Graphing Absolute Value Equations.ks-ia2 - Kuta Software - Free Printable

To solve the problem of graphing each absolute value equation, we need to understand the general behavior of absolute value functions and how transformations affect their graphs. Let's go through each equation step by step.

General Form of Absolute Value Function

The basic absolute value function is:
\[ y = |x| \]
This graph is a V-shaped curve with its vertex at the origin \((0, 0)\).

Transformations

1. Horizontal Shift: \( y = |x - h| \) shifts the graph horizontally by \( h \) units. If \( h > 0 \), it shifts right; if \( h < 0 \), it shifts left.
2. Vertical Shift: \( y = |x| + k \) shifts the graph vertically by \( k \) units. If \( k > 0 \), it shifts up; if \( k < 0 \), it shifts down.
3. Reflection: \( y = -|x| \) reflects the graph over the x-axis.
4. Combination: Combining these transformations allows us to graph more complex absolute value functions.

Graphing Each Equation

#### 1. \( y = |x - 1| \)
- Transformation: Horizontal shift right by 1 unit.
- Vertex: The vertex of \( y = |x| \) is at \((0, 0)\). Shifting right by 1 unit moves the vertex to \((1, 0)\).
- Graph: Draw a V-shaped curve with the vertex at \((1, 0)\).

#### 2. \( y = |x + 4| \)
- Transformation: Horizontal shift left by 4 units.
- Vertex: The vertex of \( y = |x| \) is at \((0, 0)\). Shifting left by 4 units moves the vertex to \((-4, 0)\).
- Graph: Draw a V-shaped curve with the vertex at \((-4, 0)\).

#### 3. \( y = |x - 2| \)
- Transformation: Horizontal shift right by 2 units.
- Vertex: The vertex of \( y = |x| \) is at \((0, 0)\). Shifting right by 2 units moves the vertex to \((2, 0)\).
- Graph: Draw a V-shaped curve with the vertex at \((2, 0)\).

#### 4. \( y = -|x - 2| \)
- Transformation: Horizontal shift right by 2 units and reflection over the x-axis.
- Vertex: The vertex of \( y = |x| \) is at \((0, 0)\). Shifting right by 2 units moves the vertex to \((2, 0)\). Reflecting over the x-axis inverts the V-shape.
- Graph: Draw an inverted V-shaped curve with the vertex at \((2, 0)\).

#### 5. \( y = -|x| - 1 \)
- Transformation: Reflection over the x-axis and vertical shift down by 1 unit.
- Vertex: The vertex of \( y = |x| \) is at \((0, 0)\). Reflecting over the x-axis moves it to \((0, 0)\), and shifting down by 1 unit moves the vertex to \((0, -1)\).
- Graph: Draw an inverted V-shaped curve with the vertex at \((0, -1)\).

#### 6. \( y = -|x - 1| \)
- Transformation: Horizontal shift right by 1 unit and reflection over the x-axis.
- Vertex: The vertex of \( y = |x| \) is at \((0, 0)\). Shifting right by 1 unit moves the vertex to \((1, 0)\). Reflecting over the x-axis inverts the V-shape.
- Graph: Draw an inverted V-shaped curve with the vertex at \((1, 0)\).

Final Answer

The graphs for each equation are as follows:

1. \( y = |x - 1| \): V-shaped curve with vertex at \((1, 0)\).
2. \( y = |x + 4| \): V-shaped curve with vertex at \((-4, 0)\).
3. \( y = |x - 2| \): V-shaped curve with vertex at \((2, 0)\).
4. \( y = -|x - 2| \): Inverted V-shaped curve with vertex at \((2, 0)\).
5. \( y = -|x| - 1 \): Inverted V-shaped curve with vertex at \((0, -1)\).
6. \( y = -|x - 1| \): Inverted V-shaped curve with vertex at \((1, 0)\).

\[
\boxed{
\begin{array}{c}
\text{1. } y = |x - 1| \\
\text{2. } y = |x + 4| \\
\text{3. } y = |x - 2| \\
\text{4. } y = -|x - 2| \\
\text{5. } y = -|x| - 1 \\
\text{6. } y = -|x - 1| \\
\end{array}
}
\]