Solve y=2(x-3)(x-1)

Question

Function

Find the vertex

Find the axis of symmetry

Rewrite in vertex form

(2, - 2)

Evaluate

y = 2 (x - 3) (x - 1)

Write the quadratic function in standard form

y = 2 x^{2} - 8 x + 6

Find the x -coordinate of the vertex by substituting a= 2 and b= - 8 into x = - \frac{b}{2 a}

x = - \frac{- 8}{2 \times 2}

Solve the equation for x

x = 2

Find the y-coordinate of the vertex by evaluating the function for x = 2

y = 2 \times 2^{2} - 8 \times 2 + 6

Calculate

More Steps

Evaluate

2 \times 2^{2} - 8 \times 2 + 6

Calculate the product

2^{3} - 8 \times 2 + 6

Multiply the numbers

2^{3} - 16 + 6

Evaluate the power

8 - 16 + 6

Calculate the sum or difference

- 2

y = - 2

Solution

(2, - 2)

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 = 2 (x - 3) (x - 1)

To test if the graph of y = 2 (x - 3) (x - 1) is symmetry with respect to the origin,substitute -x for x and -y for y

- y = 2 (- x - 3) (- x - 1)

Change the signs both sides

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

Evaluate

y = 2 (x - 3) (x - 1)

Calculate

y = 2 x^{2} - 8 x + 6

Swap the sides of the equation

2 x^{2} - 8 x + 6 = y

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

2 x^{2} - 8 x = y - 6

Multiply both sides of the equation by \frac{1}{2}

(2 x^{2} - 8 x) \times \frac{1}{2} = (y - 6) \times \frac{1}{2}

Multiply the terms

More Steps

Evaluate

(2 x^{2} - 8 x) \times \frac{1}{2}

Use the the distributive property to expand the expression

2 x^{2} \times \frac{1}{2} - 8 x \times \frac{1}{2}

Multiply the numbers

x^{2} - 8 x \times \frac{1}{2}

Multiply the numbers

x^{2} - 4 x

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

Multiply the terms

More Steps

Evaluate

(y - 6) \times \frac{1}{2}

Apply the distributive property

y \times \frac{1}{2} - 6 \times \frac{1}{2}

Use the commutative property to reorder the terms

\frac{1}{2} y - 6 \times \frac{1}{2}

Multiply the numbers

\frac{1}{2} y - 3

x^{2} - 4 x = \frac{1}{2} y - 3

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

x^{2} - 4 x + 4 = \frac{1}{2} y - 3 + 4

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

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

Add the numbers

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

Solution

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

Show Solution

Solve the equation

Solve for x

Solve for y

x = \frac{4 + 4 + 2 y}{2} x = \frac{4 - 4 + 2 y}{2}

Evaluate

y = 2 (x - 3) (x - 1)

Swap the sides of the equation

2 (x - 3) (x - 1) = y

Expand the expression

More Steps

Evaluate

2 (x - 3) (x - 1)

Multiply the terms

More Steps

Evaluate

2 (x - 3)

Apply the distributive property

2 x - 2 \times 3

Multiply the numbers

2 x - 6

(2 x - 6) (x - 1)

Apply the distributive property

2 x \times x - 2 x \times 1 - 6 x - (- 6 \times 1)

Multiply the terms

2 x^{2} - 2 x \times 1 - 6 x - (- 6 \times 1)

Any expression multiplied by 1 remains the same

2 x^{2} - 2 x - 6 x - (- 6 \times 1)

Any expression multiplied by 1 remains the same

2 x^{2} - 2 x - 6 x - (- 6)

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 x^{2} - 2 x - 6 x + 6

Subtract the terms

More Steps

Evaluate

- 2 x - 6 x

Collect like terms by calculating the sum or difference of their coefficients

(- 2 - 6) x

Subtract the numbers

- 8 x

2 x^{2} - 8 x + 6

2 x^{2} - 8 x + 6 = y

Move the expression to the left side

2 x^{2} - 8 x + 6 - y = 0

Substitute a= 2,b= - 8 and c= 6 - y into the quadratic formula x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

x = \frac{8 \pm ( - 8 ) ^{2} - 4 \times 2 ( 6 - y )}{2 \times 2}

Simplify the expression

x = \frac{8 \pm ( - 8 ) ^{2} - 4 \times 2 ( 6 - y )}{4}

Simplify the expression

More Steps

Evaluate

(- 8)^{2} - 4 \times 2 (6 - y)

Multiply the terms

More Steps

Multiply the terms

4 \times 2 (6 - y)

Multiply the terms

8 (6 - y)

