The image above is of a virtual roller coaster showing the initial path of the track. answer the following questions about its path as it goes from left to right.
(2pts) two end points of the initial incline(red) are ( , ) and ( , ) (1pt) the slope of the initial incline (red) is
(2pts) the linear equation of the initial (red ) incline would be y = x +
(2pts) two end points of the first decline(blue) are ( , ) and ( , ) (1pt) the slope of the first decline (blue) is
(2pts) the linear equation of the first decline (blue) would be y= x +
(1 pt) two end points of the second incline (green) are ( , ) and ( , ) (1 pt)the slope of the second incline(green) is
(1 pt) which incline is steepest of the three?
(1 pt) how mathematically can you tell it is the steepest?
(1 pt) for the last (orange) section of the track is the track a function
(1 pt) how do you know it is or isn’t a function for this region?
(2 pts)what is the domain for the roller coaster in total [ , ]
(2 pts) what is the range for the roller coaster in total [ , ]

See explanation

Step-by-step explanation:

The slope of the line passing through the points and is

The equation of the line is

where b is y-intercept.

Red incline:

endpoints are (0,0) and (4,6)

The slope of the line is

The equation of the line is

This line has y-intercept at (0,0), so b=0 and equation is

Blue decline:

endpoints are (4,6) and (6,2)

The slope of the line is

The equation of the line is

This line passes through the point (4,6), so its coordinates satisfy the equation:

and the equation is

Green incline:

endpoints are (6,2) and (10,5)

The slope of the line is

The equation of the line is

This line passes through the point (6,2), so its coordinates satisfy the equation:

and the equation is

The steepest slope is that of the line that is closest to being vertical. So, the steepest slope has the blue decline.

Mathematically, compare absolute values of slopes:

so the steepest is blue decline.

The orange section is not a function, because, for example, for one input value x=11 there are two different outputs.

The domain of the roller coaster is

The range of the roller coaster is

Answer 1: P1: (X1, Y1) =(0,0) and P2: (X2, Y2)=(4,6), slope=3/2

In this part we can take the point of the begining and the end of the red line. Taking into account each point has an X component and a Y component:

P1: (X1, Y1) =(0,0) and P2: (X2, Y2)=(4,6)

With this two points we can use the equation of the slope :

This is the slope of the red line

Now we have to write the equation of the line in the form

Where is the intersection point with the y-axis and can be calculated using one of the points (we will use (4,6)) and the slope ():

Hence: This is the equation of the red line

Answer 2:  P1: (X1, Y1) =(4,6) and P2: (X2, Y2)=(6,2), slope=-2

Here we will approach the problem similarly as we did in part 1:

Two end points of blue line: P1: (X1, Y1) =(4,6) and P2: (X2, Y2)=(6,2)

Calculating the slope:

This is the slope of the blue line (note the negative sign indicates it is a decline line)

Calculating the intersection point (we will use (4,6)):

Hence, the linear equation of the decline blue line is:

Answer 3: P1: (X1, Y1) =(6,2) and P2: (X2, Y2)=(10,5), slope=3/4

For the green line we have:

P1: (X1, Y1) =(6,2) and P2: (X2, Y2)=(10,5)

For the slope:

This is the slope of the green line

Answer 4: Blue line is the steepest of the three.

Now we have to compare the slope of each of the three lines:

Red line:

Blue line:

Green line:

How mathematically can we tell it is the steepest?

As we can see blue line has the greatest slope (remember the negative sign only indicates the direction of the slope). This means the Blue line is the steepest of the three.

Answer 5: The orange track is not a function

How do we know it?

Graphically, a function is identified by drawing a vertical line on it, if this line cuts to the plotted line at more than one point, this is not a function.

In this sense, if we draw a vertical line in the last region of the graph, this line will cut it in more than one point. Therefore it is not a function.

Answer 6: or [0,12]

A function is represented graphically by a correspondence between the elements of the domain set and those of the range set.

What does this mean?

The Domain of a function is the set of all the real values the variable X can take and that can be seen in the graph.  

For example, in this case the roller coaster is defined for all values of X that are between 0 and 12 (both included)

Answer 7: or [0,8]

The Range of a function refers to the values taken by the "Y" function (dependent variable), since its value depends on the value we give to "X".  

Graphically, the way to determine the Range is to graph the function and see the values that "Y" takes from the bottom up. That is, the values that have the variable “y” for certain values of x.

For example, in this graph of the roller coaster the range is defined for all the values of Y between 0 and 8 (both included).

answer is option D

Step-by-step explanation:

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