Pre-Algebra Examples

${(y - \frac{33}{10})}^{2} = \frac{4}{10} \cdot (x + \frac{1225}{1000})$

Step 1

Rewrite the equation as $\frac{4}{10} \cdot (x + \frac{1225}{1000}) = {(y - \frac{33}{10})}^{2}$ .

$\frac{4}{10} \cdot (x + \frac{1225}{1000}) = {(y - \frac{33}{10})}^{2}$

Step 2

Multiply both sides of the equation by $\frac{10}{4}$ .

$\frac{10}{4} (\frac{4}{10} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3

Simplify both sides of the equation.

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Step 3.1

Simplify the left side.

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Step 3.1.1

Simplify $\frac{10}{4} (\frac{4}{10} \cdot (x + \frac{1225}{1000}))$ .

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Step 3.1.1.1

Cancel the common factor of $10$ and $4$ .

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Step 3.1.1.1.1

Factor $2$ out of $10$ .

$\frac{2 (5)}{4} (\frac{4}{10} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.1.2

Cancel the common factors.

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Step 3.1.1.1.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 5}{2 \cdot 2} (\frac{4}{10} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.1.2.2

Cancel the common factor.

$\frac{2 \cdot 5}{2 \cdot 2} (\frac{4}{10} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.1.2.3

Rewrite the expression.

$\frac{5}{2} (\frac{4}{10} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.2

Cancel the common factor of $4$ and $10$ .

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Step 3.1.1.2.1

Factor $2$ out of $4$ .

$\frac{5}{2} (\frac{2 (2)}{10} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.2.2

Cancel the common factors.

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Step 3.1.1.2.2.1

Factor $2$ out of $10$ .

$\frac{5}{2} (\frac{2 \cdot 2}{2 \cdot 5} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.2.2.2

Cancel the common factor.

$\frac{5}{2} (\frac{2 \cdot 2}{2 \cdot 5} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.2.2.3

Rewrite the expression.

$\frac{5}{2} (\frac{2}{5} \cdot (x + \frac{1225}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.3

Cancel the common factor of $1225$ and $1000$ .

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Step 3.1.1.3.1

Factor $25$ out of $1225$ .

$\frac{5}{2} (\frac{2}{5} \cdot (x + \frac{25 (49)}{1000})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.3.2

Cancel the common factors.

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Step 3.1.1.3.2.1

Factor $25$ out of $1000$ .

$\frac{5}{2} (\frac{2}{5} \cdot (x + \frac{25 \cdot 49}{25 \cdot 40})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.3.2.2

Cancel the common factor.

$\frac{5}{2} (\frac{2}{5} \cdot (x + \frac{25 \cdot 49}{25 \cdot 40})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.3.2.3

Rewrite the expression.

$\frac{5}{2} (\frac{2}{5} \cdot (x + \frac{49}{40})) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.4

Apply the distributive property.

$\frac{5}{2} (\frac{2}{5} x + \frac{2}{5} \cdot \frac{49}{40}) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.5

Combine $\frac{2}{5}$ and $x$ .

$\frac{5}{2} (\frac{2 x}{5} + \frac{2}{5} \cdot \frac{49}{40}) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.6

Cancel the common factor of $2$ .

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Step 3.1.1.6.1

Factor $2$ out of $40$ .

$\frac{5}{2} (\frac{2 x}{5} + \frac{2}{5} \cdot \frac{49}{2 (20)}) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.6.2

Cancel the common factor.

$\frac{5}{2} (\frac{2 x}{5} + \frac{2}{5} \cdot \frac{49}{2 \cdot 20}) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.6.3

Rewrite the expression.

$\frac{5}{2} (\frac{2 x}{5} + \frac{1}{5} \cdot \frac{49}{20}) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.7

Multiply $\frac{1}{5}$ by $\frac{49}{20}$ .

$\frac{5}{2} (\frac{2 x}{5} + \frac{49}{5 \cdot 20}) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.8

Multiply $5$ by $20$ .

$\frac{5}{2} (\frac{2 x}{5} + \frac{49}{100}) = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.9

Apply the distributive property.

