Algebra Examples

$y = - \frac{7}{4} x^{2}$

Step 1

Simplify $- \frac{7}{4} x^{2}$ .

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Step 1.1

Combine $x^{2}$ and $\frac{7}{4}$ .

$y = - \frac{x^{2} \cdot 7}{4}$

Step 1.2

Move $7$ to the left of $x^{2}$ .

$y = - \frac{7 x^{2}}{4}$

Step 2

Find the properties of the given parabola.

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Step 2.1

Rewrite the equation in vertex form.

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Step 2.1.1

Complete the square for $- \frac{7 x^{2}}{4}$ .

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Step 2.1.1.1

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

$a = - \frac{7}{4}$

$b = 0$

$c = 0$

Step 2.1.1.2

Consider the vertex form of a parabola.

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

Step 2.1.1.3

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

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Step 2.1.1.3.1

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

$d = \frac{0}{2 (- \frac{7}{4})}$

Step 2.1.1.3.2

Simplify the right side.

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Step 2.1.1.3.2.1

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

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Step 2.1.1.3.2.1.1

Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (- \frac{7}{4})}$

Step 2.1.1.3.2.1.2

Cancel the common factors.

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Step 2.1.1.3.2.1.2.1

Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (- \frac{7}{4})}$

Step 2.1.1.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{- \frac{7}{4}}$

Step 2.1.1.3.2.2

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (- \frac{4}{7})$

Step 2.1.1.3.2.3

Multiply $0 (- \frac{4}{7})$ .

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Step 2.1.1.3.2.3.1

Multiply $- 1$ by $0$ .

$d = 0 (\frac{4}{7})$

Step 2.1.1.3.2.3.2

Multiply $0$ by $\frac{4}{7}$ .

$d = 0$

Step 2.1.1.4

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

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Step 2.1.1.4.1

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

$e = 0 - \frac{0^{2}}{4 (- \frac{7}{4})}$

Step 2.1.1.4.2

Simplify the right side.

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Step 2.1.1.4.2.1

Simplify each term.

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Step 2.1.1.4.2.1.1

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

$e = 0 - \frac{0}{4 (- \frac{7}{4})}$

Step 2.1.1.4.2.1.2

Simplify the denominator.

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Step 2.1.1.4.2.1.2.1

Multiply $- 1$ by $4$ .

$e = 0 - \frac{0}{- 4 (\frac{7}{4})}$

Step 2.1.1.4.2.1.2.2

Combine $- 4$ and $\frac{7}{4}$ .

$e = 0 - \frac{0}{\frac{- 4 \cdot 7}{4}}$

Step 2.1.1.4.2.1.3

Multiply $- 4$ by $7$ .

$e = 0 - \frac{0}{\frac{- 28}{4}}$

Step 2.1.1.4.2.1.4

Divide $- 28$ by $4$ .

$e = 0 - \frac{0}{- 7}$

Step 2.1.1.4.2.1.5

Divide $0$ by $- 7$ .

$e = 0 - 0$

Step 2.1.1.4.2.1.6

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 2.1.1.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 2.1.1.5

Substitute the values of $a$ , $d$ , and $e$ into the vertex form $- \frac{7}{4} x^{2}$ .

$- \frac{7}{4} x^{2}$

Step 2.1.2

Set $y$ equal to the new right side.

$y = - \frac{7}{4} x^{2}$

Step 2.2

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

$a = - \frac{7}{4}$

$h = 0$

$k = 0$

Step 2.3

Since the value of $a$ is negative, the parabola opens down.

Opens Down

Step 2.4

Find the vertex $(h, k)$ .

$(0, 0)$

Step 2.5

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

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Step 2.5.1

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

$\frac{1}{4 a}$

Step 2.5.2

Substitute the value of $a$ into the formula.

$\frac{1}{4 \cdot (- \frac{7}{4})}$

Step 2.5.3

Simplify.

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Step 2.5.3.1

Cancel the common factor of $1$ and $- 1$ .

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Step 2.5.3.1.1

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

$\frac{- 1 (- 1)}{4 \cdot (- \frac{7}{4})}$

Step 2.5.3.1.2

Move the negative in front of the fraction.

$- \frac{1}{4 (\frac{7}{4})}$

Step 2.5.3.2

Combine $4$ and $\frac{7}{4}$ .

