Problem 23 - Elementary Algebra

Question

For the following problems, graph the quadratic equations. \(y=\frac{1}{2} x^{2}\)

Step-by-Step Solution

Answer

Answer: (0, 0)

Step 1: Setup the table of values

To start with, we will create a table of values with x-values and their corresponding y-values. We can pick several x-values and plug them into the equation \(y=\frac{1}{2} x^{2}\) to find the corresponding y-values. For a variety of points, choose a mix of negative, zero, and positive x-values.

Step 2: Calculate y-values for each x-value

Now we will calculate the y-values by substituting each x-value into the equation \(y=\frac{1}{2} x^{2}\). For x = -2: \(y=\frac{1}{2} (-2)^{2}=2\) For x = -1: \(y=\frac{1}{2} (-1)^{2}=0.5\) For x = 0: \(y=\frac{1}{2} (0)^{2}=0\) For x = 1: \(y=\frac{1}{2} (1)^{2}=0.5\) For x = 2: \(y=\frac{1}{2} (2)^{2}=2\) So, we have the following table of values: | x | y | |---|----| |-2 | 2 | |-1 | 0.5| | 0 | 0 | | 1 | 0.5| | 2 | 2 |

Step 3: Plot the points on the graph

Now, we will plot the points on the graph using these values. 1. Start by drawing an x-axis and a y-axis. Label the x-axis with x and the y-axis with y. 2. Mark the points on the graph using the table of values as a reference. 3. The points will be (-2, 2), (-1, 0.5), (0, 0), (1, 0.5), and (2, 2).

Step 4: Sketch the curve of the quadratic

After plotting the points, we now draw the curve connecting the points. Since this is a quadratic equation, the curve will be a parabola. The parabola should be symmetric, be widest at the vertex (in this case, at the origin), and continue upward as x moves away from 0 in either direction. With these steps completed, you now have a graph of the quadratic equation \(y=\frac{1}{2} x^{2}\).

Key Concepts

Table of Values

One of the most effective methods to graph a quadratic equation is by using a table of values. This entails selecting a series of x-values and then computing the corresponding y-values. For example, in the case of the equation \(y=\frac{1}{2} x^{2}\), you can start with a mix of x-values such as -2, -1, 0, 1, and 2.

• For \(x = -2\), \(y = \frac{1}{2}(-2)^{2} = 2\)
• For \(x = -1\), \(y = \frac{1}{2}(-1)^{2} = 0.5\)
• For \(x = 0\), \(y = \frac{1}{2}(0)^{2} = 0\)
• For \(x = 1\), \(y = \frac{1}{2}(1)^{2} = 0.5\)
• For \(x = 2\), \(y = \frac{1}{2}(2)^{2} = 2\)

By systematically calculating these values, you create a simple reference table that serves as a guide when plotting points on a graph. This table is essential for ensuring both accuracy and completeness when sketching any complex curves.

Parabola

The graphical representation of a quadratic function, such as \(y=\frac{1}{2}x^{2}\), is known as a parabola. A parabola is a smooth, U-shaped curve that can either open upwards or downwards, depending on the leading coefficient. In this case, the parabola opens upwards because the leading coefficient, \(\frac{1}{2}\), is positive. Each quadratic equation will have its vertex, which is the highest or lowest point on the graph. For the equation \(y=\frac{1}{2}x^{2}\), the vertex is at the point (0, 0).As x-values move further from zero in both positive and negative directions, you will see the y-values gradually increase, creating the notable U-shape. Understanding how to plot and interpret parabolas is key to mastering quadratic equations. It's helpful to remember that these graphs are continuous, which means they are smooth and contain no breaks or negative jumps.

Symmetry in Graphs

A fascinating feature of quadratic graphs is their symmetry. This means they are mirror images on either side of a vertical line called the axis of symmetry. For the equation \(y=\frac{1}{2}x^{2}\), this line is the y-axis itself.Quadratic functions are always symmetric around their vertex. So, if you fold the graph along this axis, both sides would match perfectly. This property simplifies graphing, as you can plot points on one side and then mirror them on the other. It also exemplifies the ordered and predictable nature of quadratic functions.Understanding symmetry helps in quickly sketching or checking the accuracy of your graph. When drawing, you should always verify that each set of mirrored points, such as (-1, 0.5) and (1, 0.5), appear on opposite sides of the axis of symmetry. By recognizing and utilizing this trait, you simplify the process of graphing complex quadratic equations.