Use the given function to complete parts (a) through (e) below. \( f(x)=x^{4}-36 x^{2} \) c) Hind the \( y \)-intercept by computing \( t(0) \). d) Determine the symmetry of the graph. Even; \( y \)-axis symmetry Odd; origin symmetry e) Determine the graph of the function. A.

Evaluate the expression by following steps: - step0: Evaluate: \(x^{4}-36x^{2}\) - step1: Substitute: \(0^{4}-36\times 0^{2}\) - step2: Calculate: \(0-36\times 0^{2}\) - step3: Calculate: \(0-36\times 0\) - step4: Multiply: \(0+0\) - step5: Remove 0: \(0\) Determine whether the function \( f(x)=x^{4}-36 x^{2} \) is even or odd Function by following steps: - step0: Determine if even, odd or neither: \(f\left(x\right)=x^{4}-36x^{2}\) - step1: Use substitution: \(f\left(-x\right)=\left(-x\right)^{4}-36\left(-x\right)^{2}\) - step2: Simplify: \(f\left(-x\right)=x^{4}-36x^{2}\) - step3: Use substitution: \(f\left(-x\right)=f\left(x\right)\) - step4: The function is an even function: \(\textrm{Even}\) Let's go through the parts step by step: ### Part (c): Finding the \( y \)-intercept To find the \( y \)-intercept of the function \( f(x) = x^{4} - 36x^{2} \), we compute \( f(0) \): \[ f(0) = 0^{4} - 36 \cdot 0^{2} = 0 \] Thus, the \( y \)-intercept is \( 0 \). ### Part (d): Determining the symmetry of the graph To determine the symmetry of the function, we check if it is even, odd, or neither. - A function is **even** if \( f(-x) = f(x) \). - A function is **odd** if \( f(-x) = -f(x) \). For the function \( f(x) = x^{4} - 36x^{2} \): Since we found that \( f(x) \) is even, it has \( y \)-axis symmetry. ### Part (e): Determining the graph of the function The function \( f(x) = x^{4} - 36x^{2} \) is a polynomial of degree 4, which typically has a U-shape. Given that it is even, the graph will be symmetric about the \( y \)-axis. The function has a minimum at \( x = 0 \) and opens upwards. In summary: - The \( y \)-intercept is \( 0 \). - The function is even, indicating \( y \)-axis symmetry. - The graph of the function is a U-shape, symmetric about the \( y \)-axis.