Question

Below you are given some key features of a function. Based on these given features, create a sketch of a graph of what the function could possibly look like.

The function has three intervals on which it is increasing
The function has two intervals on which it is decreasing
The domain of the function is [−7, 5]
The function has a minimum value of -3.
The function has a maximum value of 5.

Answer 1

The graph of the function should reflect three increasing intervals and two decreasing intervals, with the domain set from −7 to 5, and the points of minimum and maximum values at −3 and 5, respectively. Starting with a low point, the function should increase, peak, decrease to the minimum, rise to the maximum, and finally follow a decrease before a last increase as it approaches the end of the domain.

Step-by-step explanation:

To create a sketch of a graph for a function with the provided characteristics, we'll use interval notation and incorporate the given information about increasing and decreasing intervals, the domain, and the minimum and maximum values.

Firstly, establish the domain of the function on the x-axis from −7 to 5, inclusive. Next, identify key points where the function changes from increasing to decreasing or vice versa. These points will help in constructing the intervals.

Since the function has three increasing intervals, two decreasing intervals, a minimum value of −3, and a maximum value of 5, let's start the graph at −7 with a low point (as it cannot be the minimum just yet). The function should first increase, then peak (below the maximum), and decrease to reach the minimum value of −3. Following this, the function should increase again, reaching the maximum value of 5, and then decrease slightly before a final increase towards 5 on the x-axis.