Question

The graph of the function \(f\) is to be transformed as described. Find the function for the transformed graph. \(f(x)=\frac{\sqrt{x}}{x^{2}+1} ;\) stretched vertically by a factor of 3

Short answer

The function for the transformed graph after stretching it vertically by a factor of 3 is: \(g(x) = \frac{3\sqrt{x}}{x^2 + 1}\).

Step-by-step solution

Recall the vertical stretching rule

: When a graph is vertically stretched by a factor of k, the transformation can be represented by the following rule: If \(g(x) = kf(x)\), then the graph of g(x) is the vertical stretch of the graph of f(x) by a factor of k.

Apply the vertical stretching rule to the given function

: We need to find a new function, g(x), that represents the vertical stretching of f(x) by a factor of 3. Using the rule from Step 1, we have: g(x) = 3f(x).

Replace f(x) with the given function and simplify

: We know that \(f(x) = \frac{\sqrt{x}}{x^2 + 1}\). Substitute this into the equation for g(x): g(x) = 3 * \(\frac{\sqrt{x}}{x^2 + 1}\). Now, simplify the expression by distributing the 3: g(x) = \(\frac{3\sqrt{x}}{x^2 + 1}\).

State the function representing the transformed graph

: The function for the transformed graph after stretching it vertically by a factor of 3 is: g(x) = \(\frac{3\sqrt{x}}{x^2 + 1}\).