Question

Design a Roller Coaster Portfolio

Answer 1

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 $y=3.57 x$

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

$m=\frac{\Delta y}{\Delta x}$

where

$\Delta y$ is the change in the y-coordinate

$\Delta x$ is the change in the x-coordinate

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

$\Delta y = 250 - 0 = 250\\\Delta x = 70 - 0 = 70$

Therefore, the slope in this part is

$m=\frac{250}{70}=3.57$

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

$y=mx \rightarrow y = 3.57 x$

3)

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

$m=\frac{\Delta y}{\Delta x}$

where

$\Delta y$ is the change in the y-coordinate

$\Delta x$ is the change in the x-coordinate

For the green hill, we have:

$\Delta y = 60-200 = -140$

$\Delta x = 300-200 = 100$

So the rate of change is

$m=\frac{-140}{100}=-1.4$

4)

As before, the rate of change of the hill is

$m=\frac{\Delta y}{\Delta x}$

For the blue hill, we have:

$\Delta y = 30-230 = -300$

$\Delta x = 200-60 = 140$

So the rate of change is

$m=\frac{-300}{140}=-2.1$

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:

$|m| = 1.4$

For the blue hill:

$|m|=2.1$

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.

Learn more about functions:

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#LearnwithBrainly

Answer 2

Further explanation

Set A to set B is said to be a function if each member of set A pairs is exactly one member of set B

So, one value of x is only assigned to one value of y

If y is a function of x, then y = f (x)

Rate of Change function shows the comparison between 2 objects that change

If we use the notation function, the rate of Change from a function from a to b can be formulated as:

$\large {\boxed {\boxed {\bold {ROC = \frac {f (a) -f (b)} {a-b}}}}$

ROC = rate of change

the domain and range of this roller coaster

The Domain is the X values, 0-360

written as (0,360)

The range is the Y values, 0-250

written as (0,250)

2. a. The slope of the initial climb

from points 0,0 and 70,250

slope = 250/70 = 3.57

b. equation from point 70,250

y-y1 = m (x-x1)

y-250 = 25/7 (x-70)

y = 25 / 7x-250 + 250

y = 25 / 7x

3. rate of change from (200, 200) to (300, 60) --- green hill

ROC = 60-200 / 300-200

ROC = -140/100

ROC = -1.4

4. rate of change from (200, 230) to (60, 30) - blue hill

ROC = 30-230 / 60-200

ROC = -200 / -140

ROC = 1.42

5. The greater the slope, the steeper the line

the blue hill is steeper than the green hill because the slope value of the blue hill is greater (steepness is not determined from the negative slope value)

6.the roller coaster is not a function, because there are 2 domain values ​​that are paired with the codomain for example points 200,200 and 200,230