Algebra 2 "Alg II Interpreting Functions" - Public Release - MSDE

Correct Answerf of n equals 5 plus 3 open parenthesis n minus 1 close parenthesis, for n greater than or equal to 1.

Explanation

To determine the explicit rule for the sequence defined by the recursive rule, we need to understand how the sequence grows and generalize the pattern. The initial term is 5, and each successive term is obtained by adding 3 to the previous term:

f of 1 equals 5 (This is given.)
f of 2 equals 3 plus f of 1 equals 3 plus 5 equals 15.
f of 3 equals 3 plus f of 2 equals 3 plus 8 equals 11.

So the sequence start: 5, 8, 11, 14, 17, . . . The terms form an arithmetic sequence, and the common difference between consecutive terms is 3, since: f of n minus f of n minus 1 equals 3. The general formula for an arithmetic sequence is: f of n equals f of 1 plus open parenthesis n minus 1 close parenthesis times d where: f of 1 is the first term of the sequence (which is 5), d is the common difference (which is 3), and n is the position of the term. Substitute f of 1 equals 5 and d equals 3 into the formula:f of n equals 5 plus open parenthesis n minus 1 close parenthesis times 3, equals 5 plus 3 open parenthesis n minus 1 close parenthesis.

Correct Answerf of n equals 5 open parenthesis 3 close parenthesis raised to the n minus 1 power.

Explanation

The recursive rule says the initial term is 5 and each term is three times the previous term. We can use the rule to calculate the first few terms of the sequence:

f of 1 equals 5 (This is given.)
f of 2 equals 3 times f of 1 equals 3 times 5 equals 15.
f of 3 equals 3 times f of 2 equals 3 times 15 equals 45.

So, the sequence starts: 5, 15, 45, 135, . . . . Looking at the sequence, we can see that each term is obtained by multiplying the previous term by 3. This indicates geometric growth. The general formula for a geometric sequence is: f of n equals f of 1 times r raised to the power of n minus 1 where: f of 1 is the first term (which is 5), r is the common ratio (which is 3, because each term is 3 times the previous one), and n is the position in the sequence.

We substitute f of 1 equals 5 and r equals 3 into the formula:5 times 3 raised to the power of n minus 1

Correct Answer

The sine function is responsible for the up-and-down variation in the number of cars parked in the garage. The sine function increases and decreases periodically, and we can determine whether the number of cars is increasing or decreasing by looking at the slope of the sine function over each interval. During the first interval (7:00 a.m. to 9:00 a.m.), the sine function is increasing, so the number of cars increases. During the second and third intervals (1:00 p.m. to 4:00 p.m. and 5:00 p.m. to 8:00 p.m.), the sine function is decreasing, so the number of cars decreases.

Understand the Function

The number of cars in the garage y, x hours after 6:00 a.m. is modeled by the function:y, equals 180 plus 130 sine, open parenthesis, the fraction with a numerator of x and a denominator of 12, end fraction, pi, close parenthesis

The function is a sinusoidal model, with the following key components:

180: The baseline number of cars, meaning that without the sinusoidal variation, there would be 180 cars.

130: The amplitude of the sine function, meaning the number of cars oscillates up and down by 130 cars from the baseline.

sine, open parenthesis, the fraction with a numerator of x and a denominator of 12, end fraction, pi, close parenthesis The oscillation (up and down) over time. The sine function repeats every 24 hours (since the fraction with a numerator of x and a denominator of 12, end fraction, pi implies a full sine cycle over 24 hours). The number of cars increases when this part is increasing and decreases when it is decreasing.

Analyze the Function for Each Interval

To determine whether the number of cars increases or decreases, we need to evaluate how the sine function behaves within each interval. Here, the sine function mathematics expression or equation follows the general pattern of the sine curve, which starts at 0, increases to 1, decreases back to 0, then to negative one and back to 0 over its full period, which is 2 pi.

7:00 a.m. to 9:00 a.m. x equals 1 to x equals 3

At x equals 1, fraction with numerator of x and denominator of 12, end fraction, pi, equals fraction with numerator of pi and denominator of 12, and sine open parenthesis fraction with numerator of pi and denominator of 12 is a small positive value. At x equals 3, fraction with numerator of x and denominator of 12, end fraction, pi equals fraction with numerator of pi and denominator of 4, and sine, open parenthesis fraction with numerator of pi and denominator of 4 equals fraction with numerator of the square root of 2 and denominator of 2, which is approximately equal to 0 point 7 0 7. Thus, the sine function is increasing from x equals 1 to x equals 3, so the number of cars increases.

1:00 p.m. to 4:00 p.m. x equals 7 to x equals 10

At x equals 7, fraction with numerator of x and denominator of 12, end fraction, pi, equals fraction with numerator of 7 pi and denominator of 12, and sine, open parenthesis fraction with numerator of 7 pi and denominator of 12, close parenthesis, is decreasing toward 0. At x equals 10, fraction with numerator of x and denominator of 12, end fraction, pi, equals fraction with numerator of 5 pi and denominator of 6, and sine, open parenthesis fraction with numerator of 5 pi and denominator of 6, close parenthesis is still positive but closer to 0. The sine function is decreasing during this time, so the number of cars decreases.

5:00 p.m. to 8:00 p.m. x equals 11 to x equals 14

At x equals 11, fraction with numerator of x and denominator of 12, end fraction, pi, equals fraction with numerator of 11 pi and denominator of 12, and sine, open parenthesis fraction with numerator of 11 pi and denominator of 12, close parenthesis is still positive but close to 0. At x equals 14 fraction with numerator of x and denominator of 12, end fraction, pi, equals fraction with numerator of 7 pi and denominator of 6, and sine, open parenthesis fraction with numerator of 7 pi and denominator of 6, close parenthesis is negative. The sine function is decreasing during this time, so the number of cars decreases.

The recursive rules for two sequences are shown in the table, where $n$ represents a positive integer.

For each of the two sequences, identify the equation that represents the explicit rule.

Drag and drop an equation into each box.

Recursive Rule	Explicit Rule
f of 1 equals 5, f of n equals 3 plus f of open parenthesis n minus 1 close parenthesis, for n greater than 1
f of 1 equals 5, f of n equals 3 times f of open parenthesis n minus 1, for n greater than 1

Function $f$ is defined as follows.

$f (x) = {(0.5)}^{2 x}$

Describe how the value of $f (x)$ is changing.

Select from the drop-down menus to correctly complete the statement.

For every increase of 1 in the value of the value of

f (x)

by .

The graph of the polynomial function $f$ is shown in the xy-coordinate plane.

Which of the equations shown could define $f ?$

i1

Correct i2

Thinks the function is the negative of the correct function i3

Thinks the function has zeros that are the negatives of the correct zeros i4

Thinks the function has zeros that are the negatives of the correct zeros and has the negative of the function

The approximate number of cars parked in a garage $x$ hours after $6 : 00$ a.m. can be modeled by the function $y = 180 + 130 \sin (\frac{x}{12} π),$ where $0 is less than or equal to x, which is less than or equal to 16.$

For each time interval, determine whether the number of cars parked in the garage increased or decreased, based on the model.

Select the boxes to identify each description of the function. Select only one box per row.