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Algebra 2: Roller Coaster Portfolio

1) The domain is [0,300]. The range is [0,250]

2) The slope for the initial climb is 3.57. The equation of the line is  

3) The rate of change of the second hill is -1.4

4) The rate of change of the third hill is -2.1

5) The blue hill is steeper

6) It is not a function

Step-by-step explanation:

1) The domain is [0,300]. The range is [0,250]

2) The slope for the initial climb is 3.57. The equation of the line is  

3) The rate of change of the second hill is -1.4

4) The rate of change of the third hill is -2.1

5) The blue hill is steeper

6) It is not a function

Step-by-step explanation:

1)

The domain of a function is the set of values of x (the input) for which the function itself is defined.

Instead, the range of a function is the set of values of y (the output) that the function can take.

To find the domain, we have to look at the x-axis and see for which values of x the function is defined. By looking at the graph, we see that the function is defined between

x = 0 and x = 300

So, the domain is [0,300].

To find the range, we have to look at the y-axis and see for which values of y the function has an output. By looking at the graph, we see that the function has values of y between

y = 0 and x = 250

So, the range is [0,250].

2)

The slope of a function in a certain range is given by

where

is the change in the y-coordinate

is the change in the x-coordinate

For the initial climb (pink line), we have:

Therefore, the slope in this part is

We can now write the equation of the line in the form

y = mx + b

where b is the y-intercept: since it is zero, the line has simply the form

3)

Again, to calculate the slope of the hill, we use:

where

is the change in the y-coordinate

is the change in the x-coordinate

For the green hill, we have:

So the rate of change is

4)

As before, the rate of change of the hill is

For the blue hill, we have:

So the rate of change is

5)

To know which hill is steeper, we need to compare the magnitude of their rate of change.

In fact, both hills have rate of change negative - because we are going down along the slope. Therefore, we have to consider the magnitude of their slope.

For the green hill:

For the blue hill:

We see that the blue hill has a greater slope (in magnitude): therefore, the blue hill is steeper.

6)

A function is defined as a mapping (operation between two sets of variables) in which to one value of x (the input) corresponds one and only one value of y (the output).

This means that a function cannot have multiple values of y for the same x (the input).

By looking at the graph, we see that this function does not respect this criterium: in fact, we see that certain values of x give multiple values of the output, y (for instance, at x=300 the function has two values). So, this is not a function.