Problem 41 - Calculus with Concepts in Calculus

Question

Sketch the graph of the equation. In each case determine whether the graph is that of a function. $$ x^{2}=y^{2} $$

Step-by-Step Solution

Answer

The graph is two intersecting lines and is not a function.

Step 1: Understand the Equation

The equation given is \(x^2 = y^2\). This equation states that the squares of \(x\) and \(y\) are equal.

Step 2: Solve for One Variable

Rewrite the equation to isolate one variable. Take the square root of both sides to solve for \(y\):\[ y = \pm x \]This means that \(y\) can be equal to \(x\) or \(-x\).

Step 3: Interpret the Solutions

The solutions \(y = x\) and \(y = -x\) represent two lines. One line is the identity line \(y = x\), and the other is the reflection line \(y = -x\). These two lines intersect at the origin (0,0).

Step 4: Sketch the Graph

Draw the two lines, \(y = x\) and \(y = -x\), on a coordinate plane. The graph consists of both lines, intersecting at the origin and extending infinitely in both directions.

Step 5: Determine if the Graph is a Function

To determine if the graph is a function, use the vertical line test. If any vertical line intersects the graph at more than one point, it is not a function. Since any vertical line except the y-axis will intersect both lines at two points, the graph is not the graph of a function.

Key Concepts

Vertical Line Test

The vertical line test is a simple way to determine if a graph represents a function. Simply draw vertical lines through the graph. If any line crosses the graph at more than one point, then the graph does not represent a function. This is because a function can have only one output value (y-value) for each input value (x-value).

• If a graph passes this test (intersects a vertical line only once), it indeed represents a function.
• If it fails (intersects more than once), as in this case with both lines, then it is not a function.

In our scenario, for the lines represented by the equation \(x^2 = y^2\), any vertical line will intersect both lines \(y = x\) and \(y = -x\) at two different points, except at the origin, demonstrating that the graph is not that of a function.

Equation Interpretation

Understanding the meaning behind an equation is vital in graph sketching. The equation \(x^2 = y^2\) suggests a relationship between the variables x and y.

• Both the x and y variables squared equal each other, indicating symmetry about the origin.
• Breaking this relationship down helps us see that there are two possible outcomes for y: either \(y = x\) or \(y = -x\).

This interpretation of the equation allows one to correctly draw these relationships as two lines on a graph, as they represent all the points (x,y) satisfying the original equation.

Linear Equations

Two linear equations emerge from our original equation: \(y = x\) and \(y = -x\). Linear equations graph as straight lines, and in this case, they will always pass through the origin (0,0).

• \(y = x\): This is a diagonal line at a 45-degree angle through each quadrant.
• \(y = -x\): This line has a negative slope, appearing as a diagonal line from top left to bottom right.

Both lines extend infinitely in both directions and plot with a constant slope of 1 and -1, respectively, making their characteristics straightforward and easy to visualize in a coordinate plane.

Coordinate Plane

A coordinate plane is a two-dimensional space formed by two perpendicular number lines: the x-axis (horizontal) and the y-axis (vertical). To sketch the graph of \(x^2 = y^2\), understanding how these axes interact is critical.

• The x-axis represents possible input values for the function or line, while the y-axis represents the output.
• The origin (0,0) is where these two axes intersect, and it's often a key point in understanding complex equations, such as those producing more than one line through symmetry.

The lines for this equation \(y = x\) and \(y = -x\) are plotted on this plane, where their interaction at the origin highlights the symmetry and confirms our understanding from Equation Interpretation.