Algebra Examples

$f (x) = - {(\frac{4}{3})}^{2 (x - 3)} + 1$

Step 1

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

$g (x) = {(\frac{4}{3})}^{x}$

Step 2

The transformation from the first equation to the second one can be found by finding $a$ , $h$ , and $k$ for each equation.

$y = a b^{x - h} + k$

Step 3

Simplify.

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Step 3.1

Simplify terms.

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Step 3.1.1

Simplify each term.

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Step 3.1.1.1

Apply the distributive property.

$f (x) = - {(\frac{4}{3})}^{2 x + 2 \cdot - 3} + 1$

Step 3.1.1.2

Multiply $2$ by $- 3$ .

$f (x) = - {(\frac{4}{3})}^{2 x - 6} + 1$

Step 3.1.1.3

Apply the product rule to $\frac{4}{3}$ .

$f (x) = - \frac{4^{2 x - 6}}{3^{2 x - 6}} + 1$

Step 3.1.2

Combine into one fraction.

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Step 3.1.2.1

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

$f (x) = - \frac{4^{2 x - 6}}{3^{2 x - 6}} + \frac{3^{2 x - 6}}{3^{2 x - 6}}$

Step 3.1.2.2

Combine the numerators over the common denominator.

$f (x) = \frac{- 4^{2 x - 6} + 3^{2 x - 6}}{3^{2 x - 6}}$

Step 3.2

Simplify the numerator.

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Step 3.2.1

Rewrite $3^{2 x - 6}$ as ${(3^{x - 3})}^{2}$ .

$f (x) = \frac{- 4^{2 x - 6} + {(3^{x - 3})}^{2}}{3^{2 x - 6}}$

Step 3.2.2

Rewrite $4^{2 x - 6}$ as ${(4^{x - 3})}^{2}$ .

$f (x) = \frac{- {(4^{x - 3})}^{2} + {(3^{x - 3})}^{2}}{3^{2 x - 6}}$

Step 3.2.3

Reorder $- {(4^{x - 3})}^{2}$ and ${(3^{x - 3})}^{2}$ .

$f (x) = \frac{{(3^{x - 3})}^{2} - {(4^{x - 3})}^{2}}{3^{2 x - 6}}$

Step 3.2.4

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 3^{x - 3}$ and $b = 4^{x - 3}$ .

$f (x) = \frac{(3^{x - 3} + 4^{x - 3}) (3^{x - 3} - 4^{x - 3})}{3^{2 x - 6}}$

Step 4

Find $a$ , $h$ , and $k$ for $g (x) = {(\frac{4}{3})}^{x}$ .

$a = 1$

$h = 0$

$k = 0$

Step 5

Find $a$ , $h$ , and $k$ for $f (x) = - {(\frac{4}{3})}^{2 (x - 3)} + 1$ .

$a = - 1$

$h = 3$

$k = 1$

Step 6

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.

Horizontal Shift: Right $3$ Units

Step 7

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.

Vertical Shift: Up $1$ Units

Step 8

The sign of $a$ describes the reflection across the x-axis. $- a$ means the graph is reflected across the x-axis.

Reflection about the x-axis: Reflected

Step 9

The value of $a$ describes the vertical stretch or compression of the graph.

$a > 1$ is a vertical stretch (makes it narrower)

$0 < a < 1$ is a vertical compression (makes it wider)

Vertical Compression or Stretch: None

Step 10

To find the transformation, compare the two functions and check to see if there is a horizontal or vertical shift, reflection about the x-axis, and if there is a vertical stretch.

Parent Function: $g (x) = {(\frac{4}{3})}^{x}$

Horizontal Shift: Right $3$ Units

Vertical Shift: Up $1$ Units

Reflection about the x-axis: Reflected

Vertical Compression or Stretch: None

Step 11