Problem 60
Question
For the following exercises, describe how the graph of each function is a transformation of the graph of the original function \(f\). $$ g(x)=f\left(\frac{1}{5} x\right) $$
Step-by-Step Solution
Verified Answer
The graph is stretched horizontally by a factor of 5.
1Step 1: Identify the Original Function
The given function is \( g(x) = f\left(\frac{1}{5} x\right) \). The original function is \( f(x) \). We need to identify how \( g(x) \) transforms the graph of \( f(x) \).
2Step 2: Analyze the Transformation
The expression \( f\left(\frac{1}{5} x\right) \) involves a horizontal transformation. Specifically, \( \frac{1}{5} x \) indicates a horizontal stretch.
3Step 3: Determine the Type of Transformation
To transform \( f(x) \) into \( f\left(\frac{1}{5} x\right) \), we are essentially stretching the graph horizontally by a factor of 5. This is because the coefficient of \( x \) (\( \frac{1}{5} \)) is less than 1, which means stretching rather than compression.
4Step 4: Conclude with the Transformation Effect
The graph of \( g(x) = f\left(\frac{1}{5} x\right) \) is the graph of \( f(x) \) stretched horizontally by a factor of 5.
Key Concepts
Horizontal StretchGraph TransformationsCoefficient of x
Horizontal Stretch
In function transformations, a horizontal stretch affects the width of the graph. When we have a function such as \( g(x) = f\left(\frac{1}{5} x\right) \), it indicates a horizontal change. Particularly, the expression \( \frac{1}{5} x \) signifies a horizontal stretch because the coefficient is a fraction less than 1.
Consider the original graph of \( f(x) \). In a horizontal stretch, each point on the graph moves farther away from the y-axis, making the graph wider. In our example, every point on \( f(x) \) is moved 5 times further away in the horizontal direction. This is achieved by multiplying each x-value by 5, resulting in a more stretched, wider graph.
Remember, when the coefficient of \( x \) in the function is a fraction (e.g., \( \frac{1}{5} \)), it leads to a stretch. Conversely, if the coefficient were greater than 1, the graph would undergo a compression. So, always check the coefficient to determine if it's a stretch or compression.
Consider the original graph of \( f(x) \). In a horizontal stretch, each point on the graph moves farther away from the y-axis, making the graph wider. In our example, every point on \( f(x) \) is moved 5 times further away in the horizontal direction. This is achieved by multiplying each x-value by 5, resulting in a more stretched, wider graph.
Remember, when the coefficient of \( x \) in the function is a fraction (e.g., \( \frac{1}{5} \)), it leads to a stretch. Conversely, if the coefficient were greater than 1, the graph would undergo a compression. So, always check the coefficient to determine if it's a stretch or compression.
Graph Transformations
Graph transformations involve altering the basic position and form of a graph. Transformations can be vertical, horizontal, or both, changing where a graph is positioned and its shape.
For our function \( g(x) = f\left(\frac{1}{5}x\right) \), we're only seeing a horizontal transformation. In this case, transformation refers exclusively to the horizontal stretch applied to \( f(x) \). This process involves effectively pulling the graph apart along the x-axis, resulting in a wider graph.
Different transformations include:
For our function \( g(x) = f\left(\frac{1}{5}x\right) \), we're only seeing a horizontal transformation. In this case, transformation refers exclusively to the horizontal stretch applied to \( f(x) \). This process involves effectively pulling the graph apart along the x-axis, resulting in a wider graph.
Different transformations include:
- Translations: Shifts the entire graph horizontally or vertically.
- Reflections: Flips the graph across a specific axis.
- Stretches and Compressions: Expand or contract the graph along the x or y-axis.
Coefficient of x
The coefficient of \( x \) plays a crucial role in determining the nature of a graph transformation. In mathematical terms, the coefficient of \( x \) directly influences whether a graph undergoes a horizontal stretch or compression.
For the function \( g(x) = f\left(\frac{1}{5}x\right) \), the coefficient is \( \frac{1}{5} \). This coefficient is less than 1, indicating a horizontal stretch. The interpretation here is important: the smaller the coefficient relative to 1, the more pronounced the stretch will be.
Key effects of different coefficients include:
For the function \( g(x) = f\left(\frac{1}{5}x\right) \), the coefficient is \( \frac{1}{5} \). This coefficient is less than 1, indicating a horizontal stretch. The interpretation here is important: the smaller the coefficient relative to 1, the more pronounced the stretch will be.
Key effects of different coefficients include:
- If the coefficient \( c < 1 \), the graph of the function \( f(x) \) will stretch horizontally.
- If the coefficient \( c > 1 \), the graph will compress horizontally.
- If the coefficient \( c = 1 \), the graph remains unchanged in terms of width.
- Negative coefficients also indicate reflection across the y-axis along with any stretch/compression effects.
Other exercises in this chapter
Problem 60
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