Problem 134 - College Algebra Essentials

Question

What must be done to a function's equation so that its graph is shrunk horizontally?

Step-by-Step Solution

Answer

To shrink a function's graph horizontally, one must replace every x in the function's equation with \( \frac{x}{b} \) , where b is a factor greater than 1.

Step 1: Understanding transforming a function graph

Graphs of functions can be transformed in several ways, including shifting, reflecting, and scaling both horizontally and vertically. In this case, the required transformation is a horizontal shrinking or scaling.

Step 2: Applying horizontal scaling to a function

To make a function's graph to shrink horizontally, the x-values of the function need to be multiplied by a factor greater than 1. This translates mathematically to replacing every x in the function's equation with \( \frac{x}{b} \) , where \( b > 1 \) is the scaling factor.

Step 3: Formulating the transformed function

If the initial function is \( f(x) \), after the horizontal shrinking, the transformed function will be \( f\left(\frac{x}{b}\right) \) , where \( b > 1 \). The graph of this function will be a horizontal shrink of the original function's graph by a factor of \( b \). The higher the value of \( b \), the more the function's graph will be shrunk horizontally.

Key Concepts

Horizontal Scaling

Horizontal scaling is a process used in function transformations to alter the width of a graph along the x-axis. When we talk about horizontal scaling, we're referring to changes in the input (x-values) of a function.
By multiplying the x-values by a certain factor, we can manipulate the graph's width or spread. If you multiply the x by a number greater than 1, you get horizontal shrinking, which squeezes the graph towards the y-axis. If the factor is between 0 and 1, the graph stretches, spreading out away from the y-axis.

• Shrinking Factor: Multiply or replace each x in the equation with \( \frac{x}{b} \), where \( b > 1 \).
• Stretching Factor: Multiply or replace each x with \( x \cdot b \), where \( 0 < b < 1 \).

This transformation is useful in models that need adjustment to better fit data within a given range. It's important to note that horizontal transformations affect only the x-direction, leaving the y-values untouched.

Graph Shrinking

Graph shrinking is a specific type of horizontal scaling where the graph contracts towards the y-axis. This phenomenon results in a narrower appearance of the function's graph.
To achieve this, each x-coordinate in the function is adjusted by a scaling factor greater than 1, yielding an altered graph that "shrinks". For instance, if you start with a function \( f(x) \), the shrunk version would be \( f\left(\frac{x}{b}\right) \), where \( b \) represents the scale factor.
A larger value for \( b \) results in a more pronounced shrink. For example, if \( b = 3 \), each x-value is divided by 3, bringing the points closer together horizontally. A higher factor means more squeezing, making it important in scenarios demanding precision in function representation, such as graphics and data fitting.

Mathematical Transformations

Mathematical transformations involve methods to modify existing functions or shapes through various operations and changes. In mathematics, these transformations can include translations, rotations, reflections, and scalings.
The primary goal is to change the function's appearance or orientation without altering its core properties. Transformations are used extensively in fields like graphics, physics, and engineering to model scenarios and phenomena accurately.

• Translation: Moves the graph without changing its shape.
• Rotation: Turns the graph around a specific point.
• Reflection: Flips the graph over a line or axis.
• Scaling: Enlarges or shrinks the graph in either direction.

Understanding these transformations allows for better manipulation and control of function graphs, making complex equations comprehensible and manageable. These methods are fundamental in making mathematical models much more adaptable and versatile.