SOLUTION: Kuta Software Infinite Algebra 1: Graphing Absolute ... - Free Printable

Problem: Graphing Absolute Value Functions

The task is to graph each of the given absolute value functions. Let's solve and explain the solution for each equation step by step.

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#### 1. \( y = |x - 2| - 4 \)

##### Step 1: Understand the basic shape
The general form of an absolute value function is \( y = |x| \), which produces a V-shaped graph with its vertex at the origin \((0, 0)\).

##### Step 2: Identify transformations
The given function is \( y = |x - 2| - 4 \). Let's break down the transformations:
- Horizontal shift: The term \( x - 2 \) inside the absolute value indicates a horizontal shift to the right by 2 units.
- Vertical shift: The term \(-4\) outside the absolute value indicates a vertical shift downward by 4 units.

##### Step 3: Find the vertex
The vertex of the basic \( y = |x| \) graph is at \((0, 0)\). After applying the transformations:
- Horizontal shift right by 2 units: The vertex moves from \((0, 0)\) to \((2, 0)\).
- Vertical shift down by 4 units: The vertex moves from \((2, 0)\) to \((2, -4)\).

So, the vertex of \( y = |x - 2| - 4 \) is at \((2, -4)\).

##### Step 4: Plot points
To graph the function, we can plot a few points around the vertex:
- When \( x = 2 \): \( y = |2 - 2| - 4 = 0 - 4 = -4 \). Point: \((2, -4)\).
- When \( x = 1 \): \( y = |1 - 2| - 4 = |-1| - 4 = 1 - 4 = -3 \). Point: \((1, -3)\).
- When \( x = 3 \): \( y = |3 - 2| - 4 = |1| - 4 = 1 - 4 = -3 \). Point: \((3, -3)\).
- When \( x = 0 \): \( y = |0 - 2| - 4 = |-2| - 4 = 2 - 4 = -2 \). Point: \((0, -2)\).
- When \( x = 4 \): \( y = |4 - 2| - 4 = |2| - 4 = 2 - 4 = -2 \). Point: \((4, -2)\).

##### Step 5: Draw the graph
Plot the vertex \((2, -4)\) and the points \((1, -3)\), \((3, -3)\), \((0, -2)\), and \((4, -2)\). Connect these points with two straight lines to form a V-shape.

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#### 2. \( y = |x + 1| \)

##### Step 1: Understand the basic shape
The basic shape is the V-shape of \( y = |x| \).

##### Step 2: Identify transformations
The given function is \( y = |x + 1| \). Let's break down the transformation:
- Horizontal shift: The term \( x + 1 \) inside the absolute value indicates a horizontal shift to the left by 1 unit.

##### Step 3: Find the vertex
The vertex of the basic \( y = |x| \) graph is at \((0, 0)\). After applying the transformation:
- Horizontal shift left by 1 unit: The vertex moves from \((0, 0)\) to \((-1, 0)\).

So, the vertex of \( y = |x + 1| \) is at \((-1, 0)\).

##### Step 4: Plot points
To graph the function, we can plot a few points around the vertex:
- When \( x = -1 \): \( y = |-1 + 1| = |0| = 0 \). Point: \((-1, 0)\).
- When \( x = -2 \): \( y = |-2 + 1| = |-1| = 1 \). Point: \((-2, 1)\).
- When \( x = 0 \): \( y = |0 + 1| = |1| = 1 \). Point: \((0, 1)\).
- When \( x = -3 \): \( y = |-3 + 1| = |-2| = 2 \). Point: \((-3, 2)\).
- When \( x = 1 \): \( y = |1 + 1| = |2| = 2 \). Point: \((1, 2)\).

##### Step 5: Draw the graph
Plot the vertex \((-1, 0)\) and the points \((-2, 1)\), \((0, 1)\), \((-3, 2)\), and \((1, 2)\). Connect these points with two straight lines to form a V-shape.

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#### 3. \( y = |x| + 1 \)

##### Step 1: Understand the basic shape
The basic shape is the V-shape of \( y = |x| \).

##### Step 2: Identify transformations
The given function is \( y = |x| + 1 \). Let's break down the transformation:
- Vertical shift: The term \(+1\) outside the absolute value indicates a vertical shift upward by 1 unit.

##### Step 3: Find the vertex
The vertex of the basic \( y = |x| \) graph is at \((0, 0)\). After applying the transformation:
- Vertical shift up by 1 unit: The vertex moves from \((0, 0)\) to \((0, 1)\).

So, the vertex of \( y = |x| + 1 \) is at \((0, 1)\).

##### Step 4: Plot points
To graph the function, we can plot a few points around the vertex:
- When \( x = 0 \): \( y = |0| + 1 = 0 + 1 = 1 \). Point: \((0, 1)\).
- When \( x = 1 \): \( y = |1| + 1 = 1 + 1 = 2 \). Point: \((1, 2)\).
- When \( x = -1 \): \( y = |-1| + 1 = 1 + 1 = 2 \). Point: \((-1, 2)\).
- When \( x = 2 \): \( y = |2| + 1 = 2 + 1 = 3 \). Point: \((2, 3)\).
- When \( x = -2 \): \( y = |-2| + 1 = 2 + 1 = 3 \). Point: \((-2, 3)\).

##### Step 5: Draw the graph
Plot the vertex \((0, 1)\) and the points \((1, 2)\), \((-1, 2)\), \((2, 3)\), and \((-2, 3)\). Connect these points with two straight lines to form a V-shape.

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#### 4. \( y = |x| + 2 \)

##### Step 1: Understand the basic shape
The basic shape is the V-shape of \( y = |x| \).

##### Step 2: Identify transformations
The given function is \( y = |x| + 2 \). Let's break down the transformation:
- Vertical shift: The term \(+2\) outside the absolute value indicates a vertical shift upward by 2 units.

##### Step 3: Find the vertex
The vertex of the basic \( y = |x| \) graph is at \((0, 0)\). After applying the transformation:
- Vertical shift up by 2 units: The vertex moves from \((0, 0)\) to \((0, 2)\).

So, the vertex of \( y = |x| + 2 \) is at \((0, 2)\).

##### Step 4: Plot points
To graph the function, we can plot a few points around the vertex:
- When \( x = 0 \): \( y = |0| + 2 = 0 + 2 = 2 \). Point: \((0, 2)\).
- When \( x = 1 \): \( y = |1| + 2 = 1 + 2 = 3 \). Point: \((1, 3)\).
- When \( x = -1 \): \( y = |-1| + 2 = 1 + 2 = 3 \). Point: \((-1, 3)\).
- When \( x = 2 \): \( y = |2| + 2 = 2 + 2 = 4 \). Point: \((2, 4)\).
- When \( x = -2 \): \( y = |-2| + 2 = 2 + 2 = 4 \). Point: \((-2, 4)\).

##### Step 5: Draw the graph
Plot the vertex \((0, 2)\) and the points \((1, 3)\), \((-1, 3)\), \((2, 4)\), and \((-2, 4)\). Connect these points with two straight lines to form a V-shape.

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Final Answer:

\[
\boxed{
\begin{array}{ll}
1. & y = |x - 2| - 4 \\
2. & y = |x + 1| \\
3. & y = |x| + 1 \\
4. & y = |x| + 2 \\
\end{array}
}
\]