Graph y=4sin(x)+3

Problem

y=4*sin(x)+3

Solution

1. Identify the parent function and its properties. The base function is y=sin(x) which has an amplitude of 1 a period of 2*π and oscillates around the x-axis (y=0.

2. Determine the amplitude from the coefficient of the sine term. In y=4*sin(x)+3 the amplitude is |4|=4 This means the graph will rise and fall 4 units from its midline.

3. Identify the vertical shift from the constant term. The +3 indicates a vertical shift upward by 3 units. The new midline (average value) of the function is y=3

4. Calculate the range using the midline and amplitude. The maximum value is 3 + 4 = 7a*n*d(t)*h*e*m*i*n*i*m*u*m*v*a*l*u*e*i*s() - 4 = -1$.

5. Determine the period and key points. Since the coefficient of x is 1 the period remains 2*π Key points occur at x=0,π/2,π,(3*π)/2,2*π

6. Plot the key points over one period.

• At x=0 y=4*sin(0)+3=3 (Midline)

• At x=π/2 y=4*sin(π/2)+3=4*(1)+3=7 (Maximum)

• At x=π y=4*sin(π)+3=3 (Midline)

• At x=(3*π)/2 y=4*sin((3*π)/2)+3=4*(−1)+3=−1 (Minimum)

• At x=2*π y=4*sin(2*π)+3=3 (Midline)

7. Sketch the curve by connecting these points with a smooth wave and extending the pattern in both directions.

Final Answer

To graph y=4*sin(x)+3 plot a sine wave with midline y=3 amplitude 4 period 2*π and range [−1,7]

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