$\frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot \frac{49}{100} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.10

Cancel the common factor of $5$ .

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Step 3.1.1.10.1

Cancel the common factor.

$\frac{5}{2} \cdot \frac{2 x}{5} + \frac{5}{2} \cdot \frac{49}{100} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.10.2

Rewrite the expression.

$\frac{1}{2} (2 x) + \frac{5}{2} \cdot \frac{49}{100} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.11

Cancel the common factor of $2$ .

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Step 3.1.1.11.1

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) + \frac{5}{2} \cdot \frac{49}{100} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.11.2

Cancel the common factor.

$\frac{1}{2} (2 x) + \frac{5}{2} \cdot \frac{49}{100} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.11.3

Rewrite the expression.

$x + \frac{5}{2} \cdot \frac{49}{100} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.12

Cancel the common factor of $5$ .

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Step 3.1.1.12.1

Factor $5$ out of $100$ .

$x + \frac{5}{2} \cdot \frac{49}{5 (20)} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.12.2

Cancel the common factor.

$x + \frac{5}{2} \cdot \frac{49}{5 \cdot 20} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.12.3

Rewrite the expression.

$x + \frac{1}{2} \cdot \frac{49}{20} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.13

Multiply $\frac{1}{2}$ by $\frac{49}{20}$ .

$x + \frac{49}{2 \cdot 20} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.1.1.14

Multiply $2$ by $20$ .

$x + \frac{49}{40} = \frac{10}{4} {(y - \frac{33}{10})}^{2}$

Step 3.2

Simplify the right side.

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Step 3.2.1

Simplify $\frac{10}{4} {(y - \frac{33}{10})}^{2}$ .

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Step 3.2.1.1

Simplify terms.

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Step 3.2.1.1.1

Cancel the common factor of $10$ and $4$ .

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Step 3.2.1.1.1.1

Factor $2$ out of $10$ .

$x + \frac{49}{40} = \frac{2 (5)}{4} {(y - \frac{33}{10})}^{2}$

Step 3.2.1.1.1.2

Cancel the common factors.

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Step 3.2.1.1.1.2.1

Factor $2$ out of $4$ .

$x + \frac{49}{40} = \frac{2 \cdot 5}{2 \cdot 2} {(y - \frac{33}{10})}^{2}$

Step 3.2.1.1.1.2.2

Cancel the common factor.

$x + \frac{49}{40} = \frac{2 \cdot 5}{2 \cdot 2} {(y - \frac{33}{10})}^{2}$

Step 3.2.1.1.1.2.3

Rewrite the expression.

$x + \frac{49}{40} = \frac{5}{2} {(y - \frac{33}{10})}^{2}$

Step 3.2.1.1.2

Combine $\frac{5}{2}$ and ${(y - \frac{33}{10})}^{2}$ .

$x + \frac{49}{40} = \frac{5 {(y - \frac{33}{10})}^{2}}{2}$

Step 3.2.1.2

Simplify the numerator.

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Step 3.2.1.2.1

To write $y$ as a fraction with a common denominator, multiply by $\frac{10}{10}$ .

$x + \frac{49}{40} = \frac{5 {(y \cdot \frac{10}{10} - \frac{33}{10})}^{2}}{2}$

Step 3.2.1.2.2

Combine $y$ and $\frac{10}{10}$ .

$x + \frac{49}{40} = \frac{5 {(\frac{y \cdot 10}{10} - \frac{33}{10})}^{2}}{2}$

Step 3.2.1.2.3

Combine the numerators over the common denominator.

$x + \frac{49}{40} = \frac{5 {(\frac{y \cdot 10 - 33}{10})}^{2}}{2}$

Step 3.2.1.2.4

Move $10$ to the left of $y$ .

$x + \frac{49}{40} = \frac{5 {(\frac{10 y - 33}{10})}^{2}}{2}$

Step 3.2.1.2.5

Apply the product rule to $\frac{10 y - 33}{10}$ .

