Identify a horizontal or vertical stretch or compression of the function f(x) by observing the equation of the function g(x) The base graph horizontally stretched by factor of 6_ The base graph vertically compressed by a factor of 6 The base graph vertically stretched by factor of 6_ The base graph horizontally compressed by factor of 6_

Identify a horizontal or vertical stretch or compression of the function f(x) by observing the equation of the function g(x) The base graph horizontally stretched by factor of 6_ The base graph vertically compressed by a factor of 6 The base graph vertically stretched by factor of 6_ The base graph horizontally compressed by factor of 6_

The Correct Answer and Explanation is:

To correctly identify the transformation, we need to look at how the function g(x)g(x)g(x) is defined in relation to the base function f(x)f(x)f(x). The most relevant comparison is based on the equation of g(x)g(x)g(x). Since you haven’t provided the specific equation for g(x)g(x)g(x), let’s walk through what each possible transformation would look like based on typical function transformation rules.

Function Transformation Rules:

1. Horizontal stretch/compression:
   • g(x)=f(bx)g(x) = f(bx)g(x)=f(bx)
     • If ∣b∣>1|b| > 1∣b∣>1, the graph is horizontally compressed by a factor of 1b\frac{1}{b}b1​.
     • If 0<∣b∣<10 < |b| < 10<∣b∣<1, the graph is horizontally stretched by a factor of 1b\frac{1}{b}b1​.
2. Vertical stretch/compression:
   • g(x)=a⋅f(x)g(x) = a \cdot f(x)g(x)=a⋅f(x)
     • If ∣a∣>1|a| > 1∣a∣>1, the graph is vertically stretched by a factor of aaa.
     • If 0<∣a∣<10 < |a| < 10<∣a∣<1, the graph is vertically compressed by a factor of aaa.

Let’s analyze your options assuming one of these transformations:

If the equation is g(x)=f(x/6)g(x) = f(x/6)g(x)=f(x/6), then:

• This is equivalent to g(x)=f(16x)g(x) = f\left(\frac{1}{6}x\right)g(x)=f(61​x), which matches the horizontal stretch rule where b=16b = \frac{1}{6}b=61​, and 1b=6\frac{1}{b} = 6b1​=6.
• So, the base graph is horizontally stretched by a factor of 6.

Correct Answer:

The base graph is horizontally stretched by a factor of 6

Explanation (300 words):

Transformations of functions allow us to understand how a graph changes when the function’s input or output is altered. When analyzing the equation of a transformed function g(x)g(x)g(x) in relation to the base function f(x)f(x)f(x), we use the transformation rules to determine if the graph has been stretched or compressed either horizontally or vertically.

If the function is written as g(x)=f(x/6)g(x) = f(x/6)g(x)=f(x/6), it indicates a change in the input of the function. The value xxx is divided by 6 before being passed into the original function fff. This type of transformation affects the horizontal aspect of the graph. Specifically, replacing xxx with x6\frac{x}{6}6x​ or equivalently x=6ux = 6ux=6u, where uuu is the original value, stretches the graph horizontally. In this case, the graph becomes six times wider, as each x-value must now be six times larger to achieve the same y-output as in the original function.

Thus, this represents a horizontal stretch by a factor of 6. It is important not to confuse this with vertical transformations, which involve multiplying the entire function output by a constant. Only changes to the input (inside the function’s parentheses) affect the horizontal behavior of the graph. This concept is crucial in graphing transformations and understanding function behavior in mathematics.