Write an Absolute Value Function from the Given Graph - Algebra-2

1. Fundamental Concepts

Absolute Value Function Form: The general form of an absolute value function is , where is the vertex of the V-shaped graph, and a determines the vertical stretch, compression, or reflection (direction of opening).
Graph Features: The graph of an absolute value function is a V-shape with a vertex (the "corner" of the V) as its key feature. The vertex’s coordinates and the value of a uniquely define the function.

2. Key Concepts

Vertex Identification: The vertex of the graph is the starting point. It is the point where the graph changes direction (the minimum or maximum point of the V-shape).
Role of a:
- Sign of a: Determines the direction of opening. If , the graph opens upward; if , it opens downward (reflected over the x-axis).
- Magnitude of a: Determines vertical stretch or compression. If , the graph is vertically stretched (narrower than the parent function ); if , it is vertically compressed (wider than the parent function).
Relationship to Parent Function: All absolute value graphs are transformations of the parent function (vertex at , , opens upward).

3. Examples

Easy

Problem 1: Write the equation for the given graph.

Step-by-Step Solution:

The graph shows a shift to the right by 3 units.
The base function is y = |x|.
Apply the horizontal shift transformation: y = |x - 3| .

Problem 2: A graph of an absolute value function has a vertex at and opens upward with no stretch or compression (same width as ). What is its equation?
Solution: The vertex , so , . Since there’s no stretch/compression and it opens upward, .
Equation: .

Medium

Problem 1: Write the equation for the given graph.

Step-by-Step Solution:

The graph shows a reflection over the x-axis and a vertical shift up by 2 units.
The base function is y = |x| .
Apply the reflection transformation: y = -|x| .
Apply the vertical shift transformation: y = -|x| + 2 .

Validation: Substituting into the original and transformed equations confirms the transformations.

Problem 2: A graph of an absolute value function has a vertex at , opens downward, and is vertically stretched by a factor of 2. What is its equation?
Solution: Vertex , so , . Opens downward means (stretch factor 2).
Equation: , simplified to .

Hard: Determine a from a Point on the Graph

Problem: A graph of an absolute value function has a vertex at . It passes through the point and opens upward. Find its equation.

Solution:

Vertex , so the function is .
Substitute the point into the equation: .
Simplify: → → . Equation: .

4. Problem-Solving Techniques

Step 1: Locate the Vertex: Identify the vertex from the graph (the "corner" of the V-shape). This gives the values of h and k in the general form.
Step 2: Determine the Direction of Opening: Check if the graph opens upward () or downward () by observing the "direction" of the V-shape.
Step 3: Find a Using a Point on the Graph: Pick a clear point that lies on the graph (other than the vertex). Substitute x, y, h, and k into , then solve for a.
Step 4: Write the Equation: Substitute a, h, and k back into the general form and simplify if needed.
Verification: To ensure accuracy, substitute the vertex and the chosen point back into the equation to confirm they satisfy it.