$x + \frac{49}{40} = \frac{5 \frac{{(10 y - 33)}^{2}}{10^{2}}}{2}$

Step 3.2.1.2.6

Raise $10$ to the power of $2$ .

$x + \frac{49}{40} = \frac{5 \frac{{(10 y - 33)}^{2}}{100}}{2}$

Step 3.2.1.3

Simplify terms.

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Step 3.2.1.3.1

Combine $5$ and $\frac{{(10 y - 33)}^{2}}{100}$ .

$x + \frac{49}{40} = \frac{\frac{5 {(10 y - 33)}^{2}}{100}}{2}$

Step 3.2.1.3.2

Reduce the expression $\frac{5 {(10 y - 33)}^{2}}{100}$ by cancelling the common factors.

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Step 3.2.1.3.2.1

Factor $5$ out of $5 {(10 y - 33)}^{2}$ .

$x + \frac{49}{40} = \frac{\frac{5 ({(10 y - 33)}^{2})}{100}}{2}$

Step 3.2.1.3.2.2

Factor $5$ out of $100$ .

$x + \frac{49}{40} = \frac{\frac{5 {(10 y - 33)}^{2}}{5 \cdot 20}}{2}$

Step 3.2.1.3.2.3

Cancel the common factor.

$x + \frac{49}{40} = \frac{\frac{5 {(10 y - 33)}^{2}}{5 \cdot 20}}{2}$

Step 3.2.1.3.2.4

Rewrite the expression.

$x + \frac{49}{40} = \frac{\frac{{(10 y - 33)}^{2}}{20}}{2}$

Step 3.2.1.4

Multiply the numerator by the reciprocal of the denominator.

$x + \frac{49}{40} = \frac{{(10 y - 33)}^{2}}{20} \cdot \frac{1}{2}$

Step 3.2.1.5

Combine.

$x + \frac{49}{40} = \frac{{(10 y - 33)}^{2} \cdot 1}{20 \cdot 2}$

Step 3.2.1.6

Multiply.

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Step 3.2.1.6.1

Multiply ${(10 y - 33)}^{2}$ by $1$ .

$x + \frac{49}{40} = \frac{{(10 y - 33)}^{2}}{20 \cdot 2}$

Step 3.2.1.6.2

Multiply $20$ by $2$ .

$x + \frac{49}{40} = \frac{{(10 y - 33)}^{2}}{40}$

Step 4

Subtract $\frac{49}{40}$ from both sides of the equation.

$x = \frac{{(10 y - 33)}^{2}}{40} - \frac{49}{40}$

Step 5

Simplify $\frac{{(10 y - 33)}^{2}}{40} - \frac{49}{40}$ .

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Step 5.1

Combine the numerators over the common denominator.

$x = \frac{{(10 y - 33)}^{2} - 49}{40}$

Step 5.2

Simplify the numerator.

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Step 5.2.1

Rewrite $49$ as $7^{2}$ .

$x = \frac{{(10 y - 33)}^{2} - 7^{2}}{40}$

Step 5.2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 10 y - 33$ and $b = 7$ .

$x = \frac{(10 y - 33 + 7) (10 y - 33 - 7)}{40}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Add $- 33$ and $7$ .

$x = \frac{(10 y - 26) (10 y - 33 - 7)}{40}$

Step 5.2.3.2

Factor $2$ out of $10 y - 26$ .

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Step 5.2.3.2.1

Factor $2$ out of $10 y$ .

$x = \frac{(2 (5 y) - 26) (10 y - 33 - 7)}{40}$

Step 5.2.3.2.2

Factor $2$ out of $- 26$ .

$x = \frac{(2 (5 y) + 2 (- 13)) (10 y - 33 - 7)}{40}$

Step 5.2.3.2.3

Factor $2$ out of $2 (5 y) + 2 (- 13)$ .

$x = \frac{2 (5 y - 13) (10 y - 33 - 7)}{40}$

Step 5.2.3.3

Subtract $7$ from $- 33$ .

$x = \frac{2 (5 y - 13) (10 y - 40)}{40}$

Step 5.2.3.4

Factor $10$ out of $10 y - 40$ .

