Problem 16 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of \(g(x)=|x+3|-4\) by first sketching \(h(x)=|x|\) and then translating.

Step-by-Step Solution

Answer

Translate \(|x|\) left 3 units and down 4 units.

Step 1: Understand the basic graph

The function to start with is the absolute value function, which is \(h(x) = |x|\). The graph of \(h(x) = |x|\) is a V-shaped graph that opens upwards with the vertex at the origin (0,0). Identify this shape and the position.

Step 2: Translate horizontally

The expression inside the absolute value function in \(g(x) = |x+3| - 4\) is \(x+3\). This means we need to perform a horizontal translation of the \(h(x) = |x|\) graph. Replace \(x\) with \(x+3\) which shifts the graph 3 units to the left. Now the vertex of \(|x+3|\) is at (-3,0).

Step 3: Translate vertically

Next, adjust for the '-4' outside the absolute value to complete the translation. This shifts the graph down by 4 units. The vertex of the graph \(g(x) = |x+3| - 4\) is now at (-3,-4).

Step 4: Sketch the final graph

Draw the translated V-shaped graph with the new vertex at (-3, -4). The arms of the 'V' rise upwards at 45-degree angles from the vertex.

Key Concepts

Function Transformations

Function transformations help us understand how changes to a function's equation affect its graph. They can drastically change the position, shape, and orientation of a graph on a coordinate plane. Here are some fundamental transformations:

• Translation: This moves the entire graph horizontally or vertically without altering its shape or orientation.
• Reflection: This flips the graph over a line, such as the x-axis or y-axis, creating a mirror image.
• Stretching/Shrinking: This alters the graph's shape in a specific direction, either making it wider or narrower.

In the exercise, we focus on translation, observing how these movements modify the graph of the absolute value function to obtain the desired graph of the transformed function.

Horizontal Translation

Horizontal translation involves shifting a graph left or right along the x-axis. When analyzing expressions like \(x + c\), the term \(c\) indicates the movement direction and magnitude:

• If \(x + c\), move the graph \(|c|\) units left if \(c > 0\).
• If \(x - c\), move the graph \(|c|\) units right if \(c > 0\).

In the given function \(|x + 3|\), the graph of \(h(x) = |x|\) shifts 3 units left.
Hence, the vertex initially at the origin (0,0) moves to (-3,0). This is a crucial step, as it sets the stage for further transformations. Remember to always check the signs to determine the correct direction of movement.

Vertical Translation

Vertical translation shifts a graph up or down along the y-axis. This movement is dictated by the constant term outside the function. In the exercise, \(|x+3| - 4\), changes occur vertically:

• A positive constant, say \(d\), shifts the graph \(|d|\) units up.
• A negative constant, such as \(-4\) in our case, shifts it \(|-4| = 4\) units down.

Following the horizontal shift, the graph's new vertex at (-3,0) descends 4 units to (-3,-4). This adjustment completes the transformation, aligning the graph accurately on the coordinate plane. Vertical translations are straightforward: look to the constant, then shift accordingly.

Graph Sketching

Once all transformations are understood, sketching the graph is a simple process. Begin with the function \(h(x) = |x|\), a V-shaped graph centered at the origin. Apply each transformation step-by-step:

• First: Shift left 3 units from (0,0) to (-3,0).
• Then: Shift down 4 units to arrive at the final vertex position of (-3,-4).

This produces a translated V-shape with its arms extending upward at 45-degree angles from the vertex. Be sure to maintain the V's symmetry during sketching. Finally, label your axes and vertex clearly for clarity. Understanding graph transformations allows for accurate sketching of any given function, providing a visual representation of its behavior.