Problem 56 - College Algebra

Question

For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). $$ g(x)=6 f(x) $$

Step-by-Step Solution

Answer

The graph of \(g(x)\) is a vertical stretch of the graph of \(f(x)\) by a factor of 6.

Step 1: Understand the Original Function

The original function is given as \(f(x)\). The graph of this function is our base which will be transformed to achieve the graph of \(g(x)\).

Step 2: Identify the Transformation

The function \(g(x) = 6f(x)\) represents a vertical scaling or stretching of the graph of the original function \(f(x)\).

Step 3: Analyze the Effect of the Transformation

Multiplying \(f(x)\) by 6 means that every output of the function is scaled by a factor of 6. This stretches the graph vertically, making it 6 times taller at every point. If a point on \(f(x)\) at \(x = a\) is \(y = b\), on \(g(x)\), the point at \(x = a\) is \(y = 6b\).

Step 4: Describe the Graph Changes

The whole graph of \(f(x)\) is stretched vertically. Peaks become 6 times higher and valleys become 6 times deeper, but the x-coordinates remain unchanged.

Key Concepts

Vertical Stretching

When we talk about vertical stretching in the context of graph transformations, we refer to the change in the vertical size of a graph. Imagine taking a rubber band that represents your graph and pulling it upwards or downwards, making it taller while keeping its width the same. Vertical stretching affects how tall the peaks and valleys of a graph appear.

• It happens when you multiply the function by a constant greater than 1.
• All points on the graph are moved further from the x-axis, increasing their distance from the origin in the vertical aspect.
• The x-coordinates remain unchanged, meaning the graph only changes in the y-direction.

In our example with the function transformation from \(f(x)\) to \(g(x) = 6f(x)\), the graph of \(f(x)\) is stretched vertically by a factor of 6. This means every y-value is now six times its original value, making the peaks significantly higher and the valleys much deeper.

Function Transformation

Function transformation involves changing the appearance of the graph in some way by altering the function. This can include shifting, reflecting, or stretching the graph. Each type of transformation has its own rules and results in specific changes to the graph.

• Shifting: Moving the graph horizontally or vertically, without resizing it.
• Reflecting: Flipping the graph across an axis.
• Stretching: Changing the size of the graph in one direction (like vertical stretching).

In the case of \(g(x) = 6f(x)\), the transformation is a vertical stretch. We're not shifting or reflecting, just altering how each point on the graph relates to the x-axis by stretching it vertically.

Scaling Factor

The scaling factor in function transformation is the constant by which you multiply the function. It determines how much you stretch or compress the graph. A scaling factor greater than 1 enlarges the graph (a stretch), while a factor between 0 and 1 compresses it.

• For vertical changes, this factor is applied directly to the function, affecting the y-coordinates.
• Larger scaling factors result in greater stretching of the graph.
• In our example, the scaling factor is 6, leading to a vertical stretch.

It's important to understand that the scaling factor only alters the size or height of the graph, and does not affect the horizontal placement. In \(g(x) = 6f(x)\), each output is multiplied by 6, illustrating a complete stretching of the graph by this factor.