What is the $y$-intercept of the function $f(x)=4-5 x$ ?
To find the y-intercept of a function \(f(x)\), we determine the point where the graph of the function intersects the y-axis. By definition, any point on the y-axis has an x-coordinate of \(0\). Therefore, the y-intercept is found by evaluating the function at \(x = 0\). The given function is \(f(x) = 4 - 5x\).
To calculate the y-intercept, we substitute \(x = 0\) into the expression for \(f(x)\):
\[f(0) = 4 - 5(0)\]
Performing the multiplication, we have \(5 \times 0 = 0\).
Substituting this back into the equation:
\[f(0) = 4 - 0 = 4\]
Thus, the function evaluates to 4 when \(x\) is 0.
The y-intercept occurs at the point \((0, 4)\) on the coordinate plane. The y-intercept value itself is the constant term in the linear equation when written in the slope-intercept form \(f(x) = mx + b\), where \(b\) represents the y-intercept. In our function \(f(x) = -5x + 4\), the value of \(b\) is clearly 4. Therefore, the y-intercept is 4.
The y-intercept is a fundamental concept in modeling real-world linear relationships. For example, if a water tank starts with 4 gallons of water and drains at a rate of 5 gallons per minute, the volume of water \(V\) after \(t\) minutes can be modeled by \(V(t) = 4 - 5t\). The y-intercept, 4, represents the initial amount of water in the tank before any draining begins (at time \(t = 0\)). This helps engineers and students understand initial conditions in various dynamic systems.