Problem 38 - Beginning and Intermediate Algebra

Question

Use the transformation techniques to graph each of the following functions. $$h(x)=|x+1|-5$$

Step-by-Step Solution

Answer

The given function is \(h(x)=|x+1|-5\). We can transform the basic function \(y=|x|\) with the following steps: first, apply a horizontal translation shifting 1 unit to the left (resulting in \(y=|x+1|\)); second, apply a vertical translation shifting 5 units down (resulting in \(h(x)=|x+1|-5\)). Plot the transformed function to obtain the graph of \(h(x)=|x+1|-5\), a V-shaped graph shifted 1 unit left and 5 units down from \(y=|x|\).

Step 1: Identify the basic function and the transformations

Based on the given function, \(h(x)=|x+1|-5\), we start with the basic function of \(y=|x|\). We can represent the given function as the basic function \(y=|x|\) modified by two transformations: 1. Horizontal translation: Shift to the left by 1 unit (from \(|x|\) to \(|x+1|\)). 2. Vertical translation: Shift downward by 5 units (from \(|x+1|\) to \(|x+1|-5\)).

Step 2: Apply the horizontal translation

To apply the horizontal translation, shift each point on the graph of the basic function, \(y=|x|\), to the left by 1 unit. The points on the shifted graph will now be \((x-1, |x|)\) and the new function becomes \(y=|(x-1)+1|\), i.e \(y=|x+1|\).

Step 3: Apply the vertical translation

Apply the vertical translation to the function with the horizontal translation, which is \(y=|x+1|\), by shifting down each point by 5 units. The points on the transformed graph will now be \((x, |x+1|-5)\). The final function is \(h(x)=|x+1|-5\).

Step 4: Graph the function

Now that we have identified and applied these transformations, we can plot the transformed function, \(h(x)=|x+1|-5\), on a coordinate plane. The key points to plot include the vertex (the point at which the graph reaches a minimum value), and other points on either side. The graph of the final function, \(h(x)=|x+1|-5\), is a V-shaped graph shifted 1 unit to the left and 5 units downward from the basic absolute value function \(y=|x|\).

Key Concepts

Absolute Value Function

The absolute value function is a fundamental building block in algebra. It is expressed as \( y = |x| \), where \(|x|\) represents the distance of \(x\) from zero on the number line.
This makes all values of \(|x|\) non-negative, or positive, thus creating a characteristic V-shape when graphed. Key points to remember about the absolute value function:

• The graph of \( y = |x| \) is symmetric about the y-axis.
• It has a vertex at the origin (0,0).
• The slope is positive on both sides of the vertex.

The absolute value function is the starting point in many function transformations, as seen in steps involving horizontal or vertical shifts.

Horizontal Translation

Horizontal translation involves shifting the graph of a function left or right on the coordinate plane. For functions like \( y = |x| \), adding or subtracting a constant inside the absolute value results in a horizontal shift.
In our example, the function \( h(x) = |x + 1| \) indicates a horizontal translation. To understand this:

• Replacing \(x\) with \(x + 1\) means shifting the graph 1 unit to the left.
• If it were \(x - 1\), the shift would be 1 unit to the right instead.

This transformation keeps the shape of the graph unchanged, only altering its position along the x-axis.

Vertical Translation

Vertical translation shifts the graph of a function up or down on the coordinate plane. In the case of the absolute value function, adjusting a constant outside the \(|x|\) affects this translation.
Specifically, the function \( h(x) = |x + 1| - 5 \).The vertical translation can be broken down as:

• Here, subtracting 5 indicates the graph is shifted 5 units downward.
• This impacts the y-values, moving each point directly along the y-axis.

Vertical translations do not alter the horizontal position (or the vertex if on the line of symmetry), only the vertical displacement.

Graphing Functions

Graphing functions is crucial as it visually represents mathematical relationships. For our function \( h(x) = |x+1|-5 \), it involves first graphing \( y = |x| \).
Then, applying the transformations sequentially:- With a horizontal shift to the left by 1 unit.- Then a vertical shift downward by 5 units.To plot this:

• Identify the new vertex, which shifts from (0,0) to (-1,-5).
• Draw the characteristic V-shape, ensuring symmetry about the vertical axis at the new position.

Ensure key points are marked accurately to display the translation effects. This process highlights the transformed function's minimum value and direction, maintaining the absolute value structure.