Problem 94 - Algebra and Trigonometry Real Mathematics, Real People

Question

Describe how the graph of \(g\) is related to the graph of \(f.\) $$g(x)=f(2 x)$$

Step-by-Step Solution

Answer

The graph of g(x) = f(2x) will be a horizontal compression of the graph of the function f(x) by a factor of 2. This means if a point (p, q) is on f(x), then (p/2, q) will be on g(x).

Step 1: Identify the transformation

Here, the given function g(x) is equal to f(2x). In this type of function, the '2' is inside the parentheses, which means it will affect the x-values of the function. The multiplication of x by a factor of 2 inside the function compresses the graph horizontally.

Step 2: Describe the transformation

The graph of g(x) = f(2x) will be a horizontal compression (or shrink) of the graph of the function f(x) by a factor of 2. This means all x-values in function f(x) will be halved in function g(x). So, if a point (p, q) lies on the graph of f(x), then the point (p/2, q) will lie on the graph of g(x) due to the horizontal compression.

Key Concepts

Horizontal Compression

Horizontal compression is a key graph transformation concept.
When we talk about compressing a function horizontally, we refer to the transformation applied to the x-values within a function.
This makes the graph narrower than the original.In the context of the function \( g(x) = f(2x) \), we see horizontal compression in action. Here, every x-value in the function \( f(x) \) is multiplied by 2, effectively compressing the graph along the x-axis.
Consequently, if a point \((p, q)\) lies on the graph of \( f(x) \), its corresponding point on \( g(x) \) would be \((p/2, q)\).

• This means that the graph of \( g(x) \) appears compressed towards the y-axis.
• The graph maintains the same height (y-values) but changes its width (x-values).

Understanding this concept is crucial when manipulating graphs, as it allows you to predict how the graph will change just by looking at the term inside the function.

Function Notation

Function notation helps us understand and work with functions conveniently. When you see an expression like \( g(x) = f(2x) \), it tells you how \( g(x) \) relates to \( f(x) \).
Instead of expressing \( g \) with a complex formula, you define it using an existing function \( f \).
In this notation:

• The letter \( f \) represents a predefined function rule.
• The argument inside the brackets, such as \( 2x \), indicates how to modify \( f \).

This approach makes transformations much easier to handle.
You can immediately ascertain that the changes affect x-values because of the direct manipulation occurring inside \( f(2x) \).
Function notation is powerful because it gives you a compact way to express transformations without restating large equations.

X-Values Transformation

When addressing transformations, x-values play an integral role, especially in horizontal changes.
Since transformations often modify where points appear on the graph, the x-values guide these shifts.For the given problem \( g(x) = f(2x) \), the x-values are transformed such that each value is halved compared to the original function \( f(x) \).
This results in a horizontal compression.

• Original x-values are affected by the factor inside the function.
• Every x-location on \( f(x) \) maps to a new location on \( g(x) \) by dividing by that factor.

If you have a point \((x, y)\) on \( f(x) \), its transformation under \( g(x) \) will be \((x/2, y)\).
This graphical shift is crucial for understanding how functions behave under different transformations, allowing better insight into their geometric properties.