Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Use transformations of graphs to sketch a graph of \(y=f(x)\) by hand. $$f(x)=|x+2|-3$$

Step-by-Step Solution

Answer

Start with \( |x| \), shift left 2 units, then down 3 units.

Step 1: Understand the Base Function

The given function is \( f(x) = |x+2| - 3 \). The base function we start with is \( g(x) = |x| \), which is a V-shaped graph with its vertex at the origin (0,0).

Step 2: Horizontal Shift

The function \( |x+2| \) represents a horizontal shift of the base function \( |x| \) to the left by 2 units. The vertex of the base function \( g(x) = |x| \) moves from (0,0) to (-2,0).

Step 3: Vertical Shift

Next, we address the vertical component \( |x+2| - 3 \). This expression indicates a vertical shift downward by 3 units. Thus, the vertex at (-2,0) from the previous step will now be at (-2,-3).

Step 4: Graph the Transformed Function

Plot the transformed vertex (-2,-3) on the coordinate plane. Since the shape of the graph remains a V-shape, sketch the lines with slopes of 1 and -1 respectively emanating from the vertex. This will give the graph of \( f(x) = |x+2| - 3 \).

Key Concepts

Absolute Value Function

The absolute value function is one of the fundamental functions used in graph transformations. It is represented by the expression \( g(x) = |x| \). This function creates a V-shaped graph that is symmetric around the y-axis. The vertex of this graph is located at the origin, (0, 0), making it the starting reference point for many transformations.

• **Graph Shape:** The graph is V-shaped. It looks like the point of a pencil.
• **Symmetry:** The line of symmetry is the y-axis.
• **Domain and Range:** The domain is all real numbers, and the range is all non-negative numbers, i.e., \( y \geq 0 \).

Understanding this basic shape is crucial because all transformations will modify the absolute value graph in predictable ways.

Horizontal Shift

A horizontal shift in a graph occurs when the graph moves left or right on the coordinate plane. It doesn’t change the shape of the graph, only its position horizontally. In the function \( f(x) = |x+2| \), the addition of 2 inside the absolute value impacts the horizontal shift.
When you see \( |x+\text{c}| \), think about the opposite direction:

• If c > 0, the graph shifts to the left by \( c \) units.
• If c < 0, the graph shifts to the right by \( |c| \) units.

For \( |x+2| \), since 2 is positive, the graph of \( |x| \) shifts left by 2 units. Therefore, the vertex moves from (0,0) to (-2,0). Just remember: move the vertex the opposite way you might initially think!

Vertical Shift

A vertical shift occurs when all points of a graph move up or down without affecting the shape. In \( f(x) = |x+2| - 3 \), the "-3" outside of the absolute value indicates a vertical shift.

• **Positive values shift the graph up.**
• **Negative values shift it down.**

Here, because the value is -3, the graph moves downward by 3 units. The vertex that was moved to (-2,0) in the horizontal shift step now moves to (-2,-3). This vertical adjustment is straightforward: simply adjust the vertex by the indicated amount.

Vertex

The vertex is a crucial point of reference, particularly for the absolute value function. For the base function \( |x| \), the vertex starts at the origin, (0,0). However, transformations shift the vertex to new locations.
In the function \( f(x) = |x+2| - 3 \):

• **Start:** The base vertex is (0,0).
• **Horizontal adjustment:** Move it left 2 units to (-2,0).
• **Vertical adjustment:** Then, move it down 3 units to (-2,-3).

The final transformed vertex forms the cornerstone of the new graph, where the V-shape extends symmetrically upwards.