Parent Functions and Their Transformations worksheet featuring graphs and transformation analysis.

Problem Analysis

The task involves identifying parent functions, describing transformations applied to these functions, and writing the equations of transformed functions. Additionally, there are questions about specific transformations between given functions.

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Part 1: Identifying Parent Functions, Describing Transformations, and Writing Transformation Equations

#### Graph 1
- Parent Function: The graph resembles a V-shape, which is characteristic of the absolute value function.
- Function Name: \( f(x) = |x| \)
- Transformation:
- The graph is shifted horizontally to the right by 2 units.
- The graph is also reflected over the x-axis.
- Transformation Equation:
- Shifting right by 2 units: \( f(x-2) \)
- Reflecting over the x-axis: \( -f(x-2) \)
- Therefore, the equation is: \( g(x) = -|x-2| \)

#### Graph 2
- Parent Function: The graph resembles a square root function.
- Function Name: \( f(x) = \sqrt{x} \)
- Transformation:
- The graph is shifted horizontally to the right by 3 units.
- The graph is also shifted vertically up by 2 units.
- Transformation Equation:
- Shifting right by 3 units: \( f(x-3) \)
- Shifting up by 2 units: \( f(x-3) + 2 \)
- Therefore, the equation is: \( g(x) = \sqrt{x-3} + 2 \)

#### Graph 3
- Parent Function: The graph resembles a cubic function.
- Function Name: \( f(x) = x^3 \)
- Transformation:
- The graph is reflected over the y-axis.
- The graph is also shifted vertically down by 2 units.
- Transformation Equation:
- Reflecting over the y-axis: \( f(-x) \)
- Shifting down by 2 units: \( f(-x) - 2 \)
- Therefore, the equation is: \( g(x) = (-x)^3 - 2 \) or \( g(x) = -x^3 - 2 \)

#### Graph 4
- Parent Function: The graph resembles a reciprocal function.
- Function Name: \( f(x) = \frac{1}{x} \)
- Transformation:
- The graph is stretched vertically by a factor of 3.
- The graph is also shifted vertically up by 1 unit.
- Transformation Equation:
- Stretching vertically by 3: \( 3f(x) \)
- Shifting up by 1 unit: \( 3f(x) + 1 \)
- Therefore, the equation is: \( g(x) = \frac{3}{x} + 1 \)

#### Graph 5
- Parent Function: The graph resembles a linear function.
- Function Name: \( f(x) = x \)
- Transformation:
- The graph is stretched vertically by a factor of 2.
- The graph is also shifted vertically up by 1 unit.
- Transformation Equation:
- Stretching vertically by 2: \( 2f(x) \)
- Shifting up by 1 unit: \( 2f(x) + 1 \)
- Therefore, the equation is: \( g(x) = 2x + 1 \)

#### Graph 6
- Parent Function: The graph resembles a quadratic function.
- Function Name: \( f(x) = x^2 \)
- Transformation:
- The graph is compressed vertically by a factor of \(\frac{1}{2}\).
- The graph is also shifted vertically up by 3 units.
- Transformation Equation:
- Compressing vertically by \(\frac{1}{2}\): \( \frac{1}{2}f(x) \)
- Shifting up by 3 units: \( \frac{1}{2}f(x) + 3 \)
- Therefore, the equation is: \( g(x) = \frac{1}{2}x^2 + 3 \)

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Part 2: Describing Specific Transformations

#### Question 7
- Given Functions: \( f(x) = -9(x-7)^2 + 9 \) and \( g(x) = -9(x-7)^2 + 6 \)
- Transformation:
- The only difference between \( f(x) \) and \( g(x) \) is the constant term at the end.
- \( f(x) \) is shifted vertically down by 3 units to become \( g(x) \).
- Answer: The transformation is a vertical shift downward by 3 units.

#### Question 8
- Given Functions: \( f(x) = -5|x| + 6 \) and \( g(x) = -5|x+2| + 7 \)
- Transformation:
- The term inside the absolute value changes from \( |x| \) to \( |x+2| \), indicating a horizontal shift.
- The constant term changes from 6 to 7, indicating a vertical shift.
- Horizontal shift: \( |x| \to |x+2| \) means a shift to the left by 2 units.
- Vertical shift: \( +6 \to +7 \) means a shift upward by 1 unit.
- Answer: The transformation is a horizontal shift to the left by 2 units and a vertical shift upward by 1 unit.

#### Question 9
- Given Functions: \( f(x) = 3(x)^3 + 1 \) and \( g(x) = 6(x)^3 + 1 \)
- Transformation:
- The coefficient of \( x^3 \) changes from 3 to 6, indicating a vertical stretch.
- The constant term remains the same, so there is no vertical shift.
- Vertical stretch: \( 3(x)^3 \to 6(x)^3 \) means a stretch by a factor of 2.
- Answer: The transformation is a vertical stretch by a factor of 2.

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Final Answers

1. \( g(x) = -|x-2| \)
2. \( g(x) = \sqrt{x-3} + 2 \)
3. \( g(x) = -x^3 - 2 \)
4. \( g(x) = \frac{3}{x} + 1 \)
5. \( g(x) = 2x + 1 \)
6. \( g(x) = \frac{1}{2}x^2 + 3 \)

7. Vertical shift downward by 3 units
8. Horizontal shift to the left by 2 units and vertical shift upward by 1 unit
9. Vertical stretch by a factor of 2

\[
\boxed{
\begin{array}{c}
1. g(x) = -|x-2| \\
2. g(x) = \sqrt{x-3} + 2 \\
3. g(x) = -x^3 - 2 \\
4. g(x) = \frac{3}{x} + 1 \\
5. g(x) = 2x + 1 \\
6. g(x) = \frac{1}{2}x^2 + 3 \\
7. \text{Vertical shift downward by 3 units} \\
8. \text{Horizontal shift to the left by 2 units and vertical shift upward by 1 unit} \\
9. \text{Vertical stretch by a factor of 2}
\end{array}
}
\]