Problem 20 - Calculus with Concepts in Calculus

Question

Sketch the graph. List the intercepts and describe the symmetry (if any) of the graph. $$ |x|=2 $$

Step-by-Step Solution

Answer

The graph consists of two points at \((2, 0)\) and \((-2, 0)\), with y-axis symmetry.

Step 1: Understanding the Equation

The equation given is \(|x| = 2\). This equation involves the absolute value of \(x\). Absolute value equations represent distances from zero on the number line for each point, giving both positive and negative solutions.

Step 2: Solve the Absolute Value Equation

We solve \(|x| = 2\) by considering two cases: \(x = 2\) and \(x = -2\). These are the points where the expression inside the absolute value equals the value specified on the other side of the equation.

Step 3: Sketch the Graph

On a coordinate plane, plot the points found in the previous step: \(x = 2\) and \(x = -2\). Since these are vertical, you would just plot two dots on the x-axis at these locations. This graph is simply two points, not a continuous line or curve.

Step 4: Identify the Intercepts

The points identified as solutions are both x-intercepts. Thus, the intercepts are \((2, 0)\) and \((-2, 0)\). There are no y-intercepts because the only points plotted lie exclusively on the x-axis.

Step 5: Determine Symmetry

The graph is symmetric with respect to the y-axis because the points \((2, 0)\) and \((-2, 0)\) are mirror images across the y-axis. This satisfies the condition for y-axis symmetry.

Key Concepts

Intercepts in Graphs

Intercepts are special points where a graph touches or crosses the axes on a coordinate plane. In the context of the equation \(|x| = 2\), the graph is composed of two distinct points. These points are the solutions we found when solving the absolute value equation: \(x = 2\) and \(x = -2\).

• **X-intercepts** occur where the graph crosses the x-axis. In this case, the intercepts are at coordinates \((2, 0)\) and \((-2, 0)\).
• **Y-intercepts** occur when the graph crosses the y-axis. Here, there are no y-intercepts because neither of the point solutions for \(|x| = 2\) touches the y-axis.

Understanding where the graph interacts with the axes can help you quickly identify important characteristics of the function, such as the extent along the x-axis and whether or not the graph covers a wider range on the y-axis. Remember: intercepts give us a clear snapshot of where the graph makes key contact with the axes.

Symmetry in Graphs

Symmetry in graphs provides insight into the visual structure or balance of a graph. It means that parts of the graph mirror each other in a specific way. For the equation \(|x| = 2\), we observe symmetry, as it is inherently characterized by its even distribution across the y-axis.

• Y-axis symmetry: A graph is symmetric with respect to the y-axis if replacing \(x\) with \(-x\) results in an equivalent expression. For \(|x| = 2\), this is true because both solutions \((2, 0)\) and \((-2, 0)\) are mirror images about the y-axis.
• This type of symmetry simplifies our sketching process and analytical understanding of the graph's behavior. It shows balance and often implies that changes to one side of the graph occur equally on the opposite side.

Recognizing symmetry helps to quickly sketch graphs and to analyze functions effectively, especially when dealing with equations involving absolute values like in our example.

Graph Sketching

Sketching graphs involves visually plotting the key points and characteristics of an equation. For the equation \(|x| = 2\), this process is straightforward. Absolute value equations often yield solutions that are symmetrical and focused on critical points.

- **Identify Solutions**: Start by identifying all the solutions to the equation. In our example, they are \(x = 2\) and \(x = -2\). - **Plot Points**: Place these points accurately on a coordinate plane. Since they are both on the x-axis, plot them as open points at the coordinates \((2, 0)\) and \((-2, 0)\).

When sketching absolute value graphs, remember:

• There will usually be some form of symmetry in the graph, often across the y-axis.
• Understanding and plotting the intercepts gives you a significant start.

Graph sketching becomes intuitive with practice. It’s essential to begin with pinpointing crucial points and leverage symmetry to fill out the rest. This approach is particularly helpful when dealing with equations that do not form continuous graphs like \(|x| = 2\), which consists solely of two points.