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Step 5.2.3.4.1

Factor $10$ out of $10 y$ .

$x = \frac{2 (5 y - 13) (10 (y) - 40)}{40}$

Step 5.2.3.4.2

Factor $10$ out of $- 40$ .

$x = \frac{2 (5 y - 13) (10 y + 10 \cdot - 4)}{40}$

Step 5.2.3.4.3

Factor $10$ out of $10 y + 10 \cdot - 4$ .

$x = \frac{2 (5 y - 13) (10 (y - 4))}{40}$

$x = \frac{2 (5 y - 13) \cdot 10 (y - 4)}{40}$

Step 5.2.3.5

Multiply $10$ by $2$ .

$x = \frac{20 (5 y - 13) (y - 4)}{40}$

Step 5.3

Cancel the common factor of $20$ and $40$ .

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Step 5.3.1

Factor $20$ out of $20 (5 y - 13) (y - 4)$ .

$x = \frac{20 ((5 y - 13) (y - 4))}{40}$

Step 5.3.2

Cancel the common factors.

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Step 5.3.2.1

Factor $20$ out of $40$ .

$x = \frac{20 ((5 y - 13) (y - 4))}{20 \cdot 2}$

Step 5.3.2.2

Cancel the common factor.

$x = \frac{20 ((5 y - 13) (y - 4))}{20 \cdot 2}$

Step 5.3.2.3

Rewrite the expression.

$x = \frac{(5 y - 13) (y - 4)}{2}$

Step 6

Find the properties of the given parabola.

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Step 6.1

Rewrite the equation in vertex form.

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Step 6.1.1

Complete the square for $\frac{(5 y - 13) (y - 4)}{2}$ .

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Step 6.1.1.1

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

$a = 1$

$b = 0$

$c = 0$

Step 6.1.1.2

Consider the vertex form of a parabola.

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

Step 6.1.1.3

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

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Step 6.1.1.3.1

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

$d = \frac{0}{2 \cdot 1}$

Step 6.1.1.3.2

Cancel the common factor of $0$ and $2$ .

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Step 6.1.1.3.2.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 \cdot 1}$

Step 6.1.1.3.2.2

Cancel the common factors.

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Step 6.1.1.3.2.2.1

Factor $2$ out of $2 \cdot 1$ .

$d = \frac{2 (0)}{2 (1)}$

Step 6.1.1.3.2.2.2

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 \cdot 1}$

Step 6.1.1.3.2.2.3

Rewrite the expression.

$d = \frac{0}{1}$

Step 6.1.1.3.2.2.4

Divide $0$ by $1$ .

$d = 0$

Step 6.1.1.4

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

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Step 6.1.1.4.1

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

$e = 0 - \frac{0^{2}}{4 \cdot 1}$

Step 6.1.1.4.2

Simplify the right side.

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Step 6.1.1.4.2.1

Simplify each term.

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Step 6.1.1.4.2.1.1

Raising $0$ to any positive power yields $0$ .

$e = 0 - \frac{0}{4 \cdot 1}$

Step 6.1.1.4.2.1.2

Multiply $4$ by $1$ .

$e = 0 - \frac{0}{4}$

Step 6.1.1.4.2.1.3

Divide $0$ by $4$ .

$e = 0 - 0$

Step 6.1.1.4.2.1.4

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 6.1.1.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 6.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $y^{2}$ .

$y^{2}$

Step 6.1.2

Set $x$ equal to the new right side.

$x = y^{2}$

Step 6.2

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

$a = 1$

$h = 0$

$k = 0$

Step 6.3

Since the value of $a$ is positive, the parabola opens right.

Opens Right

Step 6.4

Find the vertex $(h, k)$ .

$(0, 0)$

Step 6.5

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

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Step 6.5.1

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

$\frac{1}{4 a}$

Step 6.5.2

Substitute the value of $a$ into the formula.

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

Step 6.5.3

Cancel the common factor of $1$ .

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Step 6.5.3.1

Cancel the common factor.

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

Step 6.5.3.2

Rewrite the expression.

