To determine the shape of the graph that Rhianna will obtain from a polynomial equation, let's analyze the options:
1. A repeating wave:
- Repeating wave patterns are typically associated with trigonometric functions like sine and cosine. These functions have regular oscillations and are not characteristic of polynomial equations.
2. A line that bounces back and forth over the x-axis:
- Polynomial equations can indeed create graphs that cross the x-axis multiple times. The degree of the polynomial determines the number of possible turning points (where the graph changes direction), and these often result in a graph that may seem to "bounce" back and forth over the x-axis.
3. A J-shaped line:
- This shape is usually indicative of exponential functions, which have rapid growth or decay. Polynomial equations generally do not exhibit this simple, curved shape.
4. A horizontally stretched S-shaped line:
- This is characteristic of a cubic polynomial function (degree 3) or higher-order polynomials that have similar shapes. While this is possible for polynomials, it does not capture the general tendency for higher-degree polynomials that can cross the x-axis multiple times.
Given the choices, the option that best fits the behavior of polynomial equations, which can exhibit multiple crossings at the x-axis and changing directions, is:
A line that bounces back and forth over the x-axis
Thus, the graph of Rhianna's polynomial equation will have the shape of a line that bounces back and forth over the x-axis.
