Two Graphs for Each Relationship

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Investigate the behavior of a Line as the Slope Changes.

  • To explore the graph, check the box in the lower left corner of the applet. observe that the two points of the graph are the first and last pair of coordinates in the table. ​
  • Now, grab upper point on the graph and move it around. observe that the coordinates in the table change to match the new points. ​
  • Check the equation box in the applet and continue to move the line. The equation of the line is constantly changing with the line.
  • Answer the questions that follow.

1. What stays the same and what changes in the table?

2. What stays the same and what changes in the equation?

3. What stays the same and what changes in the graph?

Choose one row from the table above and write it here.

1. To what does this row correspond to on the graph?

Do not move the point.

Choose three rows from the table, other than the origin. Record x and y, and compute y/x.

1. What do you notice? What does this have to do with the equation of the line?

2. Do not move the point. Check the box to view the coordinates . What are the coordinates of this point? What does this correspond to in the table? What does this correspond to in the equation?

Drag the point to a different location.

Record the equation of the line, the coordinates of three points, and the value of y/x.

1. Based on your observations, summarize any connections you see between the table, characteristics of the graph, and the equation.𝜋

2. The graph of an equation of the form y=kx , where k is a positive number, is a line through (0,0) and the point 1 = k. Name at least one line through (0,0) that cannot be represented by an equation like this.

3. If you could draw the graphs of all of the equations of this form in the same coordinate plane, what would it look like?

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Comparing Two Proportional Relationships on a Graph.

Andre and Jada were in a hot dog eating contest. Andre ate 10 hot dogs in 3 minutes. Jada ate 12 hot dogs in 5 minutes.

The points shown on the first set of axes display information about Andre’s and Jada’s consumption.

(1). Which point indicates Andre’s consumption?

(2). Which indicates Jada’s consumption? Label them.

(3). Draw two lines: one through the origin and Andre’s point, and one through the origin and Jada’s point.

(4). Write an equation for Andre’s line. Use  to represent time in minutes, and h  to represent number of hot dogs. to represent time in minutes, and  to represent number of hot dogs.

(5). For each equation, what does the constant of proportionality tell you?

The points shown on the second set of axes display information about Andre’s and Jada’s consumption.

(6). Which point indicates Andre’s consumption?

(7). Which indicates Jada’s consumption? Label them.

(8). Draw lines from the origin through each of the two points.

(9). Write an equation for each line.

(10). What does the constant of proportionality tell you in each case?

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Writing an Equation for a Proportional Relationship.

(A). A trail mix recipe asks for 4 cups of raisins for every 6 cups of peanuts. There is proportional relationship between the amount of raisins, r (cups), and the amount of peanuts, p (cups), in this recipe.

(1). Write the equation for the relationship that has constant of proportionality greater than 1. 

(2). Write the equation for the relationship that has constant of proportionality less than 1.

(3). Graph each equation that you wrote on the graph below.

(4). Give the graph a title. Then, label the axes with the quantities in your situation.

(B). At the supermarket you can fill your own honey bear container. A customer buys 12 oz of honey for $5.40.

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Give the graph a title. Then, label the axes with the quantities in your situation.

Choose a point on the graph. What do the coordinates represent in your situation?

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(1). The graph of an equation of the form y = kx, where  k  is a positive number, is a line through  (0, 0)  and the point  (1, k). Name at least one line through  (0, 0) that cannot be represented by an equation like this.

(2). If you could draw the graphs of all of the equations of this form in the same coordinate plane, what would it look like?

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