Precalculus Examples

${(x - 2)}^{2} = 24 (y - 1)$

Step 1

Rewrite the equation in vertex form.

Tap for more steps...

Step 1.1

Isolate $y$ to the left side of the equation.

Tap for more steps...

Step 1.1.1

Rewrite the equation as $24 (y - 1) = {(x - 2)}^{2}$ .

$24 (y - 1) = {(x - 2)}^{2}$

Step 1.1.2

Divide each term in $24 (y - 1) = {(x - 2)}^{2}$ by $24$ and simplify.

Tap for more steps...

Step 1.1.2.1

Divide each term in $24 (y - 1) = {(x - 2)}^{2}$ by $24$ .

$\frac{24 (y - 1)}{24} = \frac{{(x - 2)}^{2}}{24}$

Step 1.1.2.2

Simplify the left side.

Tap for more steps...

Step 1.1.2.2.1

Cancel the common factor of $24$ .

Tap for more steps...

Step 1.1.2.2.1.1

Cancel the common factor.

$\frac{24 (y - 1)}{24} = \frac{{(x - 2)}^{2}}{24}$

Step 1.1.2.2.1.2

Divide $y - 1$ by $1$ .

$y - 1 = \frac{{(x - 2)}^{2}}{24}$

Step 1.1.3

Add $1$ to both sides of the equation.

$y = \frac{{(x - 2)}^{2}}{24} + 1$

Step 1.1.4

Reorder terms.

$y = \frac{1}{24} \cdot {(x - 2)}^{2} + 1$

Step 1.2

Complete the square for $\frac{1}{24} \cdot {(x - 2)}^{2} + 1$ .

Tap for more steps...

Step 1.2.1

Simplify the expression.

Tap for more steps...

Step 1.2.1.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1.1

Rewrite ${(x - 2)}^{2}$ as $(x - 2) (x - 2)$ .

$\frac{1}{24} \cdot ((x - 2) (x - 2)) + 1$

Step 1.2.1.1.2

Expand $(x - 2) (x - 2)$ using the FOIL Method.

Tap for more steps...

Step 1.2.1.1.2.1

Apply the distributive property.

$\frac{1}{24} \cdot (x (x - 2) - 2 (x - 2)) + 1$

Step 1.2.1.1.2.2

Apply the distributive property.

$\frac{1}{24} \cdot (x \cdot x + x \cdot - 2 - 2 (x - 2)) + 1$

Step 1.2.1.1.2.3

Apply the distributive property.

$\frac{1}{24} \cdot (x \cdot x + x \cdot - 2 - 2 x - 2 \cdot - 2) + 1$

Step 1.2.1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 1.2.1.1.3.1

Simplify each term.

Tap for more steps...

Step 1.2.1.1.3.1.1

Multiply $x$ by $x$ .

$\frac{1}{24} \cdot (x^{2} + x \cdot - 2 - 2 x - 2 \cdot - 2) + 1$

Step 1.2.1.1.3.1.2

Move $- 2$ to the left of $x$ .

$\frac{1}{24} \cdot (x^{2} - 2 \cdot x - 2 x - 2 \cdot - 2) + 1$

Step 1.2.1.1.3.1.3

Multiply $- 2$ by $- 2$ .

$\frac{1}{24} \cdot (x^{2} - 2 x - 2 x + 4) + 1$

Step 1.2.1.1.3.2

Subtract $2 x$ from $- 2 x$ .

$\frac{1}{24} \cdot (x^{2} - 4 x + 4) + 1$

Step 1.2.1.1.4

Apply the distributive property.

$\frac{1}{24} x^{2} + \frac{1}{24} (- 4 x) + \frac{1}{24} \cdot 4 + 1$

Step 1.2.1.1.5

Simplify.

Tap for more steps...

Step 1.2.1.1.5.1

Combine $\frac{1}{24}$ and $x^{2}$ .

$\frac{x^{2}}{24} + \frac{1}{24} (- 4 x) + \frac{1}{24} \cdot 4 + 1$

Step 1.2.1.1.5.2

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.1.1.5.2.1

Factor $4$ out of $24$ .

$\frac{x^{2}}{24} + \frac{1}{4 (6)} (- 4 x) + \frac{1}{24} \cdot 4 + 1$

Step 1.2.1.1.5.2.2

Factor $4$ out of $- 4 x$ .

$\frac{x^{2}}{24} + \frac{1}{4 (6)} (4 (- x)) + \frac{1}{24} \cdot 4 + 1$

Step 1.2.1.1.5.2.3

Cancel the common factor.

$\frac{x^{2}}{24} + \frac{1}{4 \cdot 6} (4 (- x)) + \frac{1}{24} \cdot 4 + 1$

Step 1.2.1.1.5.2.4

Rewrite the expression.

