos ^{2} ( θ ) - 2 sin ( 2 θ )}{6 cos ^{2} ( θ )}

Show Solution

Y=3x^2+2x+1

Function

Testing for symmetry

Identify the conic

Solve the equation

Rewrite the equation

Graph