Question

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

Answer 1

Part 1) The end points of the initial incline are the points (0,0) and (4,6)

Par 2) The slope of the initial incline (red) is m=1.5

Part 3) The linear equation of the initial (red ) incline would be y=1.5x

Part 4) The end points of the first decline are the points (4,6) and (6,2)

Part 5) The slope of the first decline (blue) is m=-2

Part 6) The linear equation of the first decline (blue) would be y=-2x+14

Part 7) Two end points of the second incline (green) are the points (6,2) and (10,5)

Part 8) The slope of the second incline (green) is m=0.75

Part 9) The slope of the first decline is the steepest of the three

Part 10) see the explanation

Part 11) The track is not a function

Part 12) see the explanation

Part 13) The domain is all real numbers greater than or equal to 0 and less than or equal to 12

Part 14) The range is all real numbers greater than or equal to 0 and less than or equal to 8

Explanation:

Part 1) Two end points of the initial incline(red) are ( , ) and ( , )

Observing the graph

The end points of the initial incline are the points (0,0) and (4,6)

Part 2) The slope of the initial incline (red) is

we know that

The formula to calculate the slope between two points is equal to

`m=(y2-y1)/(x2-x1)`

we have

(0,0) and (4,6)

substitute the values in the formula

`m=(6-0)/(4-0)=1.5`

Part 3) The linear equation of the initial (red ) incline would be y = __x +__

we know that

The initial red incline represent a proportional relationship, because is a line that passes through the origin

therefore

The linear equation is

`y=mx`

we have

`m=1.5`

substitute

`y=1.5x`

Part 4) Two end points of the first decline (blue) are ( , ) and ( , )

Observing the graph

The end points of the first decline are the points (4,6) and (6,2)

Part 5) The slope of the first decline (blue) is

we know that

The formula to calculate the slope between two points is equal to

`m=(y2-y1)/(x2-x1)`

we have

(4,6) and (6,2)

substitute the values in the formula

`m=(2-6)/(6-4)=-2`

Part 6) The linear equation of the first decline (blue) would be y= __ x + __

The equation of the line in slope intercept form is

`y=mx+b`

we have

`m=-2`

so

`y=-2x+b`

with the point (4,6) find the value of b

substitute the value of x and the value of y in the linear equation

`6=-2(4)+b`

`6=-8+b`

`<strong>b=</strong>6+8=14`

therefore

The linear equation is

`y=-2x+14`

Part 7) Two end points of the second incline (green) are ( , ) and ( , )

Observing the graph

The end points of the second incline are the points (6,2) and (10,5)

Part 8) The slope of the second incline(green) is

we know that

The formula to calculate the slope between two points is equal to

`m=(y2-y1)/(x2-x1)`

we have

(6,2) and (10,5)

substitute the values in the formula

`m=(5-2)/(10-6)=0.75`

Part 9) Which incline is steepest of the three?

we have

The slope of the initial incline (red) is m=1.5

The slope of the first decline (blue) is m=-2

The slope of the second incline (green) is m=0.75

we know that

The steepest slope is that of the line that is closest to being vertical. This could be a line that has positive slope OR negative slope...the steepest one should be determined by examining the ABSOLUTE VALUES of these three slopes.

This means that the slope of the first decline is the steepest of the three

Part 10) How mathematically can you tell it is the steepest?

we know that

The slope with highest absolute value is the steepest. Positive slope means the function is ascending left to right, negative slope means it is descending left to right.

In this problem

The slope of the first decline is the highest absolute value (absolute value is 2)

Part 11) For the last (orange) section of the track is the track a function

we know that

The vertical line test is a method that is used to determine whether a given relation is a function or not

In this problem the track section does not pass the vertical line test.

therefore

The track is not a function

Part 12) How do you know it is or isn’t a function for this region?

we know that

A function can only have one output, y, for each unique input, x.

In this problem, applying the vertical line test, the vertical line you draw intersects the graph more than once for any value of x, so the graph is not the graph of a function.

Part 13) What is the domain for the roller coaster in total

we know that

The domain is the interval -----> [0,12]

`0 \leq x\leq 12`

All real numbers greater than or equal to 0 and less than or equal to 12

Part 14) What is the range for the roller coaster in total

we know that

The range is the interval ----> [0,8]

`0 \leq y\leq 8`

All real numbers greater than or equal to 0 and less than or equal to 8