# Transformations

Level 5 to Level 6
##### Gotta-Otta Test

This diagnostic tool will help determine how ready you are for this lesson. It covers the topics you Gotta Know before starting this lesson and what you ‘Otta Know by the end of the lesson. If you score well on this test, skip to the next section.

#### Tutorial

##### Identify the Transformation Values: $a$, $k$, $d$ and $c$

The first step to being able to work with transformations is to identify the transformation values: $a$, $k$, $d$ and $c$ when a function is of the form: $y=a\cdot f[k(x-d)]+c$. They appear differently depending on which basic function is represented by $f(x)$.

#### Tutorial

##### Describing the Transformations

When you can accurately identify the transformation values, you can more easily describe what each does to the graph. The values $a$ and $c$ located outside the brackets transform the function vertically. The values $k$ and $c$ located within the brackets transform the function horizontally.

#### Tutorial

##### Graphing Transformed Functions

You can easily graph any function using our Simple Steps to Graphing. These steps can be used at almost any level including advanced functions.

#### Tutorial

##### Transforming Domain and Range

To determine the domain and range of a function is more easily be done when you understand transformations. The alternative method was understanding the machine model of a function.

## Try It On Your Own

Here are some additional questions that you can try on your own:

Knowledge & Understanding

1. State the basic function, $f(x)$, and describe the transformations:

1. $$y=\frac{1}{3}f[-(x-2)]+1$$
2. $$y=14-2f(\frac{x-3}{4})$$
3. $$y=0.1f(5-10x)$$
4. $$y=3x+9$$
5. $$y=(\frac{1}{2}x+3)^2$$
6. $$y=5[5x]^3+20$$
1. $$y=\frac{|0.5x|+14}{2}$$
2. $$y=6-\sqrt{0.25(x+4)}$$
3. $$y=-2\sqrt{9-3x}$$
4. $$y=\frac{1}{x+3}-4$$
5. $$y=\frac{-5}{x}+6$$
6. $$y=4\frac{1}{10+2x}$$

2. Suppose the graph of $f(x)$ is given by the function below. Graph the original and the following functions on the same grid:

1. $$y=\frac{1}{3}f[-(x-2)]+1$$
2. $$y=f\left[0.25(x+2)\right]-7$$
1. $$y=5-3f\left[2(x+6)\right]$$
2. $$y=-2f\left(\frac{x-3}{4}\right)-1$$

3. Graph the following functions:

1. $$y=3x+9$$
2. $$y=(\frac{1}{2}x+3)^2$$
3. $$y=5[5x]^3+20$$
4. $$y=\frac{|0.5x|+14}{2}$$
1. $$y=6-\sqrt{0.25(x+4)}$$
2. $$y=-2\sqrt{9-3x}$$
3. $$y=\frac{1}{x+3}-4$$
4. $$y=\frac{-5}{6+2x}+1$$

4. Find the domain and range of the following functions without using their graphs:

1. $$y=3x+9$$
2. $$y=(\frac{1}{2}x+3)^2$$
3. $$y=5[5x]^3+20$$
4. $$y=\frac{|0.5x|+14}{2}$$
5. $$y=6-\sqrt{0.25(x+4)}$$
1. $$y=-2\sqrt{9-3x}$$
2. $$y=\frac{1}{x+3}-4$$
3. $$y=\frac{-5}{x}+6$$
4. $$y=4\frac{1}{10+2x}$$

Thinking & Inquiry

5. Find the equation of the function given the base function $f(x)$ and described transformations:

1. A linear function with a vertical shift up 2 and a horizontal shift left 7.
2. A radical function with a vertical stretch by a factor of 2, a reflection in the y-axis and a horizontal shift left 12.
3. An absolute value function horizontal compression by a factor of 0.2 with a reflection in the x-axis, a vertical stretch by a factor of 6, a horizontal shift right 4 and a vertical shift down 11.
4. A cubic function with horizontal compression by a factor of $\frac{2}{3}$ and vertical stretch by a factor of $\frac{5}{4}$.
5. A quadratic function with a reflection in both axes, a horizontal stretch by a factor of 8 and a horizontal shift right 4.
6. A reciprocal function with a horizontal compression by a factor of 0.2, reflection in the x-axis, vertical compression by a factor of $\frac{1}{7}$, vertical shift down 5 and horizontal shift up 18.

6. Find the equation of the following graphs:

7. The domain and range of $f(x)$ is given by: $\{x \in \mathbb{R} |x \neq 2\}$ and $\{y \in \mathbb{R} |-3 \leq y \leq 6\}$ respectfully. Find the domain and range of the function $g(x)=\frac{1}{3}f\left[2(x-4)\right]+1$.

8. The function $g(x)=2f\left[-(x+3)\right]+5$. If the domain and range of $g(x)$ is $\{x \in \mathbb{R} |x \geq 4\}$ and $\{y \in \mathbb{R} |-7 \leq y \leq -1\}$ respectfully, find the domain and range of the function $f(x)$.

1a. $y=\frac{1}{3}f[-(x-2)]+1; a=\frac{1}{3}, k=-1, d=2, c=-1$; base function $f(x)$; vertical compression by a factor of $\frac{1}{3}$, reflection in the $y$-axis, horizontal shift right 2, vertical shift up 1
1b. $y=-2f\left[\frac{1}{4}(x-3)\right]+14; a=-2, k=\frac{1}{4}, d=3, c=14$; base function $f(x)$; reflection in the $x$-axis, vertical stretch by a factor of 2, horizontal stretch by a factor of 4, horizontal shift right 3, vertical shift up 14
1c. $y=0.1f\left[-10(x-\frac{1}{2})\right]; a=0.1, k=-10, d=\frac{1}{2}$; base function $f(x)$; vertical compression by a factor of $0.1$, reflection in the $y$-axis, horizontal compression by a factor of $\frac{1}{10}$, horizontal shift right $\frac{1}{2}$
1d. $y=3x+9; a=3, c=9$; base function $f(x)=x$; vertical stretch by a factor of 3, vertical shift up 9
1e. $y=\left[\frac{1}{2}(x+6)\right]^2$; base function $f(x)=x^2$; horizontal stretch by a factor of 2, horizontal shift left 6
1f. $y=5[5x]^3+20; a=5, k=5, c=20$; base function $f(x)=x^3$; vertical stretch by a factor of 5, horizontal compression by a factor of $\frac{1}{5}$, vertical shift up 20
1g. $y=\frac{1}{2}|0.5x|+7; a=\frac{1}{2}, k=0.5, c=7$; base function $f(x)=|x|$; vertical compression by a factor of $\frac{1}{2}$, horizontal stretch by a factor of 2, vertical shift up 7
1h. $y=-\sqrt{0.25(x+4)}+6; a=-1, k=0.25, d=-4, c=6$
1i. $y=-2\sqrt{-3(x-3)}; a=-2, k=-3, d=-3$
1j. $y=\frac{1}{x+3}-4; d=-3, c=-4$
1k. $y=\frac{-5}{x}+6; a=-5, c=6$
1l. $y=4\frac{1}{2(x+5)}; a=4, k=2, d=-5$