Problem 44 - 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. $$ y=16-x^{4} $$

Step-by-Step Solution

Answer

The x-intercepts are at -2 and 2; the y-intercept is at 16. The graph is symmetric about the y-axis.

Step 1: Construct the Table of Values

To create a table of values for the equation \( y = 16 - x^4 \), choose a set of x-values, substitute each into the equation, and solve for y. For example, use x-values: -2, -1, 0, 1, 2.- When \( x = -2 \), \( y = 16 - (-2)^4 = 16 - 16 = 0 \).- When \( x = -1 \), \( y = 16 - (-1)^4 = 16 - 1 = 15 \).- When \( x = 0 \), \( y = 16 - 0^4 = 16 \).- When \( x = 1 \), \( y = 16 - 1^4 = 16 - 1 = 15 \).- When \( x = 2 \), \( y = 16 - 2^4 = 16 - 16 = 0 \).Thus, the table of values created is:- \( x = -2, y = 0 \)- \( x = -1, y = 15 \)- \( x = 0, y = 16 \)- \( x = 1, y = 15 \)- \( x = 2, y = 0 \).

Step 2: Sketch the Graph

Plot the points from the table of values onto a graph. Connect the points to sketch a curve that represents the equation \( y = 16 - x^4 \). The curve will decrease rapidly as \( x \) moves away from zero on either side, creating a 'W' shaped curve.

Step 3: Find x- and y-Intercepts

The x-intercepts occur where \( y = 0 \). Based on the table:- \( x = -2 \), \( y = 0 \)- \( x = 2 \), \( y = 0 \)The y-intercept occurs where \( x = 0 \). From the table:- \( x = 0 \), \( y = 16 \).Thus, the y-intercept is at \( y = 16 \).

Step 4: Test for Symmetry

For y-axis symmetry, substitute \( -x \) into the equation to check if the equation remains unchanged; \( y = 16 - (-x)^4 = 16 - x^4 \). Since the equation is unchanged, it is symmetric with respect to the y-axis.For origin symmetry, substitute both \(-x\) and \(-y\): \( -y = 16 - x^4 \) or \( y' = x^4 - 16 \). Since this differs from the original equation, the graph is not symmetric with respect to the origin. The symmetry is along the y-axis.

Key Concepts

Table of Values

Creating a table of values is a foundational step in understanding how a polynomial function behaves. For the function \( y = 16 - x^4 \), selecting a series of \( x \)-values and calculating the corresponding \( y \)-values allows us to visualize changes and trends. By substituting given \( x \)-values into the function, we can compute the corresponding \( y \)-values and create ordered pairs. For instance, given the \( x \)-values \(-2, -1, 0, 1, 2\), when plugged into the equation:

• For \( x = -2 \), \( y = 0 \)
• For \( x = -1 \), \( y = 15 \)
• For \( x = 0 \), \( y = 16 \)
• For \( x = 1 \), \( y = 15 \)
• For \( x = 2 \), \( y = 0 \)

This table of values helps us sketch the graph and identify critical points such as intercepts and symmetry.

x- and y-Intercepts

The intercepts of a graph are crucial as they represent the points where the graph crosses the axes. These points provide vital information about the polynomial's behavior near key value changes.
For x-intercepts, we find where the function equals zero. Solving \( 16 - x^4 = 0 \), or from our table, this occurs at \( x = -2 \) and \( x = 2 \), as both result in \( y = 0 \).
For the y-intercept, substitute \( x = 0 \) into the function. Here, \( y = 16 \), indicating that the graph crosses the y-axis at \( y = 16 \). Recognizing these points on a graph helps in creating an accurate representation of the polynomial function.

Symmetry

Symmetry makes graphing functions simpler because it indicates that one part of the graph is a mirror image of another. For the equation \( y = 16 - x^4 \), testing for symmetry helps in efficient graph plotting.

• Y-axis Symmetry: Substitute \( -x \) in place of \( x \). If the equation remains the same, as in \( 16 - (-x)^4 = 16 - x^4 \), it is symmetric along the y-axis. Our function is indeed y-axis symmetric.
• Origin Symmetry: Substitute both \(-x\) and \(-y\). For our function, this would be \(-y = 16 - x^4\). As it differs from the original, the function is not symmetric about the origin.

Understanding symmetry provides insights into how the function's graph forms and repeats, allowing you to predict portions of the graph with ease.

Polynomial Equations

Polynomial equations are expressions consisting of variables raised to non-negative integer powers and coefficients. They can provide a wide range of beautiful and complex curves when graphed.

• In our example, \( y = 16 - x^4 \), the function is a fourth-degree polynomial, indicating a quartic equation.
• Characteristics of this polynomial include a symmetrical 'W'-shape when graphed, due to its even degree and the dominantly negative highest power term, \(-x^4\).
• The features of any polynomial equation depend on terms' degrees and coefficients, which affect its intercepts, vertices, and end behavior.

Knowing how to approach polynomial equations with these considerations in mind aids in efficient problem-solving and graphing strategies.