Problem 45 - Beginning and Intermediate Algebra

Question

Use the transformation techniques to graph each of the following functions. $$g(x)=-|x-1|+3$$

Step-by-Step Solution

Answer

To graph the given function \(g(x)=-|x-1|+3\), we apply the following transformations to the basic absolute value function \(f(x)=|x|\): 1. Horizontal shift 1 unit to the right. 2. Reflection over the x-axis. 3. Vertical shift 3 units up. After applying these transformations, the graph of the function will have its vertex at the point (1, 3) and a downward facing V-shape.

Step 1: Identify the basic function

The basic function is the absolute value function, which is given as: $$f(x) = |x|$$ It has a V-shaped graph with its vertex at the origin.

Step 2: Identify the transformations

The given function is: $$g(x) = -|x - 1| + 3$$ The transformations applied to the basic absolute value function can be identified as follows: - \(x-1\) on the inside of the absolute value function implies a horizontal shift 1 unit to the right. - The negative sign implies a reflection over the x-axis. - The \(+3\) outside the absolute value function implies a vertical shift 3 units up.

Step 3: Apply the horizontal shift

Shift the graph of the basic absolute value function 1 unit to the right. This means the vertex will be at the point (1, 0) instead of (0, 0).

Step 4: Apply the reflection

Reflect the graph over the x-axis. This means that the V-shape of the graph will now open downwards.

Step 5: Apply the vertical shift

Shift the graph 3 units up. The vertex will now be at the point (1, 3), and the function will maintain its downward V-shape.

Step 6: Graph the function

Plot the vertex point (1, 3) and sketch the downward V-shape based on the reflections and shifts applied. The final graph of the function should have its vertex at the point (1, 3) with a downward facing V-shape.

Key Concepts

Transformation Techniques

Transformation techniques are powerful tools used to modify the graph of a function. These techniques involve altering the position or orientation of the graph by applying various shifts and reflections.
They help in visualizing complex functions by using a fundamental or simpler function as the base model.In the case of absolute value functions, the basic function is often the V-shaped graph of \(f(x) = |x|\) with its vertex at the origin (0, 0).
Transformations allow us to shift, reflect, and stretch or compress this basic graph to better understand how different values affect it. Understanding these modifications can make graphing functions much easier and allow for quick visualization of changes.

Horizontal Shift

A horizontal shift involves moving the graph of a function left or right along the x-axis.
This is achieved by altering the input, or the x-variable, of the function.For the absolute value function \(g(x) = -|x-1|+3\), the expression \(x-1\) inside the absolute value sign indicates a horizontal shift.
The rule is to interpret \(x-c\) as a shift to the right by \(c\) units, and \(x+c\) as a shift to the left by \(c\) units.

• \(x-1\), in this case, means the graph is shifted right by 1 unit.
• The vertex originally at (0, 0) moves to (1, 0), maintaining the V-shape.

Understanding horizontal shifts is crucial for accurately positioning a graph on the coordinate plane.

Reflection over x-axis

Reflection over the x-axis involves flipping the graph of a function upside down.
For absolute value functions, this changes the orientation of the "V" shape.In our function \(g(x) = -|x-1|+3\), the negative sign outside the absolute value signifies this reflection.
Instead of the arms of the "V" pointing upwards, they point downwards.

• Initially, the basic function \(|x|\) opens upwards forming a positive V-shape.
• After the reflection, the graph becomes an upside-down V-shape.

Reflections are critical in transforming the behavior of the graph in relation to the x-axis.

Vertical Shift

A vertical shift moves the entire graph up or down along the y-axis.
This is controlled by adding or subtracting a constant value from the function.For \(g(x) = -|x-1|+3\), the \(+3\) indicates a vertical shift upwards by 3 units.
Vertical shifts do not alter the shape of the graph; instead, they change the position vertically.

• The vertex, originally at (1, 0) after horizontal shift and reflection, is moved up to (1, 3).
• The entire graph shifts, maintaining its downward orientation.

Understanding vertical shifts helps in situating the graph properly in relation to y-values, completing the transformation process.