Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Describe the sequence of transformations from \(f(x)=|x|\) to \(g .\) Then sketch the graph of \(g\) by hand. Verify with a graphing utility. \(g(x)=-|x|+3\)

Step-by-Step Solution

Answer

The transformation from \(f(x)=|x|\) to \(g(x)=-|x|+3\) involves reflecting the base graph over the x-axis (due to the negative sign) and then shifting it upwards by 3 units (due to the '+3'). Thus, the graph of \(g(x)\) is a downward-opening V shape with the vertex at (0,3).

Step 1: Identify the Base and Target Functions

The basic function given is \(f(x)=|x|\), and the function to transform into is \(g(x)=-|x|+3\). Understanding these two functions is the foundation for this exercise.

Step 2: Breaking Down Transformations

The transformation from \(f(x)\) to \(g(x)\) involves a reflection across the x-axis and a vertical shift up by 3 units. The negative sign in front of \(|x|\) in \(g(x)\) signifies a reflection across the x-axis, and the '+3' shows a shift upward by 3 units.

Step 3: Sketching the Graph

Start with a sketch of \(f(x)=|x|\). The graph of \(f(x)=|x|\) is a V shape with the point of the V at the origin (0,0) and arms extending into the first and second quadrants. Next, reflect this graph across the x-axis to account for the '-' sign, creating a V shape that opens downward. Then, shift this graph up by 3 units because of the '+3', raising the lowest point of the V from (0,0) to (0,3). This is the graph of \(g(x)=-|x|+3\).

Step 4: Verify with a Graphing Utility

Any graphing software or calculator can be used to plot \(g(x)=-|x|+3\). The graph should look identical to the one drawn in the previous step, thus confirming the correctness of the transformations.

Key Concepts

Absolute value function

The absolute value function, denoted as \( f(x) = |x| \), is a fundamental mathematical function that returns the non-negative value of \( x \). It forms a V-shaped graph. The vertex, or the point of the "V", is at the origin (0,0), where both arms of the V extend into the first and second quadrants.

• The left arm of the V represents the equation \( y = -x \) for \( x < 0 \).
• The right arm represents \( y = x \) for \( x \geq 0 \).

The absolute value function is pivotal in transformations since its structure is easily altered by shifts, reflections, and stretches. Understanding this fundamental graph is crucial before tackling more complex transformations.

Reflection across the x-axis

Reflection across the x-axis changes the orientation of a graph by 'flipping' it over the x-axis. For the absolute value function \( f(x) = |x| \), the transformation into \( -|x| \) reflects the graph so that the V shape "opens" downward.

• This transformation takes every y-coordinate and multiplies it by -1.
• The vertex of the V at the origin remains at (0,0).
• Making a positive slope negative and vice versa.

The importance of this transformation lies in its ability to effectively modify the steepness and direction of the graph. By understanding reflections, one can predict how the shape and pattern of the graph will appear after being transformed.

Vertical shift

A vertical shift moves every point on a graph up or down by a certain amount. In the case of our function transition from \( f(x)= |x| \) to \( g(x) = -|x| + 3 \), there is a vertical shift upward by 3 units due to the '+3'.

• This shift moves the vertex of the V shape from (0,0) to (0,3).
• All other points on the graph are moved vertically by the same amount.

Vertical shifts are straightforward but essential in transforming functions. They help tailor the position of a graph to fit particular needs without altering its shape or orientation. This shift is added to transformations where both the reflection and the vertical shift are applied sequentially.

Graph sketching

Graph sketching is an invaluable skill that combines visualization and mathematical understanding to manually create graphs of functions. It involves interpreting transformations like reflections and shifts smoothly into a visual format.

• Begin with the basic graph, such as \( f(x) = |x| \), and apply each transformation step by step.
• Reflect the graph across the x-axis to get \( -|x| \).
• Apply the vertical shift upwards by 3 units to represent \( g(x) = -|x| + 3 \).

After graph sketching manually, confirming with a graphing utility is recommended. It helps verify accuracy and ensures transformations align perfectly. This iterative process builds confidence in recognizing and applying function transformations.