Apply the distributive property

8 \times 6 - 8 y

Multiply the numbers

48 - 8 y

(- 8)^{2} - (48 - 8 y)

Rewrite the expression

8^{2} - (48 - 8 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

8^{2} - 48 + 8 y

Evaluate the power

64 - 48 + 8 y

Subtract the numbers

16 + 8 y

x = \frac{8 \pm 16 + 8 y}{4}

Simplify the radical expression

More Steps

Evaluate

16 + 8 y

Factor the expression

8 (2 + y)

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

8 \times 2 + y

Evaluate the root

More Steps

Evaluate

8

Write the expression as a product where the root of one of the factors can be evaluated

4 \times 2

Write the number in exponential form with the base of 2

2^{2} \times 2

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

2^{2} \times 2

Reduce the index of the radical and exponent with 2

22

22 \times 2 + y

Calculate the product

More Steps

Evaluate

2 \times 2 + y

The product of roots with the same index is equal to the root of the product

2 (2 + y)

Calculate the product

4 + 2 y

2 4 + 2 y

x = \frac{8 \pm 2 4 + 2 y}{4}

Separate the equation into 2 possible cases

x = \frac{8 + 2 4 + 2 y}{4} x = \frac{8 - 2 4 + 2 y}{4}

Simplify the expression

More Steps

Evaluate

x = \frac{8 + 2 4 + 2 y}{4}

Divide the terms

More Steps

Evaluate

\frac{8 + 2 4 + 2 y}{4}

Rewrite the expression

\frac{2 ( 4 + 4 + 2 y )}{4}

Cancel out the common factor 2

\frac{4 + 4 + 2 y}{2}

x = \frac{4 + 4 + 2 y}{2}

x = \frac{4 + 4 + 2 y}{2} x = \frac{8 - 2 4 + 2 y}{4}

Solution

More Steps

Evaluate

x = \frac{8 - 2 4 + 2 y}{4}

Divide the terms

More Steps

Evaluate

\frac{8 - 2 4 + 2 y}{4}

Rewrite the expression

\frac{2 ( 4 - 4 + 2 y )}{4}

Cancel out the common factor 2

\frac{4 - 4 + 2 y}{2}

x = \frac{4 - 4 + 2 y}{2}

x = \frac{4 + 4 + 2 y}{2} x = \frac{4 - 4 + 2 y}{2}

Show Solution

Rewrite the equation

r = \frac{sin ( θ ) + 8 cos ( θ ) - 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{4 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 8 cos ( θ ) + 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{4 cos ^{2} ( θ )}

Evaluate

y = 2 (x - 3) (x - 1)

Move the expression to the left side

y - 2 x^{2} + 8 x = 6

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

sin (θ) \times r - 2 (cos (θ) \times r)^{2} + 8 cos (θ) \times r = 6

Factor the expression

- 2 cos^{2} (θ) \times r^{2} + (sin (θ) + 8 cos (θ)) r = 6

Subtract the terms

- 2 cos^{2} (θ) \times r^{2} + (sin (θ) + 8 cos (θ)) r - 6 = 6 - 6

Evaluate

- 2 cos^{2} (θ) \times r^{2} + (sin (θ) + 8 cos (θ)) r - 6 = 0

Solve using the quadratic formula

r = \frac{- sin ( θ ) - 8 cos ( θ ) \pm ( sin ( θ ) + 8 cos ( θ ) ) ^{2} - 4 ( - 2 cos ^{2} ( θ ) ) ( - 6 )}{- 4 cos ^{2} ( θ )}

Simplify

r = \frac{- sin ( θ ) - 8 cos ( θ ) \pm 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{- 4 cos ^{2} ( θ )}

Separate the equation into 2 possible cases

r = \frac{- sin ( θ ) - 8 cos ( θ ) + 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{- 4 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 8 cos ( θ ) - 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{- 4 cos ^{2} ( θ )}

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

r = \frac{sin ( θ ) + 8 cos ( θ ) - 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{4 cos ^{2} ( θ )} r = \frac{- sin ( θ ) - 8 cos ( θ ) - 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{- 4 cos ^{2} ( θ )}

Solution

r = \frac{sin ( θ ) + 8 cos ( θ ) - 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{4 cos ^{2} ( θ )} r = \frac{sin ( θ ) + 8 cos ( θ ) + 1 + 15 cos ^{2} ( θ ) + 8 sin ( 2 θ )}{4 cos ^{2} ( θ )}

Show Solution

Function

Testing for symmetry

Identify the conic

Solve the equation

Rewrite the equation

Graph