$\frac{1}{4}$

Step 6.6

Find the focus.

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Step 6.6.1

The focus of a parabola can be found by adding $p$ to the x-coordinate $h$ if the parabola opens left or right.

$(h + p, k)$

Step 6.6.2

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

$(\frac{1}{4}, 0)$

Step 6.7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

$y = 0$

Step 6.8

Find the directrix.

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Step 6.8.1

The directrix of a parabola is the vertical line found by subtracting $p$ from the x-coordinate $h$ of the vertex if the parabola opens left or right.

$x = h - p$

Step 6.8.2

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

$x = - \frac{1}{4}$

Step 6.9

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{1}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{1}{4}$

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{1}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{1}{4}$

Step 7

Select a few $x$ values, and plug them into the equation to find the corresponding $y$ values. The $x$ values should be selected around the vertex.

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Step 7.1

Substitute the $x$ value $1$ into $f (x) = \frac{\sqrt{40 x + 49}}{10} + \frac{33}{10}$ . In this case, the point is $(1, 4.24339811)$ .

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Step 7.1.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \frac{\sqrt{40 (1) + 49}}{10} + \frac{33}{10}$

Step 7.1.2

Simplify the result.

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Step 7.1.2.1

Combine the numerators over the common denominator.

$f (1) = \frac{\sqrt{40 (1) + 49} + 33}{10}$

Step 7.1.2.2

Simplify each term.

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Step 7.1.2.2.1

Multiply $40$ by $1$ .

$f (1) = \frac{\sqrt{40 + 49} + 33}{10}$

Step 7.1.2.2.2

Add $40$ and $49$ .

$f (1) = \frac{\sqrt{89} + 33}{10}$

Step 7.1.2.3

The final answer is $\frac{\sqrt{89} + 33}{10}$ .

$y = \frac{\sqrt{89} + 33}{10}$

Step 7.1.3

Convert $\frac{\sqrt{89} + 33}{10}$ to decimal.

$= 4.24339811$

Step 7.2

Substitute the $x$ value $1$ into $f (x) = - \frac{\sqrt{40 x + 49}}{10} + \frac{33}{10}$ . In this case, the point is $(1, 2.35660188)$ .

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Step 7.2.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = - \frac{\sqrt{40 (1) + 49}}{10} + \frac{33}{10}$

Step 7.2.2

Simplify the result.

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Step 7.2.2.1

Combine the numerators over the common denominator.

$f (1) = \frac{- \sqrt{40 (1) + 49} + 33}{10}$

Step 7.2.2.2

Simplify each term.

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Step 7.2.2.2.1

Multiply $40$ by $1$ .

$f (1) = \frac{- \sqrt{40 + 49} + 33}{10}$

Step 7.2.2.2.2

Add $40$ and $49$ .

$f (1) = \frac{- \sqrt{89} + 33}{10}$

Step 7.2.2.3

Simplify with factoring out.

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Step 7.2.2.3.1

Factor $- 1$ out of $- \sqrt{89}$ .

$f (1) = \frac{- (\sqrt{89}) + 33}{10}$

Step 7.2.2.3.2

Rewrite $33$ as $- 1 (- 33)$ .

$f (1) = \frac{- (\sqrt{89}) - 1 \cdot - 33}{10}$

Step 7.2.2.3.3

Factor $- 1$ out of $- (\sqrt{89}) - 1 (- 33)$ .

$f (1) = \frac{- (\sqrt{89} - 33)}{10}$

Step 7.2.2.3.4

Simplify the expression.

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Step 7.2.2.3.4.1

Rewrite $- (\sqrt{89} - 33)$ as $- 1 (\sqrt{89} - 33)$ .

$f (1) = \frac{- 1 (\sqrt{89} - 33)}{10}$

Step 7.2.2.3.4.2

Move the negative in front of the fraction.

$f (1) = - \frac{\sqrt{89} - 33}{10}$

Step 7.2.2.4

The final answer is $- \frac{\sqrt{89} - 33}{10}$ .

$y = - \frac{\sqrt{89} - 33}{10}$

Step 7.2.3

Convert $- \frac{\sqrt{89} - 33}{10}$ to decimal.

