Question

Use analytical methods to find the following points of intersection. 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 two equations equal to each other

To find the points of intersection, set the two equations equal to each other and solve for x: \(x^{2} = -x^{2} + 8x\)

Simplify the equation

Add both x terms to one side of the equation: \(2x^{2} - 8x = 0\)

Factor the equation

Factor out the common factor of 2x: \(2x(x - 4) = 0\)

Find the x-values of the intersection points

Now, we'll set both factors equal to 0 and solve for x: \(2x = 0 \Rightarrow x = 0\) \(x - 4 = 0 \Rightarrow x = 4\)

Find the corresponding y-values

Plug the x-values we found into either equation to find the corresponding y-values: For \(x = 0\), \(y = 0^2 = 0\) So, the first point of intersection is \((0,0)\). For \(x=4\), \(y = 4^2 = 16\) So, the second point of intersection is \((4,16)\). These points of intersection, \((0,0)\) and \((4,16)\), are where the parabolas \(y=x^2\) and \(y=-x^2+8x\) intersect.

Key concepts

Analytical Methods

Analytical methods are mathematical techniques used to find exact solutions to problems. In the context of finding the intersection of curves, these methods involve setting two equations equal and solving for the required variables. For this particular exercise, we need to find where the two given parabolas intersect by finding values of \(x\) and \(y\) where the equations \(y = x^{2}\) and \(y = -x^{2} + 8x\) are equal.

Here's how you approach it:

• Equate the two expressions for \(y\) to eliminate it from the equation: \(x^{2} = -x^{2} + 8x\).
• This gives a single-variable quadratic equation. Solving this equation will provide the \(x\)-coordinates of the intersection points.

Using analytical methods usually includes techniques such as substitution or elimination when dealing with systems of equations.
In this case, after equating the two parabolas, you simplify and solve the resulting quadratic equation to identify the \(x\) values.

Parabolas

Parabolas are fundamental concepts in mathematics, particularly in algebra and geometry. A parabola is a U-shaped curve that can open upwards, downwards, left, or right. Mathematically, they are described by quadratic equations of the form \(y = ax^2 + bx + c\).

In the given exercise, we are analyzing two different parabolas:

• The first is \(y = x^2\), which is a simple upward-opening parabola with its vertex at the origin \((0, 0)\).
• The second is \(y = -x^{2} + 8x\), which opens downwards due to the negative sign in front of \(x^2\), and is shifted away from the origin.

Understanding the shape and position of these parabolas helps us visually verify where they might intersect. The process involves using their equations to find where they share common points, called intersection points.

Factoring Equations

Factoring is a mathematical process used to simplify equations and solve them by expressing them as a product of their factors. It is particularly useful in solving quadratic equations in the form \(ax^2 + bx + c = 0\).

In our exercise, after equating the two different forms of \(y\), we arrive at the expression: \(2x^2 - 8x = 0\). Here, factoring helps in solving for \(x\):

• First, factor out the greatest common factor, which is \(2x\): \(2x(x - 4) = 0\).
• Next, set each factor equal to zero: \(2x = 0\) and \(x - 4 = 0\).
• Solve each equation to find the values of \(x\), i.e., \(x = 0\) and \(x = 4\).

This method turns a complex equation into simpler ones, making it easier to find solutions. Understanding factoring is essential for solving quadratic equations efficiently and finding roots that might represent solutions or intersection points.