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The absolute value function is a fundamental tool in mathematics that is represented by the equation \( y = |x| \). This function generates a distinctive 'V' shaped graph with its vertex at the origin (0,0). The key feature of this graph is its symmetry about the y-axis, meaning it reflects evenly on both sides. The graph increases linearly as it moves away from the vertex in both directions. It is crucial to understand this base function as it serves as the starting point for more complex graph transformations.

• The vertex is at (0,0).
• The graph opens upwards.
• The slope is 1 in both directions.

This understanding forms a foundation for learning vertical shifts and horizontal scaling of functions.

A vertical shift involves moving the graph of a function up or down without altering its shape. In the context of the absolute value function, consider the graph \( y_1 = 3 - |x| \). This equation shows a vertical shift because the entire graph moves upwards by 3 units. This results in the vertex moving from (0,0) to (0,3).
A vertical shift is straightforward:

• Adding a constant to the function (e.g., \(+3 \)) moves the graph up.
• Subtracting a constant moves the graph down.

The shape of the graph itself doesn't change; it simply relocates higher or lower on the y-axis. Understand this as a simple yet powerful transformation that can redefine the position of any function’s graph.

Horizontal scaling changes the width of the graph by stretching or compressing it along the x-axis. In the function \( y_2 = 3 - |3x| \), we see a horizontal compression. This compresses the graph towards the y-axis due to the factor of 3 multiplied inside the absolute value, meaning that what would normally happen at \( x = 3 \) now occurs at \( x = 1 \). This squeezes the graph into a more narrow shape.
Contrast this with the function \( y_3 = 3 - \left|\frac{1}{3}x\right| \), which represents a horizontal stretch. Here, the graph extends horizontally due to the factor of \( \frac{1}{3} \) inside the absolute value, so effects that occur at \( x = 1 \) in the original graph now occur at \( x = 3 \).
Key things to remember about horizontal scaling:

• Multiplying the input by a factor greater than 1 compresses the graph.
• Multiplying by a fraction stretches the graph.

Both transformations maintain the y-intercept which is affected by vertical shifts, such as moving to the point (0,3).

Using a graphing calculator can be an effective way to visualize graph transformations like vertical shifts and horizontal scaling. With these tools, you can observe how equations like \( y_1 = 3 - |x| \), \( y_2 = 3 - |3x| \), and \( y_3 = 3 - \left|\frac{1}{3}x\right| \) transform from their base forms.
Here's how you can utilize a graphing calculator:

• Enter each function into the calculator to see how the graph is affected by the transformations. Watch how the graph moves or stretches.
• Adjust the viewing window to better observe these changes. Ensure the domain and range are set wide enough to capture all transformations.

Seeing these transformations on a calculator empowers learners to grasp how different algebraic manipulations affect graphical representation. It helps solidify their understanding by connecting algebraic theory to visual graphs.