Algebra Examples

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

Step 1

The parent function is the simplest form of the type of function given.

$g (x) = x^{2}$

Step 2

The transformation being described is from $g (x) = x^{2}$ to $f (x) = \frac{7}{2} x^{2}$ .

$g (x) = x^{2} \to f (x) = \frac{7}{2} x^{2}$

Step 3

Find the vertex form of $f (x) = \frac{7}{2} x^{2}$ .

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Step 3.1

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

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

Step 3.2

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

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Step 3.2.1

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

$a = \frac{7}{2}$

$b = 0$

$c = 0$

Step 3.2.2

Consider the vertex form of a parabola.

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

Step 3.2.3

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

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Step 3.2.3.1

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

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

Step 3.2.3.2

Simplify the right side.

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Step 3.2.3.2.1

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

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Step 3.2.3.2.1.1

Factor $2$ out of $0$ .

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

Step 3.2.3.2.1.2

Cancel the common factors.

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Step 3.2.3.2.1.2.1

Cancel the common factor.

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

Step 3.2.3.2.1.2.2

Rewrite the expression.

$d = \frac{0}{\frac{7}{2}}$

Step 3.2.3.2.2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2.3.2.3

Multiply $0$ by $\frac{2}{7}$ .

$d = 0$

Step 3.2.4

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

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Step 3.2.4.1

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

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

Step 3.2.4.2

Simplify the right side.

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Step 3.2.4.2.1

Simplify each term.

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Step 3.2.4.2.1.1

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

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

Step 3.2.4.2.1.2

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

$e = 0 - \frac{0}{\frac{4 \cdot 7}{2}}$

Step 3.2.4.2.1.3

Multiply $4$ by $7$ .

$e = 0 - \frac{0}{\frac{28}{2}}$

Step 3.2.4.2.1.4

Divide $28$ by $2$ .

$e = 0 - \frac{0}{14}$

Step 3.2.4.2.1.5

Divide $0$ by $14$ .

$e = 0 - 0$

Step 3.2.4.2.1.6

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Step 3.2.4.2.2

Add $0$ and $0$ .

$e = 0$

Step 3.2.5

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

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

Step 3.3

Set $y$ equal to the new right side.

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

Step 4

The horizontal shift depends on the value of $h$ . The horizontal shift is described as:

$f (x) = f (x + h)$ - The graph is shifted to the left $h$ units.

$f (x) = f (x - h)$ - The graph is shifted to the right $h$ units.

In this case, $h = 0$ which means that the graph is not shifted to the left or right.

Horizontal Shift: None

Step 5

The vertical shift depends on the value of $k$ . The vertical shift is described as:

$f (x) = f (x) + k$ - The graph is shifted up $k$ units.

$f (x) = f (x) - k$ - The graph is shifted down $k$ units.

In this case, $k = 0$ which means that the graph is not shifted up or down.

Vertical Shift: None

Step 6

The graph is reflected about the x-axis when $f (x) = - f (x)$ .

Reflection about the x-axis: None

Step 7

The graph is reflected about the y-axis when $f (x) = f (- x)$ .

Reflection about the y-axis: None

Step 8

Compressing and stretching depends on the value of $a$ .

When $a$ is greater than $1$ : Vertically stretched

When $a$ is between $0$ and $1$ : Vertically compressed

Vertical Compression or Stretch: Stretched

Step 9

Compare and list the transformations.

Parent Function: $g (x) = x^{2}$

Horizontal Shift: None

Vertical Shift: None

Reflection about the x-axis: None

Reflection about the y-axis: None

Vertical Compression or Stretch: Stretched

Step 10