Problem 61 - College Algebra

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). $$ g(x)=3 f(-x) $$

Step-by-Step Solution

Answer

Reflect across y-axis, then vertically stretch by a factor of 3.

Step 1: Identifying the Original Function

The original function is given as \( f(x) \). The transformation will be based on the changes applied to this function.

Step 2: Applying Reflection Across the Y-Axis

The term \(-x\) inside the function \( f(-x) \) indicates a reflection of the function \( f(x) \) across the y-axis. This means that for every positive value of \( x \), the function will reflect its output as if it were on the opposite side (negative value) of the y-axis.

Step 3: Vertical Stretch by a Factor of 3

The multiplication by 3 in \( 3f(-x) \) extends the graph vertically. This transformation stretches every y-value of \( f(-x) \) by a factor of 3, making the graph taller by three times its original height.

Step 4: Combining Transformations

The complete transformation involves both reflecting the original graph \( f(x) \) across the y-axis to get \( f(-x) \) and then stretching it vertically by a factor of 3 to obtain \( 3f(-x) \). Combine these two transformations to visualize the new graph.

Key Concepts

Reflection Across the Y-Axis

In graph transformations, reflecting a function across the y-axis involves changing the input variable from positive to negative. When you see a function written as \( f(-x) \), you're dealing with reflection across the y-axis. This change means that for every \( x \) along the horizontal axis, you flip its position to the opposite side. Think of the y-axis as a mirror. The graph of the function remains the same shape, but each point's reflection is now on the opposite side of the line.

Here's how you can visualize it:

• Consider the point \( (a, b) \) on the original graph. After reflection, this point moves to \( (-a, b) \).
• The entire left side of the graph swaps with the right side.

Reflecting across the y-axis does not change the function's domain or range. It's simply a matter of flipping its perspective horizontally.

Vertical Stretch

A vertical stretch occurs when each point along a function's graph is pulled farther away from the x-axis. In mathematical terms, this is achieved by multiplying the output of the function by a constant factor. In the expression \( 3f(-x) \), the factor of 3 signifies a vertical stretch.

Here's how the stretching works:

• Every point on the graph \((x, y)\) is transformed to \((x, 3y)\).
• If a point had a y-value of 2, after the stretch, it becomes 6, making the graph appear three times taller.

A vertical stretch does alter the range of the function because it affects how high and low the graph can go. However, the domain remains unchanged. This kind of transformation pushes the graph upwards and downwards, maintaining the same overall shape but stretching it vertically.

Function Transformations

Graph transformations like reflections and stretches are all under the umbrella of function transformations. They allow us to systematically alter the appearance of a graph while keeping its fundamental properties intact. Understanding these transformations helps us predict and describe how a function's graph will behave under various manipulations.

Function transformations include:

• Translations: Moving the graph up, down, left, or right.
• Reflections: Flipping the graph across the x-axis or y-axis.
• Stretches and Shrinks: Stretching or compressing vertically or horizontally.

By combining different transformations, you can envision how complex changes affect the behavior and representation of functions graphically. In the example of \( g(x) = 3f(-x) \), reflecting and stretching are combined to transform the original graph into something new yet predictable using these principles.