1.Graph the function f(x) = (x + 3)3 by hand and describe the end behavior.
To graph the function f(x) = (x + 3)³ by hand and describe the end behavior, follow these steps:
1. Start by finding the x-intercept(s), if any. To do this, set f(x) = 0 and solve for x. In this equation, (x + 3)³ = 0, so x + 3 = 0, which gives x = -3. Therefore, the x-intercept is -3.
2. Next, find the y-intercept by plugging in x = 0 into the equation. When x = 0, f(x) = (0 + 3)³ = 27. Therefore, the y-intercept is (0, 27).
3. Make a table of x and f(x) values. Choose a few x-values on both sides of the x-intercept, such as -4, -2, -1, 1, and 2. Calculate the corresponding f(x) values by substituting these x-values into the equation. For example, when x = -4, f(x) = (-4 + 3)³ = (-1)³ = -1. Repeat this process for the other x-values and complete the table.
x | f(x)
-----|-----
-4 | -1
-2 | 1
-1 | 0
1 | 8
2 | 27
4. Plot the points from the table on a graph. In this case, you will have points (-4, -1), (-2, 1), (-1, 0), (1, 8), and (2, 27).
5. Connect the plotted points smoothly to sketch the graph of f(x) = (x + 3)³. The graph will pass through the points and continue beyond them.
Now, let's discuss the end behavior of the graph. As x approaches negative infinity (i.e., left-side of the graph), the function f(x) will also approach negative infinity. As x approaches positive infinity (i.e., right-side of the graph), f(x) will approach positive infinity. In other words, the ends of the graph will head downwards towards negative infinity on the left and upwards towards positive infinity on the right.