Problem 11 - Precalculus : Building Concepts and Connections

Question

Identify the underlying basic function, and use transformations of the basic function to sketch the graph of the given function. $$H(x)=|x-2|+1$$

Step-by-Step Solution

Answer

The function \(H(x)=|x-2|+1\) is an absolute value function that has been shifted 2 units to the right and 1 unit upwards.

Step 1: Recognize the basic function

The underlying basic function in this case is the absolute value function \(f(x)=|x|\). It gives the magnitude of a real number, a non-negative value of x.

Step 2: Identify the horizontal shift

The function \(H(x)=|x-2|\) is a horizontal shift of the basic function to the right by 2 units. It can be achieved by replacing \(x\) with \((x-2)\) in the basic function.

Step 3: Identify the vertical shift

The function \(H(x)=|x-2|+1\) is a vertical shift of the function obtained in Step 2 upward by 1 unit. It can be achieved by adding 1 to the function obtained in Step 2.

Step 4: Sketch the graph

With these two transformations (a shift to the right by 2 units and a shift upwards by 1 unit), the final graph can now be sketched.

Key Concepts

Transformations

In the world of functions and graphs, transformations help us understand how a basic graph can change position or shape. These transformations might include shifting, stretching, or reflecting, just to name a few. In this context, we're focusing on transformations related to shifts.

Consider the absolute value function, which is the basic function in our exercise. It's the graph of a "V" shape, represented by the equation \(f(x) = |x|\). Transformations allow us to move this "V" shape around the coordinate plane. By changing the equation, we apply transformations such as sliding, which can move the graph in different directions. Understanding these transformations is key to mastering graph adjustments.

Graph Sketching

Graph sketching involves drawing the graph of a function based on our understanding of its form and transformations. It starts with identifying the basic function—in this case, the absolute value function \(f(x)=|x|\). This function has a characteristic "V" shape centered at the origin.

To sketch the graph, observe the transformations applied. Visualize the movement of the graph when shifts occur. The horizontal and vertical shifts direct us on how the graph will change its standard position.

• The vertex of the basic absolute value graph, originally at (0, 0), is later adjusted by the transformations.
• Always plot key points first, then connect them smoothly to maintain their shape.

With practice, graph sketching becomes a straightforward process of identifying transformations and depicting them visually.

Horizontal Shifts

A horizontal shift happens when we move the entire graph left or right. In the absolute value function, this shift adjusts the x-coordinate in the equation. For our example, the function transforms from \(f(x)=|x|\) to \(H(x)=|x-2|\).

The expression \(|x-2|\) indicates the graph's horizontal shift. The subtraction inside the absolute value moves the graph to the right by 2 units. This rule is counterintuitive: to move right, we subtract within the function.

• Replacing \(x\) with \(x-a\) shifts the graph right by \(a\) units.
• Replacing \(x\) with \(x+a\) shifts the graph left by \(a\) units.

Horizontal shifts involve changing the graph's central point horizontally without affecting its shape.

Vertical Shifts

Vertical shifts involve sliding the graph up or down. This transformation is performed by adjusting the function outside of its main component. Looking at \(H(x)=|x-2|+1\), we see a vertical shift applied to the function.

The added \(+1\) signifies that the graph is moved upward by 1 unit. Unlike horizontal shifts, vertical shifts directly add or subtract a constant from the function. This shifts the entire graph vertically:

• Adding a constant shifts the graph upwards by that amount.
• Subtracting a constant shifts it downwards.

In this case, the graph has already been shifted horizontally; the next step is to apply its vertical shift by adding 1 to every y-coordinate of the previous step's graph.