$= 2.35660188$

Step 7.3

Substitute the $x$ value $2$ into $f (x) = \frac{\sqrt{40 x + 49}}{10} + \frac{33}{10}$ . In this case, the point is $(2, 4.43578166)$ .

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Step 7.3.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = \frac{\sqrt{40 (2) + 49}}{10} + \frac{33}{10}$

Step 7.3.2

Simplify the result.

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Step 7.3.2.1

Combine the numerators over the common denominator.

$f (2) = \frac{\sqrt{40 (2) + 49} + 33}{10}$

Step 7.3.2.2

Simplify each term.

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Step 7.3.2.2.1

Multiply $40$ by $2$ .

$f (2) = \frac{\sqrt{80 + 49} + 33}{10}$

Step 7.3.2.2.2

Add $80$ and $49$ .

$f (2) = \frac{\sqrt{129} + 33}{10}$

Step 7.3.2.3

The final answer is $\frac{\sqrt{129} + 33}{10}$ .

$y = \frac{\sqrt{129} + 33}{10}$

Step 7.3.3

Convert $\frac{\sqrt{129} + 33}{10}$ to decimal.

$= 4.43578166$

Step 7.4

Substitute the $x$ value $2$ into $f (x) = - \frac{\sqrt{40 x + 49}}{10} + \frac{33}{10}$ . In this case, the point is $(2, 2.16421833)$ .

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Step 7.4.1

Replace the variable $x$ with $2$ in the expression.

$f (2) = - \frac{\sqrt{40 (2) + 49}}{10} + \frac{33}{10}$

Step 7.4.2

Simplify the result.

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Step 7.4.2.1

Combine the numerators over the common denominator.

$f (2) = \frac{- \sqrt{40 (2) + 49} + 33}{10}$

Step 7.4.2.2

Simplify each term.

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Step 7.4.2.2.1

Multiply $40$ by $2$ .

$f (2) = \frac{- \sqrt{80 + 49} + 33}{10}$

Step 7.4.2.2.2

Add $80$ and $49$ .

$f (2) = \frac{- \sqrt{129} + 33}{10}$

Step 7.4.2.3

Simplify with factoring out.

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Step 7.4.2.3.1

Factor $- 1$ out of $- \sqrt{129}$ .

$f (2) = \frac{- (\sqrt{129}) + 33}{10}$

Step 7.4.2.3.2

Rewrite $33$ as $- 1 (- 33)$ .

$f (2) = \frac{- (\sqrt{129}) - 1 \cdot - 33}{10}$

Step 7.4.2.3.3

Factor $- 1$ out of $- (\sqrt{129}) - 1 (- 33)$ .

$f (2) = \frac{- (\sqrt{129} - 33)}{10}$

Step 7.4.2.3.4

Simplify the expression.

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Step 7.4.2.3.4.1

Rewrite $- (\sqrt{129} - 33)$ as $- 1 (\sqrt{129} - 33)$ .

$f (2) = \frac{- 1 (\sqrt{129} - 33)}{10}$

Step 7.4.2.3.4.2

Move the negative in front of the fraction.

$f (2) = - \frac{\sqrt{129} - 33}{10}$

Step 7.4.2.4

The final answer is $- \frac{\sqrt{129} - 33}{10}$ .

$y = - \frac{\sqrt{129} - 33}{10}$

Step 7.4.3

Convert $- \frac{\sqrt{129} - 33}{10}$ to decimal.

$= 2.16421833$

Step 7.5

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 4.24 \\ 1 & 2.36 \\ 2 & 4.44 \\ 2 & 2.16 \end{array}$

Step 8

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: $(0, 0)$

Focus: $(\frac{1}{4}, 0)$

Axis of Symmetry: $y = 0$

Directrix: $x = - \frac{1}{4}$

$\begin{array}{c} x & y \\ 0 & 0 \\ 1 & 4.24 \\ 1 & 2.36 \\ 2 & 4.44 \\ 2 & 2.16 \end{array}$

Step 9