Solve y=3x^2+2x+1 - Socratic AI (ScanMath)

Pergunta

Function

Find the vertex

Find the axis of symmetry

Rewrite in vertex form

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(- \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

Mais Passos

Evaluate

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

Multiply the terms

Mais Passos

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

Mais Passos

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}

Solução

(- \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

N \overset{a}{˜} o h \overset{a}{ˊ} simetria em rela \overset{c}{¸} \overset{a}{˜} o \overset{a}{ˋ} origem.

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

Mais Passos

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

Solução

N \overset{a}{˜} o h \overset{a}{ˊ} simetria em rela \overset{c}{¸} \overset{a}{˜} o \overset{a}{ˋ} origem.

Mostrar solução

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 + \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

Mais Passos

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

Mais Passos

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

Mais Passos

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}

Solução

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

Mostrar solução

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

Mais Passos

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

Mais Passos

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

Mais Passos

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}

Solução

Mais Passos

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

Mais Passos

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}

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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 )}

Solução

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 )}

Mostrar solução

Gráfico

Y=3x^2+2x+1

Function

Testing for symmetry

Identify the conic

Solve the equation

Rewrite the equation

Gráfico