Question

Use analytical methods to find the following points of intersection. Use a graphing utility only to check your work. Find the point(s) of intersection of the parabolas \(y=x^{2}\) and \(y=-x^{2}+8 x\)

Short answer

Answer: The points of intersection are (0,0) and (4,16).

Step-by-step solution

Set the equations equal to each other

Since we are looking for the points where the two parabolas intersect, we can set their y-values equal to each other: \(x^2 = -x^2 + 8x\)

Solve for x

Now we can solve for x by simplifying the equation and finding the roots: \(x^2 + x^2 - 8x = 0\) \(2x^2 - 8x = 0\) \(x(2x - 8) = 0\) From this, we find two possible x-values for the points of intersection: \(x = 0\) or \(x = 4\)

Find the corresponding y-values

Next, we will plug in the x-values we found into either of the parabola equations to find their corresponding y-values. We can use the first equation, \(y = x^2\), for simplicity: When \(x = 0\), \(y = (0)^2 = 0\) When \(x = 4\), \(y = (4)^2 = 16\)

Write the points of intersection

Now we have the x and y values for the points of intersection, so we can write the points as ordered pairs: The points of intersection are \((0,0)\) and \((4,16)\). You can now use a graphing utility to verify the solution.

Key concepts

Intersections of Parabolas

When discussing intersections of parabolas, we aim to find points where two distinct parabolic curves meet each other in the xy-plane. These intersecting points are solutions that satisfy both parabolic equations simultaneously. For the parabolas given by equations \( y = x^2 \) and \( y = -x^2 + 8x \), intersections occur when the y-values of both equations are identical for the same x-value.

To find these intersections analytically, we set the two equations equal to each other:

• \( x^2 = -x^2 + 8x \)

Once they are equal, solving for x will reveal the points where the parabolas intersect. This forms the foundation of our approach to problem-solving when dealing with parabolic intersections.

Quadratic Equations

A quadratic equation is any equation that can be rearranged in the form \( ax^2 + bx + c = 0 \), where a, b, and c are constants. In our problem, after setting the parabolic equations equal, we derive the quadratic \( 2x^2 - 8x = 0 \). Here, the coefficients are \( a = 2 \), \( b = -8 \), and \( c = 0 \).

This results in a simple quadratic expression that can be easily factored. Factoring involves rewriting the quadratic equation as \( x(2x - 8) = 0 \). This step allows us to identify the potential values of x that satisfy the equation by setting each factor equal to zero:

• \( x = 0 \)
• \( 2x - 8 = 0 \)

By solving these equations separately, we find the x-values that determine the intersection points of the original parabolas.

Graphical Verification

Graphical verification is a valuable tool for confirming solutions found analytically. Once you've calculated potential intersection points using equations, you can use graphing utilities to visualize these solutions. This method not only boosts confidence in the derived answers but also enhances understanding of how parabolas behave.

For our parabolas \( y = x^2 \) and \( y = -x^2 + 8x \), plotting the curves on the same graph allows us to see the intersection points clearly. The graph will display the parabolas crossing at points \( (0,0) \) and \( (4,16) \), matching our earlier analytical findings. This graphical representation offers a visual confirmation that strengthens the reliability of analytical solutions.

Finding Roots

In the context of quadratic equations, finding roots is the process of determining the x-values that make the equation equal to zero. For the quadratic \( 2x^2 - 8x = 0 \), the roots can be found by factoring or using the quadratic formula. In our case, simple factoring was possible, leading to the equation \( x(2x - 8) = 0 \).

This factorization shows that:

• \( x = 0 \)
• \( 2x - 8 = 0 \) simplifies to \( x = 4 \)

These values of x, known as roots, indicate where the parabolas intersect on the x-axis. Subsequently, plugging these x-values into either of the original parabolic equations gives us the corresponding y-values \((0,0)\) and \((4,16)\). Therefore, finding roots is essential for solving quadratic equations and is a crucial step in determining intersection points.