Problem 16 - College Algebra

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

Plot points: \((-1, 2), (0, 1), (1, 0), (2, 1), (3, 2)\) and draw the V-shaped graph opening upwards.

Step 1: Identify the Vertex

The given function is written in the form \( y = |x - h| \). In this case, \( h = 1 \). The vertex of the absolute value function is \((h, 0)\), which is \((1, 0)\).

Step 2: Create a Table of Values

To graph the function, let's determine several points by choosing different values for \(x\) and computing \(y\). Choose \(x = -1, 0, 1, 2,\) and \(3\), and calculate the corresponding \(y\) values using \(y = |x - 1|\). This results in: \(y = |(-1) - 1| = 2\), \(y = |0 - 1| = 1\), \(y=|1 - 1| = 0\), \(y=|2 - 1| = 1\), and \(y=|3 - 1| = 2\).

Step 3: Plot the Points

Plot the points from the table: \((-1, 2)\), \((0, 1)\), \((1, 0)\), \((2, 1)\), and \((3, 2)\) on the coordinate plane.

Step 4: Graph the Function

Join the points with straight lines to form a V-shape, with the vertex at \((1, 0)\). The graph should open upwards and be symmetric around the vertical line \(x = 1\).

Step 5: Verify the Graph

Double-check that all plotted points satisfy the original equation \(y = |x - 1|\) and ensure the graph maintains the properties of the absolute value function requiring symmetry and a vertex indicative of the function's minimum point.

Key Concepts

Graphing Functions

Graphing functions is an essential aspect of understanding mathematical relationships visually. It allows us to interpret how variables interact through their graphical representations.
For the absolute value function like \( y = |x-1| \), the graph forms a characteristic "V" shape due to how absolute values behave. Absolute value essentially "flips" negative outputs into positive ones, creating symmetry around the line where the function changes direction.
Functions like these are often graphed by identifying critical points, such as when the output value is zero or when the input value changes from yielding a positive to a negative outcome. More specifically, for the absolute value function, these critical points revolve around its vertex and notable supporting values.

Vertex Identification

In absolute value functions, the vertex plays a significant role. It serves as the graph's turning point and is the minimum point if the absolute value function is in its basic upward-opening form like \( y = |x-1| \).
To identify the vertex, we look at the expression inside the absolute value. For our function, \( x-1 \), the vertex is at the point where this expression equals zero. Thus, setting \( x-1 = 0 \), we solve that \( x = 1 \). The vertex is then located at \((1, 0)\).
The vertex is crucial because it marks the horizontal position where the direction of the graph shifts, indicating where the slope changes from positive to negative or vice versa. Recognizing the vertex allows us to draw the graph accurately from this pivotal point.

Symmetry in Graphs

Symmetry is a fundamental characteristic of absolute value functions. These functions exhibit reflectional symmetry, also known as mirror symmetry, which results from the equivalence of positive and negative inputs.
For \( y = |x-1| \), the vertical line \( x = 1 \) acts as the axis of symmetry. This line passes through the vertex. Each point on the left side of \( x = 1 \) corresponds to a mirror image on the right side.

• This means: if a point \((a, b)\) exists, then its reflection \((2 - a, b)\) will also lie on the graph.

Recognizing symmetry helps in confirming the accuracy of the plotted points and checking that they are correctly mirrored around the symmetry axis.
Understanding symmetry makes it easier to predict additional points after determining the vertex and helps ensure the graph's shape is correctly represented.

Coordinate Plane Plotting

Plotting on a coordinate plane involves selecting appropriate points to represent the function visually. For functions like \( y = |x-1| \), it is important to choose a variety of \( x \) values and calculate their corresponding \( y \) values.

• Start by calculating the vertex point \((1, 0)\), as previously identified.
• Select additional \( x \) values, such as values less than, equal to, and greater than the vertex \( x \), to understand the function's behavior.
• For example, using the values \(-1, 0, 1, 2, \) and \(3\), we find the corresponding points as \((-1, 2)\), \((0, 1)\), \((1, 0)\), \((2, 1)\), and \((3, 2)\).

When all points are plotted on the coordinate plane, connecting them provides a visual representation of the function. Plotting by hand helps in mentally understanding the function's growth and behavior while cementing the visual grace typical of graphing absolute value functions.