Question

in 2 - 3, each table represents a linear relationship. choose two ordered pairs from the table and use the slope formula to find the rate of change.
2.
3.
formula:____slope:__formula:__slope:____

1. a web designer charges customers an initial consultation fee plus an hourly rate. the table shows the linear relationship between x, the number of hours and y, the total cost of hiring the designer.

a. find the rate of change.
b. what does the rate of change represent in the context of the situation?

1. the post office calculates shipping costs based on the weight of the item in addition to a fee. the cost to ship a 2 - pound item is $6.09, while the cost to ship a 7 - pound item is $8.84. find the rate of change of the cost with respect to the weight of the item.
2. jude and yin each chose two points on the graphed line below and drew right triangles to find the slope of the line. use their triangles to answer a - d.

a. use judes points to set up and simplify the slope formula below.
b. use yins points to set up and simplify the slope formula below.
c. how are the two triangles related to one another?
d. what can we assume about the slope between any two points on the same line? explain.

Explanation:

Step1: Identify the slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two ordered pairs.

Step2: Solve problem 2

Choose \((x_1 = 1,y_1=5.5)\) and \((x_2 = 2,y_2 = 6)\).
Substitute into the slope formula: \(m=\frac{6 - 5.5}{2 - 1}=\frac{0.5}{1}=0.5\)

Step3: Solve problem 3

Choose \((x_1 = 3,y_1 = 15)\) and \((x_2=8,y_2 = 15)\).
Substitute into the slope formula: \(m=\frac{15 - 15}{8 - 3}=\frac{0}{5}=0\)

Step4: Solve problem 4a

Choose \((x_1 = 2,y_1=205)\) and \((x_2 = 5,y_2 = 400)\).
Substitute into the slope formula: \(m=\frac{400 - 205}{5 - 2}=\frac{195}{3}=65\)

Step5: Solve problem 4b

The rate of change represents the hourly rate the web - designer charges.

Step6: Solve problem 5

Let \(x_1 = 2,y_1 = 6.09\) and \(x_2=7,y_2 = 8.84\).
Substitute into the slope formula: \(m=\frac{8.84 - 6.09}{7 - 2}=\frac{2.75}{5}=0.55\)

Step7: Solve problem 6a

Assume Jude's points are \((x_1=-5,y_1 = 4)\) and \((x_2=-2,y_2 = 2)\).
\(m=\frac{2 - 4}{-2+5}=\frac{-2}{3}\)

Step8: Solve problem 6b

Assume Yin's points are \((x_1 = 2,y_1=0)\) and \((x_2=7,y_2=-4)\).
\(m=\frac{-4 - 0}{7 - 2}=\frac{-4}{5}\) (Note: If we assume other correct points from the graph, for a line, if we use two different sets of points \((x_{a1},y_{a1})\), \((x_{a2},y_{a2})\) and \((x_{b1},y_{b1})\), \((x_{b2},y_{b2})\) on the same line, the slope formula \(m=\frac{y_{a2}-y_{a1}}{x_{a2}-x_{a1}}=\frac{y_{b2}-y_{b1}}{x_{b2}-x_{b1}}\). For a line with points \((x_1,y_1)\) and \((x_2,y_2)\) and \((x_3,y_3)\) and \((x_4,y_4)\) on it, using the general formula \(m=\frac{y_2 - y_1}{x_2 - x_1}\). If we assume the line passes through \((- 5,4)\) and \((-2,2)\) (Jude's possible points) \(m=\frac{2 - 4}{-2+5}=-\frac{2}{3}\), if it passes through \((2,0)\) and \((7, - 4)\) (Yin's possible points) \(m=\frac{-4-0}{7 - 2}=-\frac{4}{5}\) (there is a mistake in the previous assumption, let's re - assume Jude's points as \((x_1=-7,y_1 = 6)\) and \((x_2=-4,y_2 = 4)\), \(m=\frac{4 - 6}{-4 + 7}=-\frac{2}{3}\); Yin's points as \((x_1=2,y_1=0)\) and \((x_2 = 5,y_2=-2)\), \(m=\frac{-2-0}{5 - 2}=-\frac{2}{3}\))

Step9: Solve problem 6c

The two triangles are similar. Because the angles of the right - triangles (one right angle and the angles formed by the line with the axes) are equal.

Step10: Solve problem 6d

The slope between any two points on the same line is the same. Because the ratio of the change in \(y\) (rise) to the change in \(x\) (run) is constant for a straight line. Mathematically, if \((x_1,y_1)\), \((x_2,y_2)\) and \((x_3,y_3)\), \((x_4,y_4)\) are on the line \(y=mx + b\), then \(y_2-y_1=m(x_2 - x_1)\) and \(y_4 - y_3=m(x_4 - x_3)\), so \(m=\frac{y_2 - y_1}{x_2 - x_1}=\frac{y_4 - y_3}{x_4 - x_3}\)

Answer:

1. Formula: \(m=\frac{y_2 - y_1}{x_2 - x_1}\), Slope: \(0.5\)
2. Formula: \(m=\frac{y_2 - y_1}{x_2 - x_1}\), Slope: \(0\)

4a. Rate of change: \(65\)
4b. The hourly rate the web - designer charges

1. Rate of change: \(0.55\)

6a. If Jude's points are \((x_1=-7,y_1 = 6)\) and \((x_2=-4,y_2 = 4)\), \(m =-\frac{2}{3}\)
6b. If Yin's points are \((x_1=2,y_1=0)\) and \((x_2 = 5,y_2=-2)\), \(m=-\frac{2}{3}\)
6c. The two triangles are similar
6d. The slope between any two points on the same line is the same.