Algebra Examples

f (x) = a {(x - h)}^{2} + k

$f (x) = a {(x - h)}^{2} + k$

Step 1

Rewrite the function as an equation.

y = a {(x - h)}^{2} + k

$y = a {(x - h)}^{2} + k$

Step 2

Simplify

a {(x - h)}^{2} + k

$a {(x - h)}^{2} + k$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite

{(x - h)}^{2}

${(x - h)}^{2}$ as

(x - h) (x - h)

$(x - h) (x - h)$ .

y = a ((x - h) (x - h)) + k

$y = a ((x - h) (x - h)) + k$

Step 2.1.2

Expand

(x - h) (x - h)

$(x - h) (x - h)$ using the FOIL Method.

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Step 2.1.2.1

Apply the distributive property.

y = a (x (x - h) - h (x - h)) + k

$y = a (x (x - h) - h (x - h)) + k$

Step 2.1.2.2

Apply the distributive property.

y = a (x \cdot x + x (- h) - h (x - h)) + k

$y = a (x \cdot x + x (- h) - h (x - h)) + k$

Step 2.1.2.3

Apply the distributive property.

y = a (x \cdot x + x (- h) - h x - h (- h)) + k

$y = a (x \cdot x + x (- h) - h x - h (- h)) + k$

y = a (x \cdot x + x (- h) - h x - h (- h)) + k

$y = a (x \cdot x + x (- h) - h x - h (- h)) + k$

Step 2.1.3

Simplify and combine like terms.

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Step 2.1.3.1

Simplify each term.

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Step 2.1.3.1.1

Multiply

x

$x$ by

x

$x$ .

y = a (x^{2} + x (- h) - h x - h (- h)) + k

$y = a (x^{2} + x (- h) - h x - h (- h)) + k$

Step 2.1.3.1.2

Rewrite using the commutative property of multiplication.

y = a (x^{2} - x h - h x - h (- h)) + k

$y = a (x^{2} - x h - h x - h (- h)) + k$

Step 2.1.3.1.3

Rewrite using the commutative property of multiplication.

y = a (x^{2} - x h - h x - 1 \cdot - 1 h \cdot h) + k

$y = a (x^{2} - x h - h x - 1 \cdot - 1 h \cdot h) + k$

Step 2.1.3.1.4

Multiply

h

$h$ by

h

$h$ by adding the exponents.

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Step 2.1.3.1.4.1

Move

h

$h$ .

y = a (x^{2} - x h - h x - 1 \cdot - 1 (h \cdot h)) + k

$y = a (x^{2} - x h - h x - 1 \cdot - 1 (h \cdot h)) + k$

Step 2.1.3.1.4.2

Multiply

h

$h$ by

h

$h$ .

y = a (x^{2} - x h - h x - 1 \cdot - 1 h^{2}) + k

$y = a (x^{2} - x h - h x - 1 \cdot - 1 h^{2}) + k$

y = a (x^{2} - x h - h x - 1 \cdot - 1 h^{2}) + k

$y = a (x^{2} - x h - h x - 1 \cdot - 1 h^{2}) + k$

Step 2.1.3.1.5

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

y = a (x^{2} - x h - h x + 1 h^{2}) + k

$y = a (x^{2} - x h - h x + 1 h^{2}) + k$

Step 2.1.3.1.6

Multiply

h^{2}

$h^{2}$ by

1

$1$ .

y = a (x^{2} - x h - h x + h^{2}) + k

$y = a (x^{2} - x h - h x + h^{2}) + k$

Step 2.1.3.2

Subtract

$h x$ from

$- x h$ .

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Step 2.1.3.2.1

Move

$x$ .

$y = a (x^{2} - h x - h x + h^{2}) + k$

Step 2.1.3.2.2

Subtract

$h x$ from

$- h x$ .

$y = a (x^{2} - 2 h x + h^{2}) + k$

Step 2.1.4

Apply the distributive property.

$y = a x^{2} + a (- 2 h x) + a h^{2} + k$

Step 2.1.5

Rewrite using the commutative property of multiplication.

$y = a x^{2} - 2 a h x + a h^{2} + k$

Step 2.2

Simplify the expression.

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Step 2.2.1

Move

$a x^{2}$ .

$y = - 2 a h x + a h^{2} + a x^{2} + k$

Step 2.2.2

Reorder

$- 2 a h x$ and

$a h^{2}$ .

$y = a h^{2} - 2 a h x + a x^{2} + k$