Questions: Find the y-coordinate of the y-intercept of the polynomial function defined below. f(x)=-x(4 x^2-5)(x^2-4)

Transcript text: Find the $y$-coordinate of the $y$-intercept of the polynomial function defined below. \[ f(x)=-x\left(4 x^{2}-5\right)\left(x^{2}-4\right) \]

Solution

Solution Steps

Step 1: Understand the Problem

The $ y $-intercept of a function is the point where the graph of the function intersects the $ y $-axis. This occurs when $ x = 0 $. Therefore, to find the $ y $-coordinate of the $ y $-intercept, we need to evaluate the function $ f(x) $ at $ x = 0 $.

Step 2: Substitute $ x = 0 $ into the Function

Given the polynomial function: \[ f(x) = -x\left(4x^{2} - 5\right)\left(x^{2} - 4\right), \] we substitute $ x = 0 $ into the function: \[ f(0) = -0\left(4(0)^{2} - 5\right)\left((0)^{2} - 4\right). \]

Step 3: Simplify the Expression

Simplify the expression step by step: \[ f(0) = -0 \cdot \left(4 \cdot 0 - 5\right) \cdot \left(0 - 4\right). \] \[ f(0) = 0 \cdot (-5) \cdot (-4). \] \[ f(0) = 0. \]