Problem 28 - Algebra and Trigonometry

Question

19–44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ 8 y=x^{3} $$

Step-by-Step Solution

Answer

The graph is a cubic function centered at the origin with intercepts at (0,0) and symmetry about the origin.

Step 1: Create a Table of Values

Choose some values for \( x \) and calculate the corresponding \( y \) values using the equation \( 8y = x^3 \). For instance, if \( x = -2 \), \( y = \frac{(-2)^3}{8} = -1 \). Complete the table with values like \( x = -2, -1, 0, 1, 2 \).

Step 2: Sketch the Graph

Plot the points from your table of values on a coordinate plane. Connect these points to sketch the curve of the equation \( 8y = x^3 \). This will help visualize the function's behavior.

Step 3: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( 8y = x^3 \). Solving \( 8(0) = x^3 \) gives \( x = 0 \). Hence, the x-intercept is at (0, 0).

Step 4: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( 8y = x^3 \). Solving \( 8y = 0 \) gives \( y = 0 \). So, the y-intercept is also at (0, 0).

Step 5: Test for Symmetry

The equation \( 8y = x^3 \) is neither symmetric with respect to the x-axis nor the y-axis because it does not pass the symmetry tests (replacing \( y \) with \( -y \) or replacing \( x \) with \( -x \) respectively) but shows symmetry about the origin since changing both \( x \) and \( y \) to their negatives returns the negative of the original equation.

Key Concepts

x-intercepts

The x-intercepts of a graph are the points where the graph crosses the x-axis. At these points, the value of y is always zero. To find the x-intercepts of an equation, you set the equation equal to zero and solve for x. For the cubic equation given by \(8y = x^3\), to find the x-intercepts:

• Set \(y = 0\).
• This simplifies to \(x^3 = 0\), solving gives \(x = 0\).

Thus, the x-intercept of the equation is at the origin, (0, 0). Remember, the x-intercept can often provide crucial insights into the graph's shape and can be used to determine part of the curve for further sketching.

y-intercepts

Similarly, the y-intercepts are the points where the graph crosses the y-axis. At these points, the value of x is always zero. To find the y-intercepts, you set x equal to zero and solve the resulting equation for y. In our cubic equation \(8y = x^3\):

• Set \(x = 0\).
• This makes the equation \(8y = 0\), solving for y gives \(y = 0\).

Hence, the y-intercept is also at the origin (0, 0). Interestingly, both the x- and y-intercepts for this equation coincide, reaffirming that the origin is a critical point of intersection on the graph.

symmetry in graphs

Graph symmetry refers to a graph's balanced and proportionate shape when divided by its axes or the origin. The types of symmetry include symmetry about the x-axis, y-axis, and origin.For the equation \(8y = x^3\), we can test for:- **X-axis symmetry**: Replace \(y\) with \(-y\). The equation becomes \(8(-y) = x^3\), which is not equivalent to the original equation.- **Y-axis symmetry**: Replace \(x\) with \(-x\). It becomes \(8y = (-x)^3 = -x^3\), also not equivalent to the original.- **Origin symmetry**: Replace both x and y with their negatives, \(8(-y) = (-x)^3\). This results in \(-8y = -x^3\), which simplifies back to the original \(8y = x^3\).Thus, the graph of this equation is symmetric with respect to the origin. This means if the graph is rotated 180 degrees around the origin, it looks identical. Knowing this helps to accurately sketch the graph and predict its shape, reaffirming the graph's behavior with simplicity and precision.