level 4: open ended questions 3. sketch the graph of a single function with a domain (-∞,∞) that has all of…

level 4: open ended questions 3. sketch the graph of a single function with a domain (-∞,∞) that has all of the following characteristics: a. concave down and increasing on the interval (-∞,-4) b. concave down and decreasing on the interval (-4,2) c. concave up and decreasing on the interval (2,7) d. concave up and increasing on the interval (7,∞)

level 4: open ended questions 3. sketch the graph of a single function with a domain (-∞,∞) that has all of the following characteristics: a. concave down and increasing on the interval (-∞,-4) b. concave down and decreasing on the interval (-4,2) c. concave up and decreasing on the interval (2,7) d. concave up and increasing on the interval (7,∞)

Answer

Explanation:

Step1: Recall function - behavior rules

Increasing function has positive slope ($f'(x)>0$), decreasing has negative slope ($f'(x)<0$). Concave - down means $f''(x)<0$ and concave - up means $f''(x)>0$.

Step2: Sketch for $(-\infty,-4)$

Start from the left. Draw a curve that is rising (increasing) and bending downwards (concave - down). For example, a part of a downward - opening parabola with a positive slope in this interval.

Step3: Sketch for $(-4,2)$

The curve should start decreasing from the end - point of the previous interval and still be concave - down. So, it continues to bend downwards while sloping downwards.

Step4: Sketch for $(2,7)$

The curve is now concave - up (bending upwards) and decreasing. It should change its concavity at $x = 2$ smoothly.

Step5: Sketch for $(7,\infty)$

The curve should be increasing and concave - up. It changes from decreasing to increasing at $x = 7$ and continues to bend upwards.

Answer:

A hand - drawn graph on the provided grid following the above - described steps. The graph should have smooth transitions between the different intervals in terms of slope and concavity.