Problem 46 - Algebra 2

Question

Graph each equation. $$ y^{2}-8 x=0 $$

Step-by-Step Solution

Answer

Graphing the equation \(y^{2} - 8x = 0\) results in two parabolic curves reflected about the x-axis. The direction of the parabolas are to the right with the parabolas opening at 'x = 0'.

Step 1: Rearrange the equation

Rearrange the given equation \(y^{2}-8x=0\) to the form \(y^{2} = 8x\). This will be easier to plot since it shows that y varies directly with x.

Step 2: Get the square root of both sides

Since we are solving for 'y', we have to take the square root of both sides. The important thing to remember is that a square root has two values: positive and negative. Therefore, \(y = +\sqrt{8x}\) and \(y = -\sqrt{8x}\). These will give us two lines when plotted on a graph.

Step 3: Plot the Graph

Choose a range of x-values and calculate the corresponding y-values for both equations. Plot these points on a graph. Both curves should reflect each other across the x-axis. Because of the quadratic nature of the equation, the graph should show a parabolic type curve with an axis of symmetry along the x-axis.

Key Concepts

Parabolas

Parabolas are a common type of curve found in many quadratic equations. When you graph a quadratic equation like \(y^2 = 8x\), you get a shape called a parabola. In this specific equation, the parabolic curve opens horizontally. The orientation of the parabola depends on the equation’s format: it can open up, down, left, or right. In our exercise, since \(y^2\) is on the left side of the equation and equals a positive term on the right, it signifies that parabola is oriented sideways.

When graphing parabolas, it's essential to identify the vertex, which is the highest or lowest point, depending on how the parabola opens. For the equation \(y^2 = 8x\), the vertex is at the origin (0,0). This is because the simplest form of the equation doesn’t involve any added or subtracted terms that would shift the graph.

Parabolas have a unique characteristic, in that they have an axis of symmetry which divides the shape into two mirror-image halves.

Positive and Negative Square Roots

Square roots can be both positive and negative, which is a critical concept when dealing with equations like \(y^2 = 8x\). When we solve for \(y\), we take the square root of both sides. This means we consider both \(+\sqrt{8x}\) and \(-\sqrt{8x}\).

Having positive and negative square roots ensures that we account for all possible solutions. In graphing, they represent the top and bottom halves of your parabola. For \(y = \pm\sqrt{8x}\), the graph will demonstrate two symmetric curves parallel to the x-axis.

This concept of dual square roots is crucial because it aligns with the principle that whenever you square a positive or negative number, you end up with the same positive result. This property must be reflected in the graph as both sides, positive above and negative below, contribute to forming a symmetric parabola.

Symmetry in Graphs

Symmetry is an important aspect when graphing quadratic equations, particularly parabolas. For the equation \(y^2 = 8x\), the symmetry is evident across the x-axis. What this means in practice is that for any point \((x, y)\) on the parabola, there will be a corresponding point \((x, -y)\) that resides directly opposite on the other side of the x-axis.

When plotting the equation, this symmetry helps verify accuracy, as each point not only relies on the calculations but visually balances the parabola’s shape.

If you understand and identify the axis of symmetry, it becomes easier to predict and plot the entire parabola after determining just one set of solutions (either positive or negative). This mirroring effect is not just a mathematical curiosity; it facilitates drawing and understanding graphs and ensures that no part of the solution is overlooked.