Identify a horizontal or vertical stretch or compression of the function f(x) = 1/x by observing the equation of the function g(x) = 5/x. Answer choices: 1. A vertical compression by a factor of 5 2. A horizontal stretch by a factor of 5 3. A horizontal compression by a factor of 5 4. A vertical stretch by a factor of 5
The Correct Answer and Explanation is:
Correct Answer: 4. A vertical stretch by a factor of 5
To determine whether the function transformation is a horizontal or vertical stretch or compression, we need to analyze how the function g(x) = 5/x compares to the base function f(x) = 1/x.
Step-by-step Explanation:
The base function is:
f(x) = 1/x
The new function is:
g(x) = 5/x
This can be rewritten as:
g(x) = 5 ยท f(x)
This means we are multiplying the entire function f(x) by 5. When a function is multiplied by a constant on the outside, it causes a vertical transformation.
Specifically:
- If the constant is greater than 1, it causes a vertical stretch.
- If the constant is between 0 and 1, it causes a vertical compression.
Here, we are multiplying by 5, which is greater than 1. That means we are stretching the graph vertically. Every y-value from the original function f(x) becomes 5 times larger in g(x).
What it looks like:
Suppose f(2) = 1/2.
Then g(2) = 5/2.
The x-value remains the same, but the output (y-value) is stretched vertically to 5 times the original height.
This is not a horizontal transformation because we are not changing the x-value directly (like replacing x with 5x, which would be a horizontal compression).
Therefore, the correct transformation from f(x) = 1/x to g(x) = 5/x is a vertical stretch by a factor of 5. This makes the curve of the graph steeper by pulling it farther from the x-axis.
