\( 1 \leftarrow \quad \) List the intercepts and test for symmetry. \[ y=\frac{-4 x^{9}}{x^{2}-4} \]

Certainly! Let's analyze the function \( y = \frac{-4x^{9}}{x^{2} - 4} \) to determine its intercepts and symmetry. ### 1. Intercepts **a. X-Intercept(s):** To find the x-intercept(s), set \( y = 0 \) and solve for \( x \): \[ 0 = \frac{-4x^{9}}{x^{2} - 4} \] The fraction is zero when the **numerator** is zero (provided the denominator is not zero). So, \[ -4x^{9} = 0 \implies x^{9} = 0 \implies x = 0 \] **X-Intercept:** \( (0, 0) \) **b. Y-Intercept:** To find the y-intercept, set \( x = 0 \) and solve for \( y \): \[ y = \frac{-4(0)^{9}}{0^{2} - 4} = \frac{0}{-4} = 0 \] **Y-Intercept:** \( (0, 0) \) **Summary of Intercepts:** - **X-Intercept:** \( (0, 0) \) - **Y-Intercept:** \( (0, 0) \) ### 2. Symmetry To determine the symmetry of the function, we'll consider the following tests: **a. Symmetry about the Y-Axis (Even Function):** Replace \( x \) with \( -x \) and compare \( y(-x) \) with \( y(x) \): \[ y(-x) = \frac{-4(-x)^{9}}{(-x)^{2} - 4} = \frac{-4(-x)^{9}}{x^{2} - 4} = \frac{4x^{9}}{x^{2} - 4} \] Since \( y(-x) \neq y(x) \), the function is **not even** and does **not** have y-axis symmetry. **b. Symmetry about the Origin (Odd Function):** Check if \( y(-x) = -y(x) \): \[ -y(x) = -\left(\frac{-4x^{9}}{x^{2} - 4}\right) = \frac{4x^{9}}{x^{2} - 4} \] \[ y(-x) = \frac{4x^{9}}{x^{2} - 4} = -y(x) \] Since \( y(-x) = -y(x) \), the function is **odd** and has **origin symmetry**. **Conclusion on Symmetry:** - **Symmetric about the Origin:** Yes (the function is odd). - **Symmetric about the Y-Axis or X-Axis:** No. ### Summary - **Intercepts:** Both the x-intercept and y-intercept are at the origin \( (0, 0) \). - **Symmetry:** The graph of the function is symmetric about the origin, indicating that it is an odd function. --- **Graphical Insight (Optional):** While not requested, visualizing the function can provide additional intuition. Given that the function is odd, its graph will exhibit rotational symmetry of 180° about the origin.