The degree of the polynomial function \( f(x) \) is 3 . The roots of the equation \( f(x)=0 \) are \( -2,0 \), and 3 Which graph could be the graph of \( f(x) \) ?

Given that the polynomial function \( f(x) \) is of degree 3 and has roots at \( -2, 0, \) and \( 3 \), we can express the function in factored form as: \[ f(x) = k(x + 2)(x)(x - 3) \] where \( k \) is a non-zero constant that determines the vertical stretch or compression of the graph. ### Characteristics of the Graph: 1. **Degree**: Since the degree is 3, the graph will have the general shape of a cubic function, which means it will have one of the following behaviors: - As \( x \to -\infty \), \( f(x) \to -\infty \) if \( k > 0 \) or \( f(x) \to +\infty \) if \( k < 0 \). - As \( x \to +\infty \), \( f(x) \to +\infty \) if \( k > 0 \) or \( f(x) \to -\infty \) if \( k < 0 \). 2. **Roots**: The roots of the polynomial are: - \( x = -2 \) (crosses the x-axis) - \( x = 0 \) (crosses the x-axis) - \( x = 3 \) (crosses the x-axis) 3. **Behavior at the Roots**: - At \( x = -2 \), the graph will cross the x-axis. - At \( x = 0 \), the graph will also cross the x-axis. - At \( x = 3 \), the graph will again cross the x-axis. ### Summary of the Graph's Behavior: - The graph will start from the bottom left (if \( k > 0 \)) or top left (if \( k < 0 \)). - It will cross the x-axis at \( -2 \), \( 0 \), and \( 3 \). - The y-intercept will be at \( f(0) = k(0 + 2)(0)(0 - 3) = 0 \), confirming that the graph passes through the origin. ### Conclusion: To identify the correct graph of \( f(x) \), look for a cubic graph that: - Crosses the x-axis at \( -2 \), \( 0 \), and \( 3 \). - Has the appropriate end behavior based on the sign of \( k \). If you have specific graph options, you can compare them against these characteristics to find the correct one.