Question

Part AFor which time periods was the person‘s heart rate increasingPart BFor which time periods was the person‘s heart rate decreasingPart CHow many turning points( from increasing to decreasing or from decreasing to decreasing) occurred for the persons heart rate during the 15 minute film?Part Dsays suppose that a polynomial function is used to model the data displayed by the graph. What is the degree of the polynomial function of best fit( use the number of turning points to determine that degree)Part E For the model in part D should leading coefficient of the polynomial function be positive or negative why? Part FUse the graph to estimate the persons maximum heart rate during the 15 minute film (BPM)“ how many minutes did the maximum heart rate occur in part F” Part GUse the graph to estimate the persons minimum heart rate during the 15 minute filmAfter how many minutes did the minimum heart rate occur in part G?

Answer 1

Part A

The heart rate is increasing when the slope of the curve is positive. From the graph, we see that slope is positive in the intervals:

• 1 < x < 5

,

• 9 < x < 11

Part B

The heart rate is decreasing when the slope of the curve is negative. From the graph, we see that slope is negative in the intervals:

• 5 < x < 9

,

• 11 < x < 15

Part C

We see that the function has turning points at:

• x = 5 (from increasing to decreasing),

,

• x = 9 (from deceasing to increasing),

,

• x = 11 (from increasing to decreasing),

,

• x = 14 (from decreasing to decreasing).

So we conclude that the curve has 4 turning points.

Part D

We know that a polynomial of degree n will have at most n – 1 turning points.

In this case, we have 4 turning points, so the polynomial must be of degree 5.

Part E

The sign of the leading coefficient is determined by:

• the left and right behaviour of the graph,

,

• the degree of the polynomial.

From theory, we know that:

• If the degree of a polynomial f(x) is even and the leading coefficient is positive, then f(x) → ∞ as x → ±∞.

,

• If f(x) is an even degree polynomial with a negative leading coefficient, then f(x) → -∞ as x →±∞.

,

• If f(x) is an odd degree polynomial with a positive leading coefficient, then f(x) →-∞ as x →-∞ and f(x) →∞ as x → ∞.

,

• If f(x) is an odd degree polynomial with a negative leading coefficient, then f(x) → ∞ as x → -∞ and f(x) →-∞ as x →∞.

We see that:

• the graph falls to the left,

,

• the graph falls to the right,

,

• the degree of the polynomial is equal to 5 and it is odd.

We conclude that the leading coefficient must be positive.

Part F

From the graph, we identify the maximum heart rate as 72.

We see that it occurs at the 5 minutes.

Part G

From the graph, we identify the maximum heart rate as 54.

We see that it occurs at the 15 minutes.

Answers

Part A

D. from 1 through 5 minutes and from 9 through 11 minutes.

Part B

C. from 5 through 9 minutes and from 11 through 15 minutes.

Part C

Turning points: 4

Part D

Degree of the polynomial: 5

Part E

D. Positive because the graph falls to the left and falls to the right.

Part F

The maximum heart rate is 72 and occurs at 5 minutes.

Part G

The minimum heart rate is 54 and occurs at 15 minutes.