Problem 5 - 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 absolute value function, vertically stretched by applying a factor of 3 and reflected across the \(x\) -axis

Step-by-Step Solution

Answer

The equation is \( y = -3|x| \).

Step 1: Understanding Standard Formula

The standard form for the absolute value function is given by \( y = |x| \). Our goal is to apply a vertical stretch and reflection.

Step 2: Applying Vertical Stretch

To stretch the absolute value function vertically by a factor of 3, we multiply the whole function by 3. This leads to the equation \( y = 3|x| \).

Step 3: Reflecting Across x-axis

To reflect the function across the x-axis, multiply the entire equation by -1. This transforms the equation to \( y = -3|x| \).

Step 4: Final Equation

Combining both transformations, the final equation representing the desired transformation is \( y = -3|x| \).

Key Concepts

Vertical Stretch

A vertical stretch involves making a graph steeper or 'taller'. When we vertically stretch a function, we multiply the entire function by a positive constant greater than 1. In our situation, this factor is 3. This means each point on the graph moves three times further from the x-axis, giving it that 'stretched' look.

Consider the absolute value function, which typically looks like a 'V' centered at the origin of a graph. When vertically stretched by a factor of 3, the equation is altered from the standard form of \(y = |x|\) to \(y = 3|x|\).

Here are some key points about vertical stretching:

• The graph of the function is transformed by pulling it away from the x-axis.
• Only affects the y-values of the graph, while x-values remain unchanged.
• The larger the factor, the steeper or more stretched the graph will appear.

Reflection Across x-axis

When a function is reflected across the x-axis, every point on the graph is flipped over the x-axis to the opposite side. This can be visualized as flipping the graph upside down.

Mathematically, to achieve a reflection across the x-axis, we multiply the function by -1. This action changes the equation from \(y = 3|x|\) to \(y = -3|x|\).

Here's what happens during a reflection across the x-axis:

• The y-values of the function change sign (positive values become negative and vice versa).
• The x-values stay the same, maintaining the shape of the graph but inverting its position vertically.
• This reflection gives a 'mirror image' of the original graph across the x-axis.

Understanding how reflections work is crucial for correctly applying transformations to functions.

Absolute Value Function

The absolute value function is one of the most basic yet important functions in algebra. It is expressed as \(y = |x|\) and resembles a 'V' shape when graphed, with its vertex at the origin (0,0). The absolute value of a number is its distance from zero, so it is always non-negative.

Here are some essential points about the absolute value function:

• The graph always forms a 'V' shape.
• The vertex of the graph is the pivot point and it lies at the origin.
• It reflects points on the left of the y-axis onto the right and vice versa due to the nature of absolute value.

This function provides the base structure for many transformations, including vertical stretches and reflections. Understanding how it operates is crucial for mastering transformations. In our transformation, we began with this basic \(y = |x|\) form and applied the necessary alterations to achieve the desired effect.