Intersection problems Find the following points of intersection.


The point(s) of intersection of the parabolas y= x² and y= -x² + 8x

Verified step by step guidance

1. Step 1: Set the equations of the parabolas equal to each other to find the x-coordinates of the intersection points: \(x^2 = -x^2 + 8x\).
2. Step 2: Rearrange the equation to form a standard quadratic equation: \(2x^2 - 8x = 0\).
3. Step 3: Factor the quadratic equation: \(2x(x - 4) = 0\).
4. Step 4: Solve for x by setting each factor equal to zero: \(2x = 0\) and \(x - 4 = 0\).
5. Step 5: Substitute the x-values back into either original equation to find the corresponding y-values for the points of intersection.

Key Concepts

Finding Points of Intersection

To find the points of intersection between two curves, we set their equations equal to each other. This involves solving the resulting equation for the variable, which will yield the x-coordinates of the intersection points. Once the x-values are found, they can be substituted back into either original equation to find the corresponding y-values.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form y = ax² + bx + c. The graphs of these functions are parabolas, which can open upwards or downwards depending on the sign of the coefficient 'a'. Understanding the properties of parabolas, such as their vertex, axis of symmetry, and direction of opening, is essential for analyzing their intersections.

Solving Quadratic Equations

Solving quadratic equations can be done using various methods, including factoring, completing the square, or applying the quadratic formula. Each method has its advantages depending on the specific equation. For intersection problems, finding the roots of the resulting quadratic equation will reveal the x-coordinates of the intersection points, which are critical for determining where the two curves meet.