$\frac{x^{2}}{24} + \frac{1}{6} (- x) + \frac{1}{24} \cdot 4 + 1$

Step 1.2.1.1.5.3

Combine $\frac{1}{6}$ and $x$ .

$\frac{x^{2}}{24} - \frac{x}{6} + \frac{1}{24} \cdot 4 + 1$

Step 1.2.1.1.5.4

Cancel the common factor of $4$ .

Tap for more steps...

Step 1.2.1.1.5.4.1

Factor $4$ out of $24$ .

$\frac{x^{2}}{24} - \frac{x}{6} + \frac{1}{4 (6)} \cdot 4 + 1$

Step 1.2.1.1.5.4.2

Cancel the common factor.

$\frac{x^{2}}{24} - \frac{x}{6} + \frac{1}{4 \cdot 6} \cdot 4 + 1$

Step 1.2.1.1.5.4.3

Rewrite the expression.

$\frac{x^{2}}{24} - \frac{x}{6} + \frac{1}{6} + 1$

Step 1.2.1.2

Write $1$ as a fraction with a common denominator.

$\frac{x^{2}}{24} - \frac{x}{6} + \frac{1}{6} + \frac{6}{6}$

Step 1.2.1.3

Combine the numerators over the common denominator.

$\frac{x^{2}}{24} - \frac{x}{6} + \frac{1 + 6}{6}$

Step 1.2.1.4

Add $1$ and $6$ .

$\frac{x^{2}}{24} - \frac{x}{6} + \frac{7}{6}$

Step 1.2.2

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

$a = \frac{1}{24}$

$b = - \frac{1}{6}$

$c = \frac{7}{6}$

Step 1.2.3

Consider the vertex form of a parabola.

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

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$d = \frac{- \frac{1}{6}}{2 (\frac{1}{24})}$

Step 1.2.4.2

Simplify the right side.

Tap for more steps...

Step 1.2.4.2.1

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{6} \cdot \frac{1}{2 (\frac{1}{24})}$

Step 1.2.4.2.2

Combine $2$ and $\frac{1}{24}$ .

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2}{24}}$

Step 1.2.4.2.3

Cancel the common factor of $2$ and $24$ .

Tap for more steps...

Step 1.2.4.2.3.1

Factor $2$ out of $2$ .

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2 (1)}{24}}$

Step 1.2.4.2.3.2

Cancel the common factors.

Tap for more steps...

Step 1.2.4.2.3.2.1

Factor $2$ out of $24$ .

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Step 1.2.4.2.3.2.2

Cancel the common factor.

$d = - \frac{1}{6} \cdot \frac{1}{\frac{2 \cdot 1}{2 \cdot 12}}$

Step 1.2.4.2.3.2.3

Rewrite the expression.

$d = - \frac{1}{6} \cdot \frac{1}{\frac{1}{12}}$

Step 1.2.4.2.4

Multiply the numerator by the reciprocal of the denominator.

$d = - \frac{1}{6} (1 \cdot 12)$

Step 1.2.4.2.5

Multiply $12$ by $1$ .

$d = - \frac{1}{6} \cdot 12$

Step 1.2.4.2.6

Cancel the common factor of $6$ .

Tap for more steps...

Step 1.2.4.2.6.1

Move the leading negative in $- \frac{1}{6}$ into the numerator.

$d = \frac{- 1}{6} \cdot 12$

Step 1.2.4.2.6.2

Factor $6$ out of $12$ .

$d = \frac{- 1}{6} \cdot (6 (2))$

Step 1.2.4.2.6.3

Cancel the common factor.

$d = \frac{- 1}{6} \cdot (6 \cdot 2)$

Step 1.2.4.2.6.4

Rewrite the expression.

$d = - 1 \cdot 2$

Step 1.2.4.2.7

Multiply $- 1$ by $2$ .

$d = - 2$

Step 1.2.5

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

Tap for more steps...

Step 1.2.5.1

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

$e = \frac{7}{6} - \frac{{(- \frac{1}{6})}^{2}}{4 (\frac{1}{24})}$

Step 1.2.5.2

Simplify the right side.

Tap for more steps...

Step 1.2.5.2.1

Simplify each term.

Tap for more steps...

Step 1.2.5.2.1.1

Simplify the numerator.

Tap for more steps...

Step 1.2.5.2.1.1.1

Apply the product rule to $- \frac{1}{6}$ .

$e = \frac{7}{6} - \frac{{(- 1)}^{2} {(\frac{1}{6})}^{2}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.1.2

Raise $- 1$ to the power of $2$ .

