Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the function whose graph is the same as the graph of \(y=|x|\) but shifted down 4 units.

Short answer

The function is \( y = |x| - 4 \).

Step-by-step solution

Understand the original function

The original function given is the absolute value function: \[ y = |x| \] This graph is a V-shaped graph that intersects the origin (0,0) and opens upwards.

Determine the transformation

The problem states that the graph needs to be shifted down by 4 units. To shift a graph downward, you subtract the desired number of units from the function's output (i.e., the y-values).

Apply the shift

Subtract 4 from the original function to achieve the downward shift. Thus, the new function becomes: \[ y = |x| - 4 \]

Key concepts

absolute value function

The absolute value function, represented as \( y = |x| \), is an essential concept in mathematics. This function forms a 'V' shape on the graph.
This shape means that the function changes direction at the origin (0,0).
Here are a few key points about the absolute value function:

• Its vertex is at the origin (0,0).
• The graph opens upwards, meaning all y-values are non-negative.
• It is symmetric around the y-axis, so \( y \) is the same for \( x \) and \( -x \).

Understanding this function is crucial for graph transformations.

vertical shift

A vertical shift changes the y-values of a function by a constant amount. In this exercise, we shift \( y = |x| \) down by 4 units. To do this, we subtract 4 from the entire function.

Here's how it works:

• Start with \( y = |x| \).
• Shifting downward means subtracting because we want all y-values to be less.
• The new function will be \( y = |x| - 4 \).

By performing this subtraction, every point on the graph moves down 4 units. The 'V' shape and symmetry remain the same.

function graphing

Graphing transformations helps you understand how functions behave. Let's graph the new function \( y = |x| - 4 \):

• Start with the basic \( y = |x| \), which touches the y-axis at (0,0).
• Shift every point down by 4 units.
• The vertex of the new function is now at (0,-4).
• Retain the 'V' shape and the symmetry.

Graphing transformations helps visualize these changes. When shifting, focus on adjusting y-values while preserving the original shape.