Problem 24 - College Algebra

Question

Begin by graphing the absolute value function, \(f(x)=|x| .\) Then use transformations of this graph to graph the given function. $$ g(x)=|x|+3 $$

Step-by-Step Solution

Answer

The graph of the function \(g(x) = |x| + 3\) is the graph of the absolute value function \(f(x) = |x|\), shifted upwards by 3 units. It intersects the y-axis at the point (0,3).

Step 1: Graphing the Primary Function

First, plot the absolute value function \(f(x) = |x|\). This is a 'v' shaped graph, where when \(x < 0\), \(f(x) = -x\) and when \(x \geq 0\), \(f(x) = x\). The graph intersects the y-axis at the origin (0,0).

Step 2: Analyzing the Transformation

Next, analyze the given function \(g(x) = |x| + 3\). The '+3' suggests a vertical shift. Specifically, it will shift the original absolute function, \(f(x)\), upwards by 3 units.

Step 3: Applying the Transformation

By shifting the graph of \(f(x) = |x|\) upwards by 3 units, the graph of \(g(x) = |x| + 3\) is obtained. The graph will still be 'v' shaped but now intersects the y-axis at (0,3).

Key Concepts

Graph Transformations

Transformation is a powerful tool in mathematics for altering the appearance of a function's graph. When we talk about graph transformations, we mean changes made to a function's graph, such as shifts, stretches, or reflections. Specifically, transformations can be:

• Horizontal and vertical shifts: Moving the entire graph left, right, up, or down without changing its shape.
• Reflections: Flipping the graph over certain lines, like the x-axis or y-axis.
• Stretches and compressions: Expanding or contracting the graph's dimensions.

In the context of absolute value functions like the one discussed here, the most common transformation applied is a vertical shift. This results in the graph moving up or down along the y-axis while maintaining its v-shaped structure.

Vertical Shift

A vertical shift in the graph of a function occurs when you add or subtract a constant value. For the absolute value function, this means changing the output values by raising or lowering the entire graph. With the function \(g(x) = |x| + 3\), we perform a vertical shift upwards by 3 units.
A simple way to observe this transformation is:

• The original graph of \(f(x) = |x|\) meets the y-axis at the origin (0,0).
• By adding +3, each y-value is increased by 3, shifting all points on the graph up by 3 units.
• The vertex of the graph, which was originally at the origin, moves up to (0,3).

Despite the shift, the overall shape of the graph remains the same, retaining its iconic v-shape.

Absolute Value Graphs

Understanding absolute value graphs is fundamental in algebra. The absolute value function \(f(x) = |x|\) has a distinct v-shape. The V’s lowest point, or vertex, is at the origin when no transformations are applied. This v-shape results from how the absolute value function treats negative and positive inputs:

• For \(x \geq 0\), the function is a linear line with a positive slope, resulting in \(f(x) = x\).
• For \(x < 0\), the function reflects negative inputs positively, following \(f(x) = -x\).

Transformations like vertical shifts adjust the graph's position but don't alter this basic v-shape. In the case of \(g(x) = |x| + 3\), we still see this pattern, only the entire structure is elevated on the coordinate plane.
Recognizing these patterns helps in predicting and sketching further transformations.