Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate any points of intersection of the graphs of the equations. Check your results algebraically. $$\begin{aligned} &y=2 x^{2}\\\ &y=x^{4}-2 x^{2} \end{aligned}$$

Step-by-Step Solution

Answer

The points of intersection can be obtained by setting the two given equations equal to each other and solving for x, then substituting those x-values into either equation to obtain the y-values of the points. These values should then be verified against the intersection points observed on the graph.

Step 1: Graphical Intersection Points

Plot the given equations \(y = 2x^{2}\) and \(y = x^{4}-2x^{2}\) using a graphing tool like Desmos or GeoGebra. The points where the curves intersect are the solutions to the problem. Observe these points and make a note of them.

Step 2: Algebraic Verification

To check the obtained points algebraically, set the two equations equal to each other and solve for x. This will result in the equation \(2x^{2} = x^{4} - 2x^{2}\). Solving this equation will give the x-values of the intersection points. Substitute these x-values into either of the original equations to get the corresponding y-values. These (x, y) pairs are the intersection points.

Step 3: Compare the Points

Finally, compare the intersection points obtained algebraically with the ones observed on the graph. If they match, then the points obtained initially can be considered accurate.

Key Concepts

Exploring Points of Intersection

When you have two or more equations and want to find where they intersect, you are essentially looking for common solutions. These common solutions are known as the "points of intersection." In this context, you are trying to find where the graph of the equation \(y = 2x^2\) intersects with the graph of \(y = x^4 - 2x^2\). Points of intersection are important because they give us insight into how the solutions relate to each equation graphically.

To discover these points:

• Plot both equations on the same graph.
• Observe where their paths cross one another.
• These crossing points are your intersection points.

Once identified, these intersection points provide vital coordinates where the two graphs have identical values for both x and y. This is crucial for understanding the relationship between the equations.

Utilizing the Graphical Method

The graphical method involves using a technology tool, such as a graphing calculator or applications like Desmos or GeoGebra, to visually analyze the behavior of equations. This method gives a direct visual representation, making it easier to understand where solutions might lie.

Using the graphical method requires the following steps:

• Input the equations into your graphing tool.
• Look at the visual representation of each equation.
• Identify the intersection points, which are visually evident where the curves cross each other.

Graphing is beneficial when dealing with complex equations, as it provides an immediate way to visualize solutions without going through algebraic manipulations. The key advantage here is seeing the solutions, which can sometimes eliminate errors that occur during manual calculations. Remember to note down the coordinates (x, y) of these points for further verification.

Algebraic Verification Methods

While the graphical method provides a visual representation, algebraic verification involves solving the equations systematically to ensure the graphical observations are correct. This method confirms accuracy and can be broken down into simple steps.

Steps for algebraic verification:

• Set the two equations equal to each other: \(2x^2 = x^4 - 2x^2\).
• Rearrange the equation, if necessary, and solve for \(x\).
• The solution for each \(x\) will give the \(x\)-coordinates of the intersection points.
• Substitute these \(x\)-values back into either original equation to find the \(y\)-coordinates.

This provides a precise confirmation of where the two graphs intersect. Algebraic verification is especially useful because it eliminates any potential discrepancies that might arise from initial graphical observations. Cross-verification ensures that the points identified graphically are indeed the correct ones mathematically.