Problem 12 - A Graphical Approach to Precalculus with Limits

Question

Write an equation in \(x\) and \(y\) that results in the desired translation. Do not use a calculator. The absolute value function, shifted 1 unit downward and 5 units to the right

Step-by-Step Solution

Answer

The equation is \( y = |x - 5| - 1 \).

Step 1: Understanding the base absolute value function

The base function for an absolute value is \( y = |x| \). This is our starting point for performing transformations.

Step 2: Translating horizontally

A horizontal shift to the right by 5 units affects the \(x\)-term inside the absolute value. This shift is represented by replacing \(x\) with \(x - 5\) in the absolute value function, resulting in \(y = |x - 5|\).

Step 3: Translating vertically

A shift downward by 1 unit affects the entire function, decreasing the output value by 1. This modification translates to subtracting 1 from the whole equation, leading to \(y = |x - 5| - 1\).

Step 4: Final equation

Combine the effects of both translations. The equation \( y = |x - 5| - 1 \) reflects an absolute value function shifted 5 units to the right and 1 unit downward.

Key Concepts

Absolute value function

The absolute value function is a fundamental concept in mathematics. The basic form of an absolute value function is represented by the equation \( y = |x| \). This function creates a V-shaped graph. The point at the tip of the 'V' is the vertex, and it occurs at the origin \((0,0)\) for the basic absolute value function.

Here are some key traits of absolute value functions:

• The absolute value of a number is always non-negative, as it refers to the distance of the number on the number line from zero.
• The graph is symmetrical with respect to the y-axis, meaning it looks the same on both sides of this axis.
• As a piece of the broader family of functions, this function does not pass through any negative y-values; it 'reflects' up at its vertex.

Absolute value functions are quite versatile. They not only measure distances but also help graphically represent conditions and constraints that involve non-negative values.

Horizontal translation

A horizontal translation changes a graph's position along the x-axis, which is simply a shift of the graph left or right.

For the absolute value function, a horizontal translation can be thought of as modifying the x-component inside the absolute value. Specifically:

• To translate the graph to the right, replace \( x \) with \( x - c \), where \( c \) is the number of units you want to move.
• This makes the function \( y = |x - c| \). Moving to the right by 5 units, for instance, gives \( y = |x - 5| \).
• To shift left, use \( x + c \), reflecting a move to the left \( c \) units, resulting in \( y = |x + c| \).

These translations are fundamentally about shifting the x-values around, changing where the 'center' of the graph sits horizontally.

Vertical translation

Vertical translations move a function up or down along the y-axis. For an absolute value function, this involves simply modifying the output value.

Vertical translations are applied by adding or subtracting a constant from the entire function:

• Addition of a number \( d \) moves the graph up by \( d \) units: \( y = |x| + d \).
• Subtraction of a number moves the graph down by \( d \) units: \( y = |x| - d \).
• In our example from the original exercise, subtracting 1 moves the whole graph down one step: \( y = |x - 5| - 1 \).
• This shifts the 'V' downward, keeping its shape identical but decreasing all y-values by \( d \).

Vertical translations do not affect the x-components; they purely adjust the height of the graph along the y-axis. This makes them an important tool to customize the appearance of a given function without altering its fundamental horizontal characteristics.