Precalculus Examples

$y = - (x - 3) (x + 1)$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for $- (x - 3) (x + 1)$ .

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Step 1.1.1

Simplify the expression.

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Step 1.1.1.1

Apply the distributive property.

$(- x - - 3) (x + 1)$

Step 1.1.1.2

Multiply $- 1$ by $- 3$ .

$(- x + 3) (x + 1)$

Step 1.1.1.3

Expand $(- x + 3) (x + 1)$ using the FOIL Method.

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Step 1.1.1.3.1

Apply the distributive property.

$- x (x + 1) + 3 (x + 1)$

Step 1.1.1.3.2

Apply the distributive property.

$- x \cdot x - x \cdot 1 + 3 (x + 1)$

Step 1.1.1.3.3

Apply the distributive property.

$- x \cdot x - x \cdot 1 + 3 x + 3 \cdot 1$

Step 1.1.1.4

Simplify and combine like terms.

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Step 1.1.1.4.1

Simplify each term.

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Step 1.1.1.4.1.1

Multiply $x$ by $x$ by adding the exponents.

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Step 1.1.1.4.1.1.1

Move $x$ .

$- (x \cdot x) - x \cdot 1 + 3 x + 3 \cdot 1$

Step 1.1.1.4.1.1.2

Multiply $x$ by $x$ .

$- x^{2} - x \cdot 1 + 3 x + 3 \cdot 1$

Step 1.1.1.4.1.2

Multiply $- 1$ by $1$ .

$- x^{2} - x + 3 x + 3 \cdot 1$

Step 1.1.1.4.1.3

Multiply $3$ by $1$ .

$- x^{2} - x + 3 x + 3$

Step 1.1.1.4.2

Add $- x$ and $3 x$ .

$- x^{2} + 2 x + 3$

Step 1.1.2

Use the form $a x^{2} + b x + c$ , to find the values of $a$ , $b$ , and $c$ .

$a = - 1$

$b = 2$

$c = 3$

Step 1.1.3

Consider the vertex form of a parabola.

$a {(x + d)}^{2} + e$

Step 1.1.4

Find the value of $d$ using the formula $d = \frac{b}{2 a}$ .

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Step 1.1.4.1

Substitute the values of $a$ and $b$ into the formula $d = \frac{b}{2 a}$ .

$d = \frac{2}{2 \cdot - 1}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Cancel the common factor of $2$ .

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Step 1.1.4.2.1.1

Cancel the common factor.

$d = \frac{2}{2 \cdot - 1}$

Step 1.1.4.2.1.2

Rewrite the expression.

$d = \frac{1}{- 1}$

Step 1.1.4.2.1.3

Move the negative one from the denominator of $\frac{1}{- 1}$ .

$d = - 1 \cdot 1$

Step 1.1.4.2.2

Multiply $- 1$ by $1$ .

$d = - 1$

Step 1.1.5

Find the value of $e$ using the formula $e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.5.1

Substitute the values of $c$ , $b$ and $a$ into the formula $e = c - \frac{b^{2}}{4 a}$ .

$e = 3 - \frac{2^{2}}{4 \cdot - 1}$

Step 1.1.5.2

Simplify the right side.

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Step 1.1.5.2.1

Simplify each term.

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Step 1.1.5.2.1.1

Raise $2$ to the power of $2$ .

$e = 3 - \frac{4}{4 \cdot - 1}$

Step 1.1.5.2.1.2

Multiply $4$ by $- 1$ .

$e = 3 - \frac{4}{- 4}$

Step 1.1.5.2.1.3

Divide $4$ by $- 4$ .

$e = 3 - - 1$

Step 1.1.5.2.1.4

Multiply $- 1$ by $- 1$ .

$e = 3 + 1$

Step 1.1.5.2.2

Add $3$ and $1$ .

$e = 4$

Step 1.1.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- {(x - 1)}^{2} + 4$ .

$- {(x - 1)}^{2} + 4$

Step 1.2

Set $y$ equal to the new right side.

$y = - {(x - 1)}^{2} + 4$

Step 2

Use the vertex form, $y = a {(x - h)}^{2} + k$ , to determine the values of $a$ , $h$ , and $k$ .

$a = - 1$

$h = 1$

$k = 4$

Step 3

Find the vertex $(h, k)$ .

$(1, 4)$

Step 4