Transformations

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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).

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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.

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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.

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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.

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Practice

Try It On Your Own

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

AnswersSolutions

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).

Answers

Here are the answers:
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

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