Y-Intercept
The y-intercept of a function is the point where the graph intersects the y-axis. This happens when the input value, x, is zero. For the sine function, expressed as \(f(x) = \sin x\), we need to find \(f(0)\). Since \(\sin 0 = 0\), the y-intercept is 0.
In other words, the point (0,0) is where the graph of \(f(x) = \sin x\) crosses the y-axis.
Remember, the y-intercept gives us a helpful starting point to begin plotting the function graph.
Increasing Intervals
A function is said to be increasing on an interval if as x increases, f(x) also increases. To determine where \(f(x) = \sin x\) is increasing, we look at the derivative \(f'(x) = \cos x\).
The sine function increases where its derivative, cosine, is positive. Cosine is positive in the interval \(-\pi/2 < x < \pi/2\). This means that the sine function is increasing on this interval within the given range \(-\pi \leq x \leq \pi\).
Therefore, the graph of \(f(x) = \sin x\) is increasing between \(-\pi/2\) and \(\pi/2\), which translates to the interval \(-1.57 < x < 1.57\) in radians.
Absolute Maximum
The absolute maximum of a function is the highest point over its entire domain. For the function \(f(x) = \sin x\), the absolute maximum value is 1. This occurs at specific points where sine reaches its peak.
Specifically, within one cycle of the sine function (from \(-\pi\) to \(\pi\)), the absolute maximum of 1 occurs at \(x = \pi/2\).
It's important to note that sine function is periodic, so this maximum value of 1 repeats at intervals \(2n\pi + \pi/2\) where n is any integer.
Zeros of Function
The zeros of a function are the values of x at which \(f(x) = 0\). For the sine function, these are the points where the sine curve crosses the x-axis.
Solve \(\sin x = 0\) within the interval \(0 \leq x \leq 2\pi\). The function equals zero at \(x = 0, \pi, \) and \(2\pi\).
These points, (0, 0), (\pi, 0), and (2\pi, 0), represent the x-values where the function touches the x-axis without going above or below it.
X-Intercepts
X-intercepts are similar to zeros. They are the points where the graph of the function crosses the x-axis. For \(f(x) = \sin x\), the x-intercepts occur where the function equals zero.
We know that \(\sin x = 0\) at integer multiples of \(\pi\). Therefore, the x-intercepts for \(\sin x\) are at points \(x = k\pi\), where k is any integer.
For instance, within the range \(-2\pi \leq x \leq 2\pi\), the x-intercepts are at \(-2\pi, -\pi, 0, \pi, \text{and } 2\pi\). These points summarize where the sine wave meets the x-axis.
