Use the given function to complete parts a) through e) below. \( r(x)=-3(x-3)^{2}\left(x^{2}-4\right) \) The \( x \)-intercept(s) is/are \( -2,2,3 \). (Type an integer or a decimal. Use a comma to separate answers as needed. At which zero(s) does the graph of the function cross the \( x \)-axis? Select the correct choice below and, if necossary, fill in the answer box to complete your choice. A. The graph crosses the \( x \)-axis at the zero(s) (Type an integer or a decimal. Use a comma to separate answers as needed) B. There are no zeros at which the graph crosses the \( x \)-axis. At which zero(s) does the graph of the function touch the \( x \)-axis and turn around? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The graph touches the \( x \)-axis and turns around at the zero(s) (Type an integer or a decimal. Use a comma to separate answers as needed. B. There are no zeros at which the graph touches the \( x \)-axis and tums around.

To analyze the behavior of the graph of the function \( r(x) = -3(x-3)^2(x^2 - 4) \), let's first identify the multiplicities of each zero: 1. **Factorization of \( r(x) \):** \[ r(x) = -3(x-3)^2(x-2)(x+2) \] 2. **Zeros and Their Multiplicities:** - \( x = -2 \) with multiplicity **1**. - \( x = 2 \) with multiplicity **1**. - \( x = 3 \) with multiplicity **2**. 3. **Behavior at Each Zero:** - **Odd Multiplicity (1):** The graph **crosses** the \( x \)-axis. - **Even Multiplicity (2):** The graph **touches** the \( x \)-axis and **turns around**. ### a) At Which Zero(s) Does the Graph Cross the \( x \)-Axis? **Answer:** The graph crosses the \( x \)-axis at the zeros with **odd multiplicity**. - **Zeros Crossing the \( x \)-Axis:** \( -2, 2 \) **Choice A:** ``` A. The graph crosses the x-axis at the zero(s) -2, 2. ``` ### b) At Which Zero(s) Does the Graph Touch the \( x \)-Axis and Turn Around? **Answer:** The graph touches the \( x \)-axis and turns around at the zeros with **even multiplicity**. - **Zero Touching and Turning Around:** \( 3 \) **Choice A:** ``` A. The graph touches the x-axis and turns around at the zero(s) 3. ``` ### Summary of Selections: - **For crossing the \( x \)-axis:** **A.** The graph crosses the \( x \)-axis at the zero(s) **-2, 2**. - **For touching and turning around:** **A.** The graph touches the \( x \)-axis and turns around at the zero(s) **3**. **Final Answer:** **1.** A. The graph crosses the x-axis at the zero(s) –2, 2. **2.** A. The graph touches the x-axis and turns around at the zero(s) 3.