Assess Yourself

$(0,\ 0)$
$(4.189,\ 2.598)$
$(1.571,\ 0)$
$(2.094, - 2.598)$

The point identified is the $y$ -intercept. Find the lowest point of the graph on the interval.

The point identified is the maximum. Find the lowest point of the graph on the interval.

The point identified is the $x$ -intercept. Find the lowest point of the graph on the interval.

The problem was graphed correctly, and the minimum point was determined for the lowest value of the interval.

0.75 and 1.75 seconds
0.75 seconds
1 and 2 seconds
1 second

The function was graphed correctly, and the maximum points were found on the interval.

Only the first occurrence was used. Find all occurrences of the maximum on the interval.

The function was graphed incorrectly, but the maximum points of the graph were found.

The function was graphed incorrectly, and only one maximum point was found.

1 month
6 months
11 months
12 months

Review the graph and find the minimum value.

The maximum number of hours was identified as the minimum. Review the graph and find the minimum value.

The function was graphed correctly, and the minimum number of hours identified.

Review the graph and find the minimum value.

0 times for the minimum and 2 times for the maximum
2 times for the minimum and 2 times for the maximum
1 time for the minimum and 2 times for the maximum
3 times for the minimum and 2 times for the maximum

The function was graphed incorrectly. Re-graph the function.

All minimums on the graph need to be determined on the interval. Check the graph for all minimum values.

The function is graphed correctly, and there are 3 minimum values and 2 maximum values on the interval.

2.5 seconds
5.5 seconds
1, 4, 5, and 7 seconds.
1, 4, and 7 seconds

The minimum time was used. Find the time with a height at 5 inches.

The maximum time was used. Find the time with a height at 5 inches.

The height was included on the time. Find only the time with a height of 5 inches.

The problem was graphed correctly, and the height was determined at 5 inches.

1 seconds
6.3333 seconds
0 seconds
3.667 seconds

The point identified is the minimum value. Find the time with a height at 4 inches.

This point is outside the interval. Find the time with a height at 4 inches.

The point identified is the $y$ -intercept. Find the time with a height at 4 inches.

The problem was graphed correctly, and the correct times for the point of intersection were determined.

2.836 and 4.361 seconds
2.836 seconds
3.493 and 5.018 seconds
3.493 seconds

The problem was graphed correctly, and the correct values were determined.

Only one value was determined. Find all occurrences when the height is 2 feet below the water.

The function is graphed incorrectly. Re-graph the function and find all occurrences.

How well do you understand the concepts and skills introduced in this lesson?

Let's Practice

1. A spring that bobs up and down in the water can be modeled by the function $y = - 2(\sin x - \sin x\cos x)$ . Within the interval $0 < x < 6$ , when does the function reach a minimum value, and what is that value?

2. A mass suspended on a spring is pulled down 3 feet and represented by the function $y = 3\cos\left( 2\pi\left( x - 0.75 \right) \right)$ , with time in $x$ seconds. When does the maximum height occur on the interval $0 < x \leq 2$ ?

3. The number of hours of sunlight can be represented by the function $y = 2\cos{\left( \frac{\pi}{6}\left( x - 5 \right) \right) +}12$ , with time in $x$ months. When does the minimum number of hours of sunlight occur on the interval $0 < x < 12$ ?

4. The depth of water at a dock varies by the function $4\sin{\left( \frac{\pi}{6}\left( x - 3 \right) \right) +}8$ , with time in $x$ hours. How many times does the water have a maximum and minimum height on the interval $0 \leq x \leq 24$ ?

5. The height of a spring can be modeled by the function $y = 4\sin\left( \frac{\pi}{3}\left( x + 2 \right) + 5 \right)$ after $x$ seconds. Within the interval $0 < x < 8$ , when does the function have a height of 5 inches?

6. A marble in a maze can be modeled by the function $y = - 2\cos\left( \frac{\pi}{4}\left( x - 1 \right) + 3 \right)$ after $x$ seconds. Within the interval $0 < x < 6$ , when does the marble reach a height of 4 inches?

7. A toy that bobs below the water can be modeled by the function $y = 3(\cos x - \cos x\sin x)$ after $x$ seconds. Within the interval $0 < x < 7$ , when is the function 2 feet below the water?

Summary

How well do you understand the concepts and skills introduced in this lesson?

Let's Practice

1. A spring that bobs up and down in the water can be modeled by the function y=−2(sinx−sinxcosx) y = - 2(\sin x - \sin x\cos x) . Within the interval 0<x<6 0 < x < 6 , when does the function reach a minimum value, and what is that value?

2. A mass suspended on a spring is pulled down 3 feet and represented by the function y=3cos(2π(x−0.75)) y = 3\cos\left( 2\pi\left( x - 0.75 \right) \right) , with time in x x seconds. When does the maximum height occur on the interval 0<x≤2 0 < x \leq 2 ?

3. The number of hours of sunlight can be represented by the function y=2cos(π6(x−5))+12 y = 2\cos{\left( \frac{\pi}{6}\left( x - 5 \right) \right) +}12 , with time in x x months. When does the minimum number of hours of sunlight occur on the interval 0<x<12 0 < x < 12 ?

4. The depth of water at a dock varies by the function 4sin(π6(x−3))+8 4\sin{\left( \frac{\pi}{6}\left( x - 3 \right) \right) +}8 , with time in x x hours. How many times does the water have a maximum and minimum height on the interval 0≤x≤24 0 \leq x \leq 24 ?

5. The height of a spring can be modeled by the function y=4sin(π3(x+2)+5) y = 4\sin\left( \frac{\pi}{3}\left( x + 2 \right) + 5 \right) after x x seconds. Within the interval 0<x<8 0 < x < 8 , when does the function have a height of 5 inches?

6. A marble in a maze can be modeled by the function y=−2cos(π4(x−1)+3) y = - 2\cos\left( \frac{\pi}{4}\left( x - 1 \right) + 3 \right) after x x seconds. Within the interval 0<x<6 0 < x < 6 , when does the marble reach a height of 4 inches?

7. A toy that bobs below the water can be modeled by the function y=3(cosx−cosxsinx) y = 3(\cos x - \cos x\sin x) after x x seconds. Within the interval 0<x<7 0 < x < 7 , when is the function 2 feet below the water?