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Magazine Chapter 2: Problem 44
19鈥�44 ? Make a table of values and sketch the graph of the equation. Find the x- and y-intercepts and test for symmetry. $$ y=16-x^{4} $$
Short Answer
Expert verified
The x-intercepts are at -2 and 2; the y-intercept is at 16. The graph is symmetric about the y-axis.
Step by step solution
01
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:- \( x = -2, y = 0 \)- \( x = -1, y = 15 \)- \( x = 0, y = 16 \)- \( x = 1, y = 15 \)- \( x = 2, y = 0 \).
02
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.
03
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 \( y = 16 \).
04
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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 \( x = 2 \), \( y = 0 \)
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.
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.
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.
