Solved IS MIU ruces 2.7-Use Absolute Value Functions and | Chegg.com - Free Printable

Problem Overview:

The task involves graphing absolute value functions and comparing them to the parent function \( y = |x| \). The worksheet focuses on transformations of absolute value functions, including translations, scale changes (stretches, shrinks, and reflections), and understanding their effects on the graph.

Step-by-Step Solution:

#### 1. Parent Function of Absolute Value Functions: \( y = |x| \)

The parent function \( y = |x| \) is a V-shaped graph with the following properties:
- Vertex: The vertex of \( y = |x| \) is at the origin, \( (0, 0) \).
- Symmetry: The graph is symmetric about the y-axis.

##### Table for \( y = |x| \):
We can create a table of values for \( y = |x| \):

| \( x \) | \( y = |x| \) |
|---------|---------------|
| -2 | 2 |
| -1 | 1 |
| 0 | 0 |
| 1 | 1 |
| 2 | 2 |

##### Graph of \( y = |x| \):
- Plot the points from the table: \( (-2, 2) \), \( (-1, 1) \), \( (0, 0) \), \( (1, 1) \), \( (2, 2) \).
- Connect the points to form a V-shape with the vertex at \( (0, 0) \).

#### 2. Translation of Absolute Value Functions: \( y = |x - h| + k \)

The general form \( y = |x - h| + k \) represents a translation of the parent function \( y = |x| \):
- \( h \): Horizontal shift (right if \( h > 0 \), left if \( h < 0 \)).
- \( k \): Vertical shift (up if \( k > 0 \), down if \( k < 0 \)).

#### 3. Graph the Following Absolute Value Equations

##### a) \( y = |x + 1| - 4 \)

This equation can be rewritten as \( y = |x - (-1)| - 4 \), which indicates:
- A horizontal shift left by 1 unit (\( h = -1 \)).
- A vertical shift down by 4 units (\( k = -4 \)).

###### Steps to Graph:
1. Start with the parent function \( y = |x| \).
2. Shift the graph left by 1 unit.
3. Shift the graph down by 4 units.

###### Vertex:
The vertex of the parent function \( y = |x| \) is at \( (0, 0) \). After the transformations:
- Shifting left by 1 unit moves the vertex to \( (-1, 0) \).
- Shifting down by 4 units moves the vertex to \( (-1, -4) \).

###### Symmetry:
The graph remains symmetric about the vertical line \( x = -1 \).

###### Table for \( y = |x + 1| - 4 \):
| \( x \) | \( y = |x + 1| - 4 \) |
|---------|-----------------------|
| -3 | 0 |
| -2 | -3 |
| -1 | -4 |
| 0 | -3 |
| 1 | 0 |

###### Graph:
Plot the points and connect them to form a V-shape with the vertex at \( (-1, -4) \).

##### b) \( y = \frac{1}{2}|x| \)

This equation represents a vertical shrink of the parent function \( y = |x| \) by a factor of \( \frac{1}{2} \):
- Every \( y \)-value of the parent function is multiplied by \( \frac{1}{2} \).

###### Steps to Graph:
1. Start with the parent function \( y = |x| \).
2. Multiply each \( y \)-value by \( \frac{1}{2} \).

###### Vertex:
The vertex remains at \( (0, 0) \) because the transformation does not affect the vertex.

###### Symmetry:
The graph remains symmetric about the y-axis.

###### Table for \( y = \frac{1}{2}|x| \):
| \( x \) | \( y = \frac{1}{2}|x| \) |
|---------|--------------------------|
| -2 | 1 |
| -1 | 0.5 |
| 0 | 0 |
| 1 | 0.5 |
| 2 | 1 |

###### Graph:
Plot the points and connect them to form a V-shape that is narrower than the parent function \( y = |x| \).

#### 4. Scale Change – Stretches, Shrinks, and Reflections: \( y = a|x| \)

The general form \( y = a|x| \) represents a vertical stretch or shrink of the parent function \( y = |x| \):
- If \( |a| > 1 \), the graph is stretched vertically.
- If \( 0 < |a| < 1 \), the graph is shrunk vertically.
- If \( a < 0 \), the graph is reflected across the x-axis.

##### Summary:
- For \( y = \frac{1}{2}|x| \) (part b), since \( a = \frac{1}{2} \) and \( 0 < |a| < 1 \), the graph is shrunk vertically.

Final Answer:

\[
\boxed{
\text{See detailed steps above for graphs and explanations.}
}
\]