Usually, you’re asked to draw the graph to show one period of the function, because in this period you capture all possible values for sine before it starts repeating over and over again. The sine function has 180-degree-point symmetry about the origin. What is the amplitude of the function [latex]f(x)=7\cos(x)[/latex]? Determine the equation for the sinusoidal function in Figure 17. Step 5. Figure 7. The period is [latex]\frac{2π}{|B|}[/latex]. Because the graph of the sine function is being graphed on the x–y plane, you rewrite this as f(x) = sin x where x is the measure of the angle in radians. To determine the equation, we need to identify each value in the general form of a sinusoidal function. Sketch a graph of the y-coordinate of the point as a function of the angle of rotation. Determine the formula for the sine function in Figure 16. Draw a graph of [latex]g(x)=−2\cos(\frac{\pi}{3}x+\frac{\pi}{6})[/latex]. Determine the phase shift as [latex]\frac{C}{B}[/latex]. The London Eye is a huge Ferris wheel with a diameter of 135 meters (443 feet). If [latex]f(x) =\sin (2x)[/latex], then [latex]B= 2[/latex], so the period is [latex]π[/latex] and the graph is compressed. The equation for a sinusoidal function can be determined from a graph. Because we can evaluate the sine and cosine of any real number, both of these functions are defined for all real numbers. Let’s begin by comparing the equation to the general form [latex]y=A\sin(Bx−C)+D[/latex]. The table below lists some of the values for the cosine function on a unit circle. Is the function stretched or compressed vertically? Because A is negative, the graph descends as we move to the right of the origin. Animation: Graphing the Cosine Function Using the Unit Circle. If [latex]f(x) = \sin\left(\frac{x}{2} \right)[/latex], then [latex]B=\frac{1}{2}[/latex], so the period is [latex]4π[/latex] and the graph is stretched. While C relates to the horizontal shift, D indicates the vertical shift from the midline in the general formula for a sinusoidal function. Scroll down the page for examples Is the function stretched or compressed vertically? The table below lists some of the values for the sine function on a unit circle. Determine the midline, amplitude, period, and phase shift. Determining the amplitude and period of sine and cosine functions. Periodic functions repeat after a given value. Calculate the graph’s maximum and minimum points. The negative value of A results in a reflection across the x-axis of the sine function, as shown in Figure 10. You've already learned the basic trig graphs.But just as you could make the basic quadratic, y = x 2, more complicated, such as y = –(x + 5) 2 – 3, so also trig graphs can be made more complicated.We can transform and translate trig functions, just like you transformed and translated other functions in algebra.. Let's start with the basic sine function, f (t) = sin(t). The period is 4. Step 3. Recall from The Other Trigonometric Functions that we determined from the unit circle that the sine function is an odd function because [latex]\sin(−x)=−\sin x[/latex]. The equation shows that [latex]B=\frac{π}{2}[/latex], so the period is, [latex] \begin{align}P&=\frac{2\pi}{\frac{\pi}{2}}\\&=2\pi\times\frac{2}{\pi}\\&=4 \end{align}[/latex]. If you look at it upside down, the graph looks exactly the same. or [latex]\frac{\pi}{6}[/latex] units to the left. midline: y=0; amplitude: |A|=0.8; period: P=[latex]\frac{2π}{|B|}=\pi[/latex]; phase shift: [latex]\frac{C}{B}=0[/latex] or none. Step 4–7. Let’s start with the sine function. So far, our equation is either [latex]y=3\sin(\frac{\pi}{3}x−C)−2[/latex] or [latex]y=3\cos(\frac{\pi}{3}x−C)−2[/latex].