The graph shows the depth, y, in meters, of a shark from the surface of an ocean for a certain amount of time, x, in minutes: a graph titled distance vs. time is shown. the x axis is labeled time in minutes and shows numbers 0, 1, 2, 3, 4, 5. the y axis is labeled distance from ocean surface in meters. a straight line joins the points c at ordered pair 0,63, b at ordered pair 1, 105, a at ordered pair 2, 147, and the ordered pair 3,189. part a: describe how you can use similar triangles to explain why the slope of the graph between points a and b is the same as the slope of the graph between points a and c. (4 points) part b: what are the initial value and slope of the graph, and what do they represent? (6 points)

We are given point A as (2,147)

B(1,105)

C(0,63)

In table form it could also be written as:

     x-value             y-value

       0                        63

       1                        105

       2                       147

       3                       189

A)

slope between two points A(a,b) and B(c,d) is calculated as:

slope between A(2,147) and B (1,105) is:

similarly slope of graph between A(2,147) and C(0,63) is given as:

as

Hence, the slope of AB and Ac are equal.

B)

The initial value of the data represents the y-intercept i.e. the point where x=0.

the y-intercept is 63.

and the slope of the graph is 42.

it is a linear relationship as each time the distance is increasing by 42 meters.

i.e. there is a increase in distance by 42 meters every minute.

We are given the following data in tabulated form:

time (minute), x                               distance from surface (m), y

0                                                              63          

1                                                              105

2                                                              147

3                                                              189

We can see that for every 1 minute increase in time, there is a constant 42 m increase in distance from the surface. Therefore the equation relating the two variables is linear in the form:

y = m x + b

where m is the slope and b is the y-intercept

Part A.

We can use the similar triangles to explain why the slope at points A and B is similar as the slope at points A and C by calculating for the slope in each case. The formula for slope m is:

m = (y2 – y1) / (x2 – x1)

Slope between A (2, 147) and B (1, 105):

mAB = (147 - 105) / (2 – 1) = 42

Slope between A (2, 147) and C (0, 63):

mAC = (147 - 63) / (2 – 0) = 42

Hence mAB = mAC

Part B.

The intial value is the value of y intercept (b). The value of b is the value of y when x = 0, hence:

initial value = b = 63

The slope of the graph is m = 42 and it tells us that there is 42 m change in distance per 1 minute of time.