Use the given function to complete parts a) through e) below \( f(x)=-3(x-3)^{2}\left(x^{2}-4\right) \) 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 turns around c) Find the \( y \)-intercept by computing \( f(0) \) The \( y \)-intercept is \( \square \) d) Determine the symmetry of the graph Even, \( y \)-axis symmetry Odd, origin symmetry neither \( f(x) \) nor \( -f(x) \)

Let's analyze the function \( f(x) = -3(x-3)^{2}(x^{2}-4) \) to answer parts **a)** through **d)**. --- ### a) **Zeros Where the Graph Touches the \( x \)-Axis and Turns Around** To determine where the graph touches the \( x \)-axis and turns around, we need to find the **zeros** of the function and examine their **multiplicities**: 1. **Find the Zeros:** \[ f(x) = -3(x-3)^{2}(x^{2}-4) = 0 \] - \( x - 3 = 0 \) ⇒ \( x = 3 \) - \( x^2 - 4 = 0 \) ⇒ \( x = 2 \) or \( x = -2 \) 2. **Determine Multiplicities:** - **\( x = 3 \):** Multiplicity **2** (even) - **\( x = 2 \) and \( x = -2 \):** Multiplicity **1** each (odd) 3. **Interpretation:** - **Even Multiplicity:** The graph **touches** the \( x \)-axis and **turns around**. - **Odd Multiplicity:** The graph **crosses** the \( x \)-axis. **Conclusion:** The graph **touches the \( x \)-axis and turns around** only at \( x = 3 \). **Answer:** **A. The graph touches the \( x \)-axis and turns around at the zero(s) 3** --- ### c) **Finding the \( y \)-Intercept by Computing \( f(0) \)** The \( y \)-intercept of a function is the point where the graph crosses the \( y \)-axis, which occurs when \( x = 0 \). \[ f(0) = -3(0 - 3)^{2}(0^{2} - 4) \] \[ f(0) = -3(9)(-4) \] \[ f(0) = -3 \times 9 \times (-4) \] \[ f(0) = 108 \] **Answer:** **The \( y \)-intercept is 108** --- ### d) **Determining the Symmetry of the Graph** A function can be **even**, **odd**, or **neither** based on its symmetry: 1. **Even Function (Symmetric about the \( y \)-axis):** \[ f(-x) = f(x) \] 2. **Odd Function (Symmetric about the Origin):** \[ f(-x) = -f(x) \] 3. **Neither:** - If neither of the above conditions is satisfied. **Check for Symmetry:** \[ f(-x) = -3(-x - 3)^{2}((-x)^2 - 4) \] \[ f(-x) = -3(x + 3)^{2}(x^{2} - 4) \] Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \): - **\( f(-x) \neq f(x) \)** (Not even) - **\( f(-x) \neq -f(x) \)** (Not odd) **Conclusion:** The function is **neither even nor odd**. **Answer:** **Neither \( f(x) \) nor \( -f(x) \)** ---