$- \frac{1}{\frac{4 \cdot 7}{4}}$

Step 2.5.3.3

Simplify the expression.

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Step 2.5.3.3.1

Multiply $4$ by $7$ .

$- \frac{1}{\frac{28}{4}}$

Step 2.5.3.3.2

Divide $28$ by $4$ .

$- \frac{1}{7}$

Step 2.6

Find the focus.

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Step 2.6.1

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 2.6.2

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

$(0, - \frac{1}{7})$

Step 2.7

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

$x = 0$

Step 2.8

Find the directrix.

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Step 2.8.1

The directrix of a parabola is the horizontal line found by subtracting $p$ from the y-coordinate $k$ of the vertex if the parabola opens up or down.

$y = k - p$

Step 2.8.2

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

$y = \frac{1}{7}$

Step 2.9

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

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - \frac{1}{7})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{1}{7}$

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - \frac{1}{7})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{1}{7}$

Step 3

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 3.1

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

$f (- 2) = - \frac{7 {(- 2)}^{2}}{4}$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (- 2) = - \frac{7 \cdot 4}{4}$

Step 3.2.2

Multiply $7$ by $4$ .

$f (- 2) = - \frac{28}{4}$

Step 3.2.3

Divide $28$ by $4$ .

$f (- 2) = - 1 \cdot 7$

Step 3.2.4

Multiply $- 1$ by $7$ .

$f (- 2) = - 7$

Step 3.2.5

The final answer is $- 7$ .

$- 7$

Step 3.3

The $y$ value at $x = - 2$ is $- 7$ .

$y = - 7$

Step 3.4

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

$f (- 1) = - \frac{7 {(- 1)}^{2}}{4}$

Step 3.5

Simplify the result.

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Step 3.5.1

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

$f (- 1) = - \frac{7 \cdot 1}{4}$

Step 3.5.2

Multiply $7$ by $1$ .

$f (- 1) = - \frac{7}{4}$

Step 3.5.3

The final answer is $- \frac{7}{4}$ .

$- \frac{7}{4}$

Step 3.6

The $y$ value at $x = - 1$ is $- \frac{7}{4}$ .

$y = - \frac{7}{4}$

Step 3.7

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

$f (2) = - \frac{7 {(2)}^{2}}{4}$

Step 3.8

Simplify the result.

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Step 3.8.1

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

$f (2) = - \frac{7 \cdot 4}{4}$

Step 3.8.2

Multiply $7$ by $4$ .

$f (2) = - \frac{28}{4}$

Step 3.8.3

Divide $28$ by $4$ .

$f (2) = - 1 \cdot 7$

Step 3.8.4

Multiply $- 1$ by $7$ .

$f (2) = - 7$

Step 3.8.5

The final answer is $- 7$ .

$- 7$

Step 3.9

The $y$ value at $x = 2$ is $- 7$ .

$y = - 7$

Step 3.10

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

$f (1) = - \frac{7 {(1)}^{2}}{4}$

Step 3.11

Simplify the result.

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Step 3.11.1

One to any power is one.

$f (1) = - \frac{7 \cdot 1}{4}$

Step 3.11.2

Multiply $7$ by $1$ .

$f (1) = - \frac{7}{4}$

Step 3.11.3

The final answer is $- \frac{7}{4}$ .

$- \frac{7}{4}$

Step 3.12

The $y$ value at $x = 1$ is $- \frac{7}{4}$ .

$y = - \frac{7}{4}$

Step 3.13

Graph the parabola using its properties and the selected points.

$\begin{array}{c} x & y \\ - 2 & - 7 \\ - 1 & - \frac{7}{4} \\ 0 & 0 \\ 1 & - \frac{7}{4} \\ 2 & - 7 \end{array}$

Step 4

Graph the parabola using its properties and the selected points.

Direction: Opens Down

Vertex: $(0, 0)$

Focus: $(0, - \frac{1}{7})$

Axis of Symmetry: $x = 0$

Directrix: $y = \frac{1}{7}$

$\begin{array}{c} x & y \\ - 2 & - 7 \\ - 1 & - \frac{7}{4} \\ 0 & 0 \\ 1 & - \frac{7}{4} \\ 2 & - 7 \end{array}$

Step 5