Describe the Transformation f(x)=5/3x^2

$f (x) = \frac{5}{3} x^{2}$

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

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

The transformation being described is from $g (x) = x^{2}$ to $f (x) = \frac{5}{3} x^{2}$ .

$g (x) = x^{2} \to f (x) = \frac{5}{3} x^{2}$

Find the vertex form of $f (x) = \frac{5}{3} x^{2}$ .

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Combine $\frac{5}{3}$ and $x^{2}$ .

$y = \frac{5 x^{2}}{3}$

Complete the square for $\frac{5 x^{2}}{3}$ .

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

$a = \frac{5}{3}, b = 0, c = 0$

Consider the vertex form of a parabola.

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

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

$d = \frac{0}{2 (\frac{5}{3})}$

Simplify the right side.

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

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Factor $2$ out of $0$ .

$d = \frac{2 (0)}{2 (\frac{5}{3})}$

Cancel the common factors.

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Cancel the common factor.

$d = \frac{2 \cdot 0}{2 (\frac{5}{3})}$

Rewrite the expression.

$d = \frac{0}{\frac{5}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$d = 0 (\frac{3}{5})$

Multiply $0$ by $\frac{3}{5}$ .

$d = 0$

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

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Simplify each term.

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

$e = 0 - \frac{0}{4 (\frac{5}{3})}$

Combine $4$ and $\frac{5}{3}$ .

$e = 0 - \frac{0}{\frac{4 \cdot 5}{3}}$

Multiply $4$ by $5$ .

$e = 0 - \frac{0}{\frac{20}{3}}$

Multiply the numerator by the reciprocal of the denominator.

$e = 0 - (0 (\frac{3}{20}))$

Multiply $0$ by $\frac{3}{20}$ .

$e = 0 - 0$

Multiply $- 1$ by $0$ .

$e = 0 + 0$

Add $0$ and $0$ .

$e = 0$

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

$\frac{5 x^{2}}{3}$

Set $y$ equal to the new right side.

$y = \frac{5 x^{2}}{3}$

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

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

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

Reflection about the x-axis: None

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

Reflection about the y-axis: None

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

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

Describe the Transformation f(x)=5/3x^2

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