Problem 21
Question
Use the Vertical Line Test to determine whether y is a function of x. Describe how you can use a graphing utility to produce the given graph. $$0.25 x^{2}+y^{2}=1$$
Step-by-Step Solution
Verified Answer
'y' is not a function of 'x' in this context, because the graph of the equation \(0.25x^{2} + y^{2} = 1\) fails the Vertical Line Test.
1Step 1: Rewrite the equation
Before proceeding with the vertical line test, it is beneficial to rewrite the equation for clarity. Rearrange the given equation \(0.25x^{2} + y^{2} = 1\) to obtain \(y^{2} = 1 - 0.25x^{2}\)
2Step 2: Understanding the graph of the equation
The equation \(y^{2} = 1 - 0.25x^{2}\) is an equation for a circle centered at (0,0) with a radius of 1. When graphing this equation, note that we have two values of 'y' for each 'x' within the domain ([-2,2]). This is because \(y = \sqrt{1 - 0.25x^{2}}\) and \(y = -\sqrt{1 - 0.25x^{2}}\).
3Step 3: Perform the Vertical Line Test
To perform the vertical line test, plot vertical lines along the graph. If any line touches the graph at more than one point, it shows that for one value of 'x' there are multiple corresponding 'y' values. In this case, this is true since the graph is a circle. Therefore, 'y' is not a function of 'x'.
4Step 4: Graphing the equation
To graph the equation using a graphing utility, plot the positive and negative square root functions separately. This will give the full circle. Label the graph clearly with the x and y axes and plot points for better visual clarity.
Key Concepts
FunctionGraphing UtilityCircle EquationVertical Line Definition
Function
To determine if a relationship is a function, we need to check if each input has a unique output. In mathematical terms, a function is a relation where each input, or 'x' value, has exactly one output, or 'y' value. In the context of a graph, the Vertical Line Test is a simple method to determine if a curve represents a function. If a vertical line intersects the graph at more than one point, then it is not a function, as a single 'x' value is producing multiple 'y' values.
This test is straightforward and very useful for quick evaluations. It highlights how important it is for functions to have unique outputs, ensuring predictability and consistency in equations.
Understanding this concept is fundamental in mathematics, especially when dealing with complex equations and graph interpretations.
This test is straightforward and very useful for quick evaluations. It highlights how important it is for functions to have unique outputs, ensuring predictability and consistency in equations.
Understanding this concept is fundamental in mathematics, especially when dealing with complex equations and graph interpretations.
Graphing Utility
A graphing utility is a tool that allows us to visualize mathematical equations and geometric shapes. They are increasingly available on calculators and computer software. Graphing utilities help simplify the visualization of complex functions and relations, like circles or ellipses, which might be challenging to draw manually.
For the equation \(0.25x^{2} + y^{2} = 1\), using a graphing utility, we can graph both \(y = \sqrt{1 - 0.25x^{2}}\) and \(y = -\sqrt{1 - 0.25x^{2}}\) separately to create the full circle. This visualization helps verify the results from manual calculations, providing a clearer understanding of the relationship between 'x' and 'y'.
For the equation \(0.25x^{2} + y^{2} = 1\), using a graphing utility, we can graph both \(y = \sqrt{1 - 0.25x^{2}}\) and \(y = -\sqrt{1 - 0.25x^{2}}\) separately to create the full circle. This visualization helps verify the results from manual calculations, providing a clearer understanding of the relationship between 'x' and 'y'.
- Graphing utilities allow us to easily perform the Vertical Line Test.
- They offer visual confirmations of equations.
- These tools support mathematics education by offering quick, reliable ways to graph functions and equations.
Circle Equation
A circle equation in a Cartesian plane characterizes the set of all points that are equidistant from a central point. The general form of a circle's equation is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius.
In the exercise, the equation \(0.25x^{2} + y^{2} = 1\) can be rewritten to fit the circle formula. By rearranging, you can see that it represents a circle centered at the origin \((0, 0)\) with a radius of 1. The simplified form reveals that for each 'x' value in the range [-2,2], there are corresponding positive and negative 'y' values, outlining a full circle.
Understanding circle equations is essential, especially in geometry, where circles serve as a basis for more complex shapes and calculations.
In the exercise, the equation \(0.25x^{2} + y^{2} = 1\) can be rewritten to fit the circle formula. By rearranging, you can see that it represents a circle centered at the origin \((0, 0)\) with a radius of 1. The simplified form reveals that for each 'x' value in the range [-2,2], there are corresponding positive and negative 'y' values, outlining a full circle.
Understanding circle equations is essential, especially in geometry, where circles serve as a basis for more complex shapes and calculations.
Vertical Line Definition
The vertical line in mathematics is a straight line that runs parallel to the y-axis and perpendicular to the x-axis. Its equation is generally written as \(x = a\), where 'a' is a constant, meaning that all points on the line have the same x-coordinate.
Applying the concept of a vertical line is essential in performing the Vertical Line Test to determine if a relation is a function. By drawing these lines across the graph, one can focus on the 'x' values to check if they associate with more than one 'y' value. This understanding of vertical lines helps particularly in identifying relations, like circles, that are not functions because they fail this test.
Applying the concept of a vertical line is essential in performing the Vertical Line Test to determine if a relation is a function. By drawing these lines across the graph, one can focus on the 'x' values to check if they associate with more than one 'y' value. This understanding of vertical lines helps particularly in identifying relations, like circles, that are not functions because they fail this test.
- Vertical lines are pivotal in identifying functions visually.
- They help simplify complex graph interpretations.
- The concept supports students' understanding of functions versus non-functions.
Other exercises in this chapter
Problem 21
Show that \(f\) and \(g\) are inverse functions algebraically. Use a graphing utility to graph \(f\) and \(g\) in the same viewing window. Describe the relation
View solution Problem 21
Determine whether the equation represents \(y\) as a function of \(x.\) $$y=\sqrt{x^{2}-1}$$
View solution Problem 21
Use the point on the line and the slope of the line to find three additional points through which the line passes. (There are many correct answers.) Point (0,-9
View solution Problem 22
Evaluate the indicated function for \(f(x)=x^{2}-1\) and \(g(x)=x-2\) algebraically. If possible, use a graphing utility to verify your answer. $$(f+g)(1)$$
View solution