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

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Evaluate

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

Multiply the terms

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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}

Add or subtract the terms

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

Calculate

\frac{2}{3}

y = \frac{2}{3}

Solution

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

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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

Evaluate the power

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

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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 terms

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Evaluate

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

Reduce fractions to a common denominator

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

Reorder the terms

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

Add or subtract the terms

\frac{- 3 + 1}{9}

Calculate

\frac{- 2}{9}

Calculate

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

Substitute a= 3,b= 2 and c= 1 - y into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{- 2 \pm 2 ^{2} - 4 \times 3 ( 1 - y )}{2 \times 3}

Simplify the expression

x = \frac{- 2 \pm 2 ^{2} - 4 \times 3 ( 1 - y )}{6}

Simplify the expression

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Evaluate

2^{2} - 4 \times 3 (1 - y)

Multiply the terms

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Multiply the terms

4 \times 3 (1 - y)

Multiply the terms

12 (1 - y)

Apply the distributive property

12 - 12 y

2^{2} - (12 - 12 y)

If a negative sign or a subtraction symbol appears outside parentheses, remove the parentheses and change the sign of every term within the parentheses

2^{2} - 12 + 12 y

Evaluate the power

4 - 12 + 12 y

Subtract the terms

- 8 + 12 y

x = \frac{- 2 \pm - 8 + 12 y}{6}

Simplify the radical expression

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Evaluate

- 8 + 12 y

Factor the expression

4 (- 2 + 3 y)

The root of a product is equal to the product of the roots of each factor

4 \times - 2 + 3 y

Evaluate the root

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Evaluate

4

Write the number in exponential form with the base of 2

2^{2}

Reduce the index of the radical and exponent with 2

2

2 - 2 + 3 y

x = \frac{- 2 \pm 2 - 2 + 3 y}{6}

Separate the equation into 2 possible cases

x = \frac{- 2 + 2 - 2 + 3 y}{6} x = \frac{- 2 - 2 - 2 + 3 y}{6}

Simplify the expression

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Evaluate

x = \frac{- 2 + 2 - 2 + 3 y}{6}

Divide the terms

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Evaluate

\frac{- 2 + 2 - 2 + 3 y}{6}

Rewrite the expression

\frac{2 ( - 1 + - 2 + 3 y )}{6}

Reduce the fraction

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

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

x = \frac{- 1 + - 2 + 3 y}{3} x = \frac{- 2 - 2 - 2 + 3 y}{6}

Solution

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Evaluate

x = \frac{- 2 - 2 - 2 + 3 y}{6}

Divide the terms

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Evaluate

\frac{- 2 - 2 - 2 + 3 y}{6}

Rewrite the expression

\frac{2 ( - 1 - - 2 + 3 y )}{6}

Reduce the fraction

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

Simplify the expression

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

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

x = \frac{- 1 + - 2 + 3 y}{3} x = - \frac{1 + - 2 + 3 y}{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 x for r cos (θ) and y for r sin (θ)

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 )}

Evaluate

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Evaluate

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

Rewrite the expression

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

Rewrite the expression

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

Solution

More Steps

Evaluate

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

Rewrite the expression

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

Rewrite the expression

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

Show Solution

Graph