Problem 8 - A Graphical Approach to Precalculus with Limits

Question

Write an equation in \(x\) and \(y\) that results in the desired transformation. Do not use a calculator. The square root function, vertically shrunk by applying a factor of 0.2 and reflected across the \(x\) -axis

Step-by-Step Solution

Answer

The equation is \( y = -0.2\sqrt{x} \).

Step 1: Understanding the Base Function

The base function in this exercise is the square root function, which is represented as \( f(x) = \sqrt{x} \). Our goal is to transform this base function according to the given instructions.

Step 2: Transforming the Function Vertically and Reflecting

A vertical shrink by a factor of 0.2 is applied by multiplying the function by 0.2, changing the function from \( \sqrt{x} \) to \( 0.2\sqrt{x} \). Reflecting a function across the \(x\)-axis involves multiplying the function by \(-1\). So, the reflection of \( 0.2\sqrt{x} \) across the \(x\)-axis becomes \( y = -0.2\sqrt{x} \).

Step 3: Writing the Transformed Equation

With the transformations applied, the final equation of the transformed function is \( y = -0.2\sqrt{x} \), where the square root function is both vertically shrunk and reflected across the \(x\)-axis.

Key Concepts

Vertical Shrink

A vertical shrink is a type of transformation that changes the shape of a function by compressing it along the vertical axis. Consider the square root function, written as \( f(x) = \sqrt{x} \). In a vertical shrink, each \( y \)-value of this function is multiplied by a constant less than 1. This makes the graph plateau faster and appear closer to the \( x \)-axis.

In this exercise, we apply a vertical shrink by a factor of 0.2. What this means is that every output of the square root function is reduced to 20% of its original value. Mathematically, this transformation changes our function from \( f(x) = \sqrt{x} \) to \( f(x) = 0.2\sqrt{x} \):

• Each \( y \)-value shrinks, altering the steepness of the curve.
• The graph becomes flatter, yet still increases as \( x \) increases.

Reflection Across X-axis

Reflecting a function across the \( x \)-axis is a transformation that inverts the function's values. This means that for every point on the original graph, which had a \( y \)-coordinate, the reflected graph has a \( y \)-coordinate of negative that original value.

When reflecting the function \( 0.2\sqrt{x} \) over the \( x \)-axis, we multiply the entire function by \(-1\). Therefore, the transformed function becomes \( y = -0.2\sqrt{x} \). In practical terms:

• All positive output values flip to become negative.
• The graph is essentially a mirror image with respect to the \( x \)-axis.
• This transformation does not affect the \( x \)-intercept because it remains at the origin.

Function Transformation

Function transformation is a powerful concept in mathematics that allows us to alter a graph's position, shape, and orientation on the coordinate plane. Transformation includes actions such as translation, scaling, and reflection. In this exercise, we specifically deal with both a scaling and a reflection.

By applying the vertical shrink, we scale the graph of the square root function, \( f(x) = \sqrt{x} \), to become \( 0.2\sqrt{x} \). This compression changes the function's steepness without moving it horizontally. Then, by reflecting this scaled graph across the \( x \)-axis, we flip it to produce the final function \( y = -0.2\sqrt{x} \).

• The transformation begins with scaling, followed by a reflection.
• The order of transformations affects the final result.
• Alterations only involve the \( y \)-values, leaving \( x \)-intercepts unchanged.

Transformation thus provides a structured way to manipulate the position and appearance of functions, helping to graphically interpret changes.