$e = \frac{7}{6} - \frac{1 {(\frac{1}{6})}^{2}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.1.3

Apply the product rule to $\frac{1}{6}$ .

$e = \frac{7}{6} - \frac{1 \frac{1^{2}}{6^{2}}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.1.4

One to any power is one.

$e = \frac{7}{6} - \frac{1 \frac{1}{6^{2}}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.1.5

Raise $6$ to the power of $2$ .

$e = \frac{7}{6} - \frac{1 (\frac{1}{36})}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.1.6

Multiply $\frac{1}{36}$ by $1$ .

$e = \frac{7}{6} - \frac{\frac{1}{36}}{4 (\frac{1}{24})}$

Step 1.2.5.2.1.2

Combine $4$ and $\frac{1}{24}$ .

$e = \frac{7}{6} - \frac{\frac{1}{36}}{\frac{4}{24}}$

Step 1.2.5.2.1.3

Cancel the common factor of $4$ and $24$ .

Tap for more steps...

Step 1.2.5.2.1.3.1

Factor $4$ out of $4$ .

$e = \frac{7}{6} - \frac{\frac{1}{36}}{\frac{4 (1)}{24}}$

Step 1.2.5.2.1.3.2

Cancel the common factors.

Tap for more steps...

Step 1.2.5.2.1.3.2.1

Factor $4$ out of $24$ .

$e = \frac{7}{6} - \frac{\frac{1}{36}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 1.2.5.2.1.3.2.2

Cancel the common factor.

$e = \frac{7}{6} - \frac{\frac{1}{36}}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 1.2.5.2.1.3.2.3

Rewrite the expression.

$e = \frac{7}{6} - \frac{\frac{1}{36}}{\frac{1}{6}}$

Step 1.2.5.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$e = \frac{7}{6} - (\frac{1}{36} \cdot 6)$

Step 1.2.5.2.1.5

Cancel the common factor of $6$ .

Tap for more steps...

Step 1.2.5.2.1.5.1

Factor $6$ out of $36$ .

$e = \frac{7}{6} - (\frac{1}{6 (6)} \cdot 6)$

Step 1.2.5.2.1.5.2

Cancel the common factor.

$e = \frac{7}{6} - (\frac{1}{6 \cdot 6} \cdot 6)$

Step 1.2.5.2.1.5.3

Rewrite the expression.

$e = \frac{7}{6} - \frac{1}{6}$

Step 1.2.5.2.2

Combine the numerators over the common denominator.

$e = \frac{7 - 1}{6}$

Step 1.2.5.2.3

Subtract $1$ from $7$ .

$e = \frac{6}{6}$

Step 1.2.5.2.4

Divide $6$ by $6$ .

$e = 1$

Step 1.2.6

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $\frac{1}{24} {(x - 2)}^{2} + 1$ .

$\frac{1}{24} {(x - 2)}^{2} + 1$

Step 1.3

Set $y$ equal to the new right side.

$y = \frac{1}{24} \cdot {(x - 2)}^{2} + 1$

Step 2

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

$a = \frac{1}{24}$

$h = 2$

$k = 1$

Step 3

Find the vertex $(h, k)$ .

$(2, 1)$

Step 4

Find $p$ , the distance from the vertex to the focus.

Tap for more steps...

Step 4.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

$\frac{1}{4 a}$

Step 4.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot \frac{1}{24}}$

Step 4.3

Simplify.

Tap for more steps...

Step 4.3.1

Combine $4$ and $\frac{1}{24}$ .

$\frac{1}{\frac{4}{24}}$

Step 4.3.2

Cancel the common factor of $4$ and $24$ .

Tap for more steps...

Step 4.3.2.1

Factor $4$ out of $4$ .

$\frac{1}{\frac{4 (1)}{24}}$

Step 4.3.2.2

Cancel the common factors.

Tap for more steps...

Step 4.3.2.2.1

Factor $4$ out of $24$ .

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 4.3.2.2.2

Cancel the common factor.

$\frac{1}{\frac{4 \cdot 1}{4 \cdot 6}}$

Step 4.3.2.2.3

Rewrite the expression.

$\frac{1}{\frac{1}{6}}$

Step 4.3.3

Multiply the numerator by the reciprocal of the denominator.

$1 \cdot 6$

Step 4.3.4

Multiply $6$ by $1$ .

$6$

Step 5

Find the focus.

Tap for more steps...

Step 5.1

The focus of a parabola can be found by adding $p$ to the y-coordinate $k$ if the parabola opens up or down.

$(h, k + p)$

Step 5.2

Substitute the known values of $h$ , $p$ , and $k$ into the formula and simplify.

$(2, 7)$

Step 6