Question

2. Four different examples are given below.

Example 1	In the newspaper, Mr. Shields read that the population of Jefferson County was decreasing at a rate of 20% every year.
Example 2	Alec is paid $200 each month to mow the grass and weed the garden. In addition, he earns $6.50 per hour for the number of hours he works as a dishwasher at the local diner.
Example 3	The PTA made $2,400 in ticket sales for their annual fundraiser. Adult tickets, a, are $6.00 and children's tickets, c, are half price.
Example 4	In an arcade game, stars are based on the number of space aliens that are hit. For example, for the first alien, 1 star is given, for the second alien, 2 stars are given, for 3 aliens, 4 stars are given, for 4 aliens, 8 stars are given. Each star is multiplied by 100 to determine the players score.

Analyze each one, telling whether or not the example can be modeled by a linear or exponential function and tell any specific characteristics about each example (include common difference, common ratio, slope, etc.)

Answer 1

1. Yes, the first example can be modelled in Linear or Exponential function. Here, as the population of Jefferson County was decreasing at a rate of 20% every year, so the data points of this experiment will be scattered with a negative slope. This is because every year the population is decreasing, so if we perform a trend analysis and draw a trend line with Year on the X axis and the corresponding Population on the Y axis, it will represent a decreasing line with a negative slope.

2. Yes, the second example can also be modelled in Linear or Exponential function. Here there is a fixed component, i.e. the monthly salary and also there is a variable component, i.e. rate per hour.

So the equation can be:

Income = 200 + 6.5 * Total Hour he works as a Dishwasher at the Local Diner

Here, the slope is 6.5.

This is because, if we compare the above equation with the equation of the straight line in gradient form, i.e. y = m*x + c,

Then, c = 200 (intercept)

m = 6.5 (slope/gradient)

The above equation derived is also in linear form (straight line).

3. Yes, the third example can also be represented as a linear or exponential function.

Here the equation will be 6x + 3y = 2400 ------------------------------------ (1)

This is because, the price for adult is $6 and the price for children is halved, i.e. $3 and the total collection is $2400

"x" is the total number of tickets sold to adults and "y" is the total number of tickets sold to children.

"The above equation can also be written as 6a + 3c = 2400 (a - adult and c - children)"

Equation (1) becomes:

Y = -2x + 800 (dividing both sides by 3)

If we compare the above equation with the equation of the straight line in gradient form, i.e. y = m*x + c,

Slope = -2 (negative)

c = 800

The above equation represents a straight line (linear) with a negative slope.

4. Yes, the example four can also be represented in linear or exponential function.

Here,

The equation will be n₁*1*100 + n₂*2*100 + n₃*4*100 + n₄*8*100 = Total Score

n₁, n₂, n₃ and n₄ are the total number of times 1, 2, 4 and 8 stars are obtained.

100 is to be multiplied (as shown above) as every stars obtained needs to be multiplied by it as stated in the question.

The above equation derived is also in linear form.

End of the Solution...