Problem 41 - 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. $$ x=y^{3} $$

Step-by-Step Solution

Answer

Graph is symmetric about the origin with x- and y-intercepts at (0,0).

Step 1: Make a Table of Values

To create a table of values, choose several values for \( y \) and compute the corresponding \( x \) using the equation \( x = y^3 \). For example:- If \( y = -2 \), then \( x = (-2)^3 = -8 \).- If \( y = -1 \), then \( x = (-1)^3 = -1 \).- If \( y = 0 \), then \( x = 0^3 = 0 \).- If \( y = 1 \), then \( x = 1^3 = 1 \).- If \( y = 2 \), then \( x = 2^3 = 8 \).The table now looks like:| \( y \) | \( x = y^3 \) ||:---:|:---:|| -2 | -8 || -1 | -1 || 0 | 0 || 1 | 1 || 2 | 8 |

Step 2: Sketch the Graph

Using the table of values, plot the points on the graph. Connect them to see the curve:- Plot (-8, -2), (-1, -1), (0, 0), (1, 1), and (8, 2) on a coordinate plane.- The graph forms a cubic curve which is consistent with the equation \( x = y^3 \).

Step 3: Find the x- and y-intercepts

To find where the curve intersects the axes:- **x-intercept:** It's where the graph crosses the x-axis, which occurs when \( y = 0 \). For \( y = 0 \), \( x = 0^3 = 0 \). Therefore, the x-intercept is at (0, 0).- **y-intercept:** It's where the graph crosses the y-axis, which occurs when \( x = 0 \). For \( x = 0 \), \( y = 0 \) since \( y^3 = 0 \). Therefore, the y-intercept is at (0, 0).

Step 4: Test for Symmetry

Test the equation for symmetry by replacing variables and observing if the resulting equation is equivalent to the original:- **Symmetry about the x-axis:** Replace \( y \) with \( -y \): the equation becomes \( x = (-y)^3 = -y^3 \) which is not equivalent to \( x = y^3 \). Not symmetric about x-axis.- **Symmetry about the y-axis:** Replace \( x \) with \( -x \): the equation becomes \( -x = y^3 \), which is not equivalent. Not symmetric about y-axis.- **Symmetry about the origin:** Replace \( x \) with \( -x \) and \( y \) with \( -y \): the equation becomes \( -x = (-y)^3 = -y^3 \), equivalent. Symmetric about the origin.

Key Concepts

X-Intercepts

An x-intercept is a point where a graph crosses the x-axis. This happens when the value of y is zero. For the equation \( x = y^3 \), we set \( y = 0 \) to find the x-intercept. By substituting \( y = 0 \) into the equation, we get:\[x = 0^3 = 0\]So, the x-intercept for this cubic function is at the point \((0, 0)\).

• Key takeaway: An x-intercept tells us where the graph meets the x-axis. In this case, it happens because when \( y \) is zero, \( x \) also becomes zero.
• It's useful to note that in some functions, there may be more than one x-intercept depending on the degree and shape of the polynomial.

X-intercepts are crucial for understanding the starting points of curves and how they interact with the horizontal axis.

Y-Intercepts

The y-intercept is the point where the graph crosses the y-axis. This occurs when the value of \( x \) is zero. By setting \( x = 0 \) in the equation \( x = y^3 \), we solve for \( y \):\[0 = y^3\]The solution to this equation is \( y = 0 \), which means that the y-intercept for the curve is also at \((0, 0)\).

• Key insight: A y-intercept gives us the point where a curve meets the y-axis.
• Like the x-intercept, the y-intercept of this function is the origin. This indicates that both the x and y intercepts coincide in this case.

Knowing the y-intercept helps in sketching the graph, providing a point of reference where the curve interacts with the vertical axis.

Symmetry in Graphs

Symmetry in graphs provides insights into the balance and structure of a curve. For the equation \( x = y^3 \), we are particularly interested in three types of symmetry: x-axis, y-axis, and origin.
**Symmetry about the x-axis**: We test this by replacing \( y \) with \( -y \). The equation becomes \( x = (-y)^3 = -y^3 \). Since this new equation is not the same as the original, the graph is not symmetric about the x-axis.
**Symmetry about the y-axis**: For this test, replace \( x \) with \( -x \), resulting in \( -x = y^3 \). As this is not equivalent to \( x = y^3 \), the graph does not exhibit symmetry about the y-axis.
**Symmetry about the origin**: Here, replace both \( x \) with \( -x \) and \( y \) with \( -y \). This gives us \( -x = (-y)^3 = -y^3 \). This equation is equivalent to the original, indicating that the graph is symmetric about the origin.

• Conclusion: Symmetry about the origin suggests a balanced nature around the center of the graph, here the origin point \((0, 0)\).
• This type of symmetry helps in understanding how the graph might look when we rotate it 180 degrees around the origin.

Identifying symmetry can simplify the graphing process and provide intuitive understanding of the function's behavior.