Problem 16 - College Algebra with Corequisite Support

Question

For the following exercises, graph the absolute value function. Plot at least five points by hand for each graph. \(y=|x-1|\)

Step-by-Step Solution

Answer

The graph is V-shaped with vertex at (1, 0); key points: (-1, 2), (0, 1), (1, 0), (2, 1), (3, 2), (4, 3).

Step 1: Understanding the Absolute Value Function

The function given is \( y = |x-1| \). The absolute value function \( y = |x-a| \) reflects around the vertical line \( x = a \). This function creates a 'V' shape with the vertex at the point \( (a, 0) \) when the base form \( |x| \) is shifted horizontally by \( a \) units.

Step 2: Determine the Vertex

In the equation \( y = |x-1| \), identify the horizontal shift. Here, \( x \) is replaced by \( x - 1 \), which indicates a shift 1 unit to the right. Hence, the vertex of the graph is \( (1, 0) \).

Step 3: Choose Points and Calculate Values

Choose points around the vertex: \( x = -1, 0, 1, 2, 3, 4 \). Calculate \( y \) for each:- \( x = -1: y = |-1-1| = | -2 | = 2 \)- \( x = 0: y = |0-1| = | -1 | = 1 \)- \( x = 1: y = |1-1| = | 0 | = 0 \)- \( x = 2: y = |2-1| = | 1 | = 1 \)- \( x = 3: y = |3-1| = | 2 | = 2 \)- \( x = 4: y = |4-1| = | 3 | = 3 \)

Step 4: Plot the Points

Plot the calculated points: \((-1, 2), (0, 1), (1, 0), (2, 1), (3, 2), (4, 3)\). Make sure to accurately place these coordinates on the graph.

Step 5: Draw the Graph

Connect the points using straight lines to form the graph. Since this is an absolute value graph, start from \( (-1, 2) \) to \( (1, 0) \) with a downward slope on the left side of the vertex and upward slope from \( (1, 0) \) to \( (4, 3) \). These lines will form a 'V' shape.

Key Concepts

Vertex of Absolute Value Function

The vertex of an absolute value function is the point where the graph changes direction, creating its characteristic 'V' shape. This function typically takes the form \( y = |x-a| \), where \( (a, 0) \) is the vertex. This means the lowest or highest point of the graph occurs at the vertex, depending on whether the 'V' opens upwards or downwards.

In our exercise, we have the equation \( y = |x-1| \). Here, the vertex is easily identifiable by looking at the expression inside the absolute value. By recognizing \( x-1 \), we can determine that the graph shifts to the right of the origin by 1 unit. Hence, the vertex is located at \( (1, 0) \).

Understanding the vertex provides a foundation for sketching the rest of the graph. It is the starting point from where other values and points are calculated, making it central to accurately plotting the absolute value graph.

Horizontal Shift in Functions

Horizontal shifts are critical in understanding how graphs of functions move along the x-axis. For an absolute value function, such as \( y = |x-a| \), the graph shifts horizontally. When \( x \) is replaced by \( x-a \), the graph shifts to the right by \( a \) units. Conversely, if you see \( x+a \), the graph shifts to the left.

In the example \( y = |x-1| \), the graph is affected by a horizontal shift of 1 unit to the right. This shift alters the graph's position on the coordinate plane but does not change its shape. The 'V' shape remains intact but positions itself around the new vertex at \( (1, 0) \).

Understanding horizontal shifts is crucial, as they impact where key points and features of the graph, such as the vertex, will be plotted. It's a simple but powerful transformation that helps in accurately drawing the function's graph.

Plotting Points on a Graph

Plotting points is an essential skill in graphing functions accurately. To graph an absolute value function, like \( y = |x-1| \), you need a set of coordinate points that represent this function on the graph.

After identifying the vertex \( (1, 0) \), which is a starting point, we select several additional x-values around this vertex to calculate corresponding y-values. These points could be \( x = -1, 0, 1, 2, 3, 4 \). For each x-value:

• \( x = -1 \): \( y = |-1-1| = 2 \)
• \( x = 0 \): \( y = |0-1| = 1 \)
• \( x = 1 \): \( y = |1-1| = 0 \) (This is also the vertex)
• \( x = 2 \): \( y = |2-1| = 1 \)
• \( x = 3 \): \( y = |3-1| = 2 \)
• \( x = 4 \): \( y = |4-1| = 3 \)

Once you've calculated these points, plot them on your graph paper. After plotting these points, draw lines to connect them smoothly, forming the 'V' shape characteristic of an absolute value function graph. Plotting points is a step-by-step process that requires patience and precision to ensure the graph accurately reflects the function's behavior.