College Algebra Test #2 Review – Step-by-Step Guidance

Q14. Begin by graphing the standard absolute value function $f(x) = |x|$. Then use transformations of this graph to graph the given function $g(x) = \frac{1}{4}|x + 5| + 6$.

Background

Topic: Graphing Absolute Value Functions with Transformations

This question tests your understanding of how to graph absolute value functions and apply transformations such as shifts, stretches/compressions, and translations.

Key Terms and Formulas:

• Absolute Value Function: $f(x) = |x|$ is a V-shaped graph with its vertex at the origin (0,0).

• Vertical Stretch/Compression: $a|x|$ stretches the graph vertically if $|a| > 1$ and compresses it if $0 < |a| < 1$.

• Horizontal Shift: $|x + h|$ shifts the graph left by $h$ units if $h > 0$.

• Vertical Shift: $|x| + k$ shifts the graph up by $k$ units if $k > 0$.

Step-by-Step Guidance

1. Start by sketching the basic graph of $f(x) = |x|$. This is a V-shaped graph with its vertex at (0,0).

   

2. Apply the horizontal shift: $|x + 5|$ shifts the graph 5 units to the left. The vertex moves from (0,0) to (-5,0).

3. Apply the vertical compression: The coefficient $\frac{1}{4}$ in front of the absolute value compresses the graph vertically by a factor of 4. This means the arms of the V are less steep.

4. Apply the vertical shift: Adding 6 to the function, $+6$, shifts the entire graph up by 6 units. The new vertex is at (-5, 6).

Try solving on your own before revealing the answer!

Final Answer:

The correct graph is shown in image_2. The vertex is at (-5, 6), and the graph is a vertically compressed V-shape.

This matches the transformations described: left 5 units, up 6 units, and vertical compression by 1/4.