Which function represents g(x), the reflection of f(x) = 2(3)^x across the y-axis? A. g(x) = 2(3)^x B. g(x) = -2(3)^x C. g(x) = 2(3)^(-x) D. g(x) = 2(3)^(-x)?

Maths Questions
Short Answer
Step by Step Solution
Mathematics Questions

Which function represents g(x), the reflection of f(x) = 2(3)^x across the y-axis? A. g(x) = 2(3)^x B. g(x) = -2(3)^x C. g(x) = 2(3)^(-x) D. g(x) = 2(3)^(-x)?

Short Answer

The original function is f(x) = 3x, which is linear with a steep slope. The transformed function g(x) is obtained by reflecting f(x) across the y-axis to get g(x) = 3 – x and then applying a vertical compression to adjust the y-intercept, resulting in g(x) = ‚Öü2(3 – x).

Step-by-Step Solution

Step 1: Understand the Original Function

Start by noting the given function f(x) = 3x. This function describes a linear graph that passes through the origin. The graph has a slope of 3, indicating it rises steeply. Understanding the shape and characteristics of this function is crucial, as our transformations will reflect and modify it.

Step 2: Identify the Reflection About the Y-Axis

The graph of g(x) is a reflection of f(x) across the y-axis. The transformation rule for this reflection is expressed as f(x) ‚Üí f(-x). Thus, we replace x with -x in the function. This gives us g(x) = 3 – x, which represents the mirrored graph around the y-axis.

Step 3: Adjust the Y-Intercept and Vertical Compression

We observe that the y-intercept of the reflected function (0, 1) does not match the actual crossing point of (0, 0.5) on the graph of g(x). To correct this, we apply a vertical compression by a factor of ¼. This leads to the transformation y → ⅟2y, resulting in the final equation g(x) = ⅟2(3 Рx).

Related Concepts

Original function

The initial mathematical expression that defines the relationship between variables, in this case, f(x) = 3x, a linear function indicating a straight line graph with specific slope and y-intercept.

Reflection

A geometric transformation that creates a mirror image of a function across a specified axis; in this context, it refers to changing f(x) into g(x) by evaluating it at -x, resulting in g(x) = 3 – x.

Vertical compression

A transformation that reduces the height of a graph by a certain factor (in this case, ¼), effectively compressing it towards the x-axis, which alters the function to g(x) = ⚟2(3 Рx).

Table Of Contents
  1. Which function represents g(x), the reflection of f(x) = 2(3)^x across the y-axis? A. g(x) = 2(3)^x B. g(x) = -2(3)^x C. g(x) = 2(3)^(-x) D. g(x) = 2(3)^(-x)?

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