To help you plot the graph for the equation $Y = 2 \sin(2x + 30^\circ)$ using the table, I will outline the steps based on the values provided in the image:

Steps to plot the graph:

1. Read the data from the table:

   • The table shows values for $x$ and corresponding values of $Y = 2 \sin(2x + 30^\circ)$.
   • The table lists values of $x$ such as 0, 30, 60, 90, etc., with the corresponding $Y$ values, like 1, 1.8, and so on.

2. Plot $x$ vs $Y$:

   • Use the given $x$-values (which appear to be angles in degrees) as the horizontal axis (x-axis).
   • Use the corresponding $Y$-values as the vertical axis (y-axis).
   • Each pair of $x$ and $Y$ should be plotted as a point.

3. Connect the points:

   • After plotting all the points, draw a smooth curve to connect them. Since this is a sine function, the curve will be sinusoidal in shape.

4. Label the axes:

   • Label the x-axis with the angle values and the y-axis with the corresponding $Y$-values.

If you'd like, I can plot the graph for you. Would you like me to proceed with that? Also, if you need more details or clarification, let me know!

Related Questions:

1. What is the effect of the $+30^\circ$ phase shift in $Y = 2 \sin(2x + 30^\circ)$?
2. How would the graph change if the amplitude were 3 instead of 2?
3. What would happen if the function was $Y = 2 \cos(2x + 30^\circ)$ instead of sine?
4. How do changes in the frequency term (the coefficient of $x$) affect the graph?
5. Can you explain the general steps to sketch the graph of a transformed trigonometric function?

Tip:

Always check the period of the sine function when plotting, as it helps to predict the repeating pattern in the graph.