For the given equation, list the intercepts and test for symmetry. \[ 4 x^{2}+16 y^{2}=64 \] What are the intrent

To analyze the equation \( 4x^{2} + 16y^{2} = 64 \), we'll find the intercepts and test for symmetry. ### 1. Intercepts **a. X-Intercepts:** To find the x-intercepts, set \( y = 0 \) and solve for \( x \). \[ 4x^{2} + 16(0)^{2} = 64 \\ 4x^{2} = 64 \\ x^{2} = 16 \\ x = \pm 4 \] **X-Intercepts:** \( (4, 0) \) and \( (-4, 0) \) --- **b. Y-Intercepts:** To find the y-intercepts, set \( x = 0 \) and solve for \( y \). \[ 4(0)^{2} + 16y^{2} = 64 \\ 16y^{2} = 64 \\ y^{2} = 4 \\ y = \pm 2 \] **Y-Intercepts:** \( (0, 2) \) and \( (0, -2) \) ### 2. Testing for Symmetry We can test for symmetry with respect to the **x-axis**, **y-axis**, and the **origin**. **a. Symmetry with Respect to the Y-Axis:** Replace \( x \) with \( -x \) in the equation. \[ 4(-x)^{2} + 16y^{2} = 64 \\ 4x^{2} + 16y^{2} = 64 \] The equation remains unchanged, indicating **symmetry about the y-axis**. --- **b. Symmetry with Respect to the X-Axis:** Replace \( y \) with \( -y \) in the equation. \[ 4x^{2} + 16(-y)^{2} = 64 \\ 4x^{2} + 16y^{2} = 64 \] The equation remains unchanged, indicating **symmetry about the x-axis**. --- **c. Symmetry with Respect to the Origin:** Replace \( x \) with \( -x \) and \( y \) with \( -y \). \[ 4(-x)^{2} + 16(-y)^{2} = 64 \\ 4x^{2} + 16y^{2} = 64 \] The equation remains unchanged, indicating **symmetry about the origin**. ### 3. Summary - **X-Intercepts:** \( (4, 0) \) and \( (-4, 0) \) - **Y-Intercepts:** \( (0, 2) \) and \( (0, -2) \) - **Symmetry:** The graph is symmetric with respect to both the x-axis and y-axis, and hence symmetric about the origin.