Problem 61 - College Algebra

Question

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 the y-axis and vertically stretch by a factor of 3.

Step 1: Understanding the Transformation

The function given is \(g(x) = 3f(-x)\). First, we recognize that transformations of the graph involve shifts, stretches, or reflections of the original function \(f(x)\). We need to identify each component in the transformation.

Step 2: Horizontal Reflection

The inside of the function \(f\) is \(-x\). This indicates a reflection across the y-axis. If \(f(x)\) is the original function, then \(f(-x)\) reflects the graph over the y-axis.

Step 3: Vertical Stretch

The function \(3f(-x)\) includes a multiplication by 3 outside of the function \(f\). This affects the y-values directly, leading to a vertical stretch of the graph by a factor of 3. This means each y-coordinate on the graph of \(f(-x)\) is tripled.

Key Concepts

Horizontal Reflection

When we talk about a horizontal reflection, we are describing a specific kind of transformation that flips a graph across the y-axis. Imagine the original function, denoted as \(f(x)\). In a horizontal reflection, each point of this graph is mirrored to the opposite side of the y-axis.
For example, if you have a point \((a, b)\) on the graph of \(f(x)\), after a horizontal reflection, the point will move to \((-a, b)\) on the graph of \(f(-x)\). This means that the x-value of each point changes its sign while the y-value remains unchanged.

• This transformation does not affect the height of the graph; it only changes horizontal positioning.
• Every x-coordinate from the original function is flipped to its negative counterpart.
• It is called a *reflection* because the graph essentially mirrors itself over the vertical y-axis.

Vertical Stretch

A vertical stretch is a type of transformation that alters the y-values of a function, making the graph appear taller or flatter. When a function like \(3f(-x)\) is evaluated, the multiplier of 3 applied outside \(f(-x)\) causes this stretch.
This means that for every point \((x, y)\) on the original graph of \(f(-x)\), the new graph will have the point \((x, 3y)\). Here, every y-value is multiplied by 3, "stretching" the graph vertically.

• Only the y-values change; the x-values stay the same.
• The multiplier affects how much the graph is stretched; greater than 1 means stretching, between 0 and 1 means compressing.
• Think of vertical stretch as each point moving away from the x-axis by a consistent factor.

Graph of Function

The graph of a function is a visual representation of all the points that satisfy the function. Essentially, it shows how input values (x-coordinates) relate to output values (y-coordinates). Transformations like horizontal reflections and vertical stretches modify this graph, displaying it in a new form based on these changes.
When looking at the function \(g(x) = 3f(-x)\), we see how these transformations come together:

• First, the original function \(f(x)\) is reflected horizontally across the y-axis to become \(f(-x)\).
• Next, this reflected graph is then vertically stretched by a factor of 3.

Just visualize starting with the base graph of \(f(x)\), reflecting it across the y-axis to create a mirror image, and then pulling that image upwards and downwards to make it taller. Together, these transformations make the graph of \(g(x)\) look distinct from the original \(f(x)\).