Problem 67 - Introductory Algebra for College Students

Question

Graph \(y=2 x^{2}-8\) and \(y=-2 x^{2}+8\) in the same rectangular coordinate system. What are the coordinates of the points of intersection?

Step-by-Step Solution

Answer

The coordinates of the points of intersection between the functions are \((2, -4) \) and \( (-2, -4) \)

Step 1: Graph the Functions

The parabolic equations provided are \(y=2 x^{2}-8\) and \(y=-2 x^{2}+8\). When graphed, these equations produce two parabolas. The shape of the parabola \(y=2 x^{2}-8\) opens upwards while \(y=-2 x^{2}+8\) opens downward.

Step 2: Find Points of Intersection

To find the points at which these two graphs intersect, set the equations equal to each other and solve for x: \(2 x^{2}-8=-2 x^{2}+8\). This gives \(4 x^{2}=16\) and \(x=\pm 2\). Substitute \(x=\pm 2\) into either of the original equations to get \(y= -4\). So the graphs intersect at the points \((2, -4) \) and \( (-2, -4) \)

Step 3: Confirm With the Graphs

Plotting these points on the graph will confirm that these are indeed the points of intersection.

Key Concepts

Solving Quadratic Equations

When you encounter a quadratic equation like the ones given in the exercise, a common task is to solve for the variable, typically represented as \(x\). This involves rearranging and manipulating the equation so you can find the values of \(x\) that make the equation true. For example, given the equation \(2x^2 - 8 = -2x^2 + 8\), we want to simplify it.

- Start by getting all \(x\) terms on one side: Add \(2x^2\) to both sides to yield \(4x^2 = 16\).- Next, solve for \(x^2\): Divide both sides by 4 to get \(x^2 = 4\).- Finally, find \(x\): Take the square root of both sides to solve for \(x\), resulting in \(x = \pm 2\).The solutions \(x = 2\) and \(x = -2\) are the values where these quadratic equations are equal, and they help identify points of intersection when paired with corresponding \(y\)-values.

Parabolas

A parabola is the shape of the graph you get when you plot a quadratic equation, like \(y = 2x^2 - 8\). This curve is symmetrical and can open upwards or downwards, depending on the quadratic term (\(ax^2\)) in the equation.

- If \(a > 0\), the parabola opens upwards, creating a 'U' shape.- If \(a < 0\), it opens downwards, forming an upside-down 'U'.Each parabola has a vertex, which is the highest or lowest point of the curve, and an axis of symmetry, a vertical line through the vertex, splitting the parabola into mirror images. In the graphs of \(y = 2x^2 - 8\) and \(y = -2x^2 + 8\), the first equation represents a parabola opening upwards, while the second opens downwards.

These properties are crucial for understanding how the graphs behave and where they might intersect.

Points of Intersection

Points of intersection are where two graphs cross each other on the coordinate plane. This means that at these points, the \(x\)- and \(y\)-coordinates satisfy both equations. To find these points for the equations \(y = 2x^2 - 8\) and \(y = -2x^2 + 8\), we set the equations equal to each other and solved for \(x\).

- From the solution \(x = \pm 2\) for \(4x^2 = 16\), substitute \(x\) back into the original equation to find \(y\): For both \(x = 2\) and \(x = -2\), \(y = -4\).Thus, the points of intersection are \((2, -4)\) and \((-2, -4)\). These points are where the peaks or troughs of each parabola can visually appear to touch, confirming their intersection in a graph plot.