how to graph $2\sin\left(\frac{1}{2}(x+\frac{\pi}{3})\right)+1$

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Q. how to graph

2\sin\left(\frac{1}{2}(x+\frac{\pi}{3})\right)+1

Understand Sine Function Form: First, we need to understand the general form of the sine function, which is $y = A\sin(B(x - C)) + D$ , where $A$ is the amplitude, $B$ affects the period, $C$ is the phase shift, and $D$ is the vertical shift. In our function, $2\sin(\frac{1}{2}(x+\frac{\pi}{3}))+1$ , $A = 2$ , $B = \frac{1}{2}$ , $C = -\frac{\pi}{3}$ , and $D = 1$ . We will use these values to graph the function.
Calculate Period: Next, we calculate the period of the function. The period of a sine function is given by $2\pi/B$ . In our case, $B = 1/2$ , so the period $T$ is $2\pi/(1/2) = 4\pi$ .
Determine Phase Shift: Now, we determine the phase shift. The phase shift is given by the value of $C$ in the general form, which is subtracted from $x$ . In our function, $C = -\frac{\pi}{3}$ , so the phase shift is to the left by $\frac{\pi}{3}$ units.
Consider Vertical Shift: We also need to consider the vertical shift, which is $D$ in the general form. In our function, $D = 1$ , so the graph will be shifted up by $1$ unit.
Graph Function: To graph the function, we start by drawing the basic sine curve, then apply the transformations. We'll plot key points for one period of the sine function, starting at the phase shift. The key points occur at $0$ , $T/4$ , $T/2$ , $3T/4$ , and $T$ , where $T$ is the period. We'll then shift these points left by $\pi/3$ units and up by $1$ unit.
Determine Amplitude: The amplitude of the function is $2$ , which means the maximum value is $2$ units above the midline ( $y = 1$ ), and the minimum value is $2$ units below the midline ( $y = 1$ ). So, the maximum and minimum values of the function are $3$ and $-1$ , respectively.
Plot Key Points: Plot the points for one period: starting at the phase shift $x = -\frac{\pi}{3}$ , the sine function starts at the midline, reaches its maximum at $x = -\frac{\pi}{3} + \frac{T}{4}$ , crosses the midline again at $x = -\frac{\pi}{3} + \frac{T}{2}$ , reaches its minimum at $x = -\frac{\pi}{3} + \frac{3T}{4}$ , and returns to the midline at $x = -\frac{\pi}{3} + T$ . Adjust these $x$ -values by the phase shift and $y$ -values by the vertical shift.
Connect Points: Connect the points with a smooth, continuous curve, making sure the curve has the shape of a sine wave, with the appropriate maximum and minimum values and crossing the midline at the correct points.