Problem 12 - A Graphical Approach to College Algebra

Question

Write the equation 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: Identify the Base Function

The base function for this problem is the absolute value function, which can be written as \( y = |x| \). This is the starting point before any transformations are applied.

Step 2: Apply Horizontal Translation

To translate a function horizontally, replace \(x\) with \((x-h)\), where \(h\) is the number of units to shift. Here, we shift 5 units to the right, so \(h = 5\). This gives us the intermediate function \( y = |x - 5| \).

Step 3: Apply Vertical Translation

To move the function vertically, add or subtract a constant from the function. Moving it 1 unit downward is done by subtracting 1 from the function. Thus, we have \( y = |x - 5| - 1 \).

Step 4: Write the Final Transformed Equation

The resulting function after the translations is \( y = |x - 5| - 1 \). This equation reflects the absolute value function that has been shifted 5 units to the right and 1 unit downward.

Key Concepts

Horizontal Translation

Horizontal translation involves moving a graph left or right across the coordinate plane. It’s like sliding the entire graph horizontally without altering its shape or orientation. In mathematical terms, this is done by replacing the variable

• Replace \( x \) in the function with \( (x - h) \)
• \( h \) represents the number of units the graph shifts.

In our case, for the base absolute value function \( y = |x| \), we want to shift the graph 5 units to the right. By following the rule, we replace \( x \) with \( (x - 5) \), which gives us the function \( y = |x - 5| \). Here, \( 5 \) is positive because we move right, contrary to what might seem intuitive since \( x \) actually decreases. Think of it as shifting the input values by 5 units to the right.

Vertical Translation

Vertical translation refers to moving a graph up or down on the coordinate plane. It is achieved by adding or subtracting a number directly to/from the function:

• Adding a constant moves the graph up.
• Subtracting a constant shifts it down.

For the horizontal-shifted function \( y = |x - 5| \), we need to shift it 1 unit downward, which requires subtracting 1 from the entire function. Consequently, the equation becomes \( y = |x - 5| - 1 \). This process lowers every point on the graph by 1 unit, perfectly executing the desired downward shift.

Function Transformations

Function transformations involve changing a base function to produce a new function with alterations in its graph’s position, size, or orientation. The main types of transformations include

• Horizontal translations
• Vertical translations
• Reflections
• Stretches

The transformation process starts with identifying the base function, such as the absolute value function \( y = |x| \). By applying the horizontal and vertical translations explored, we change this base into another function: \( y = |x - 5| - 1 \). Here, the transformations involved repositioning the graph 5 units right and 1 unit down. These modifications preserve the original shape of the absolute value graph while altering its location on the coordinate plane, offering a new perspective without affecting its intrinsic characteristics.