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Lesson 8Translating to y=mx+b

Let’s see what happens to the equations of translated lines.

Learning Targets:

  • I can explain where to find the slope and vertical intercept in both an equation and its graph.
  • I can write equations of lines using y=mx+b.

8.1 Lines that Are Translations

Lines F, G, H, I and J are on a plane and intersect at different points.

The diagram shows several lines. You can only see part of the lines, but they actually continue forever in both directions.

  1. Which lines are images of line f under a translation?
  2. For each line that is a translation of f , draw an arrow on the grid that shows the vertical translation distance.

8.2 Increased Savings

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  1. Diego earns $10 per hour babysitting. Assume that he has no money saved before he starts babysitting and plans to save all of his earnings. Graph how much money, y , he has after x hours of babysitting.

  2. Now imagine that Diego started with $30 saved before he starts babysitting. On the same set of axes, graph how much money, y , he would have after x hours of babysitting.

  3. Compare the second line with the first line. How much more money does Diego have after 1 hour of babysitting? 2 hours? 5 hours? x hours?

  4. Write an equation for each line.

8.3 Translating a Line

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  1. Experiment with moving point A .
    1. Place point A in three different locations above the x -axis. For each location, write the equation of the line and the coordinates of point A .
    2. Place point A in three different locations below the x -axis. For each location, write the equation of the line and the coordinates of point A .
    3. In the equations, what changes as you move the line? What stays the same?
    4. If the line passes through the origin, what equation is displayed? Why do you think this is the case?
  2. Your teacher will give you 12 cards. There are 4 pairs of lines, A–D, showing the graph, a , of a proportional relationship and the image, h , of a under a translation. Match each line h with an equation and either a table or description. For the line with no matching equation, write one on the blank card.

Are you ready for more?

A student says that the graph of the equation y = 3(x+8) is the same as the graph of y = 3x , only translated upwards by 8 units. Do you agree? Why or why not?

Lesson 8 Summary

During an early winter storm, the snow fell at a rate of \frac12 inches per hour. We can see the rate of change, \frac12 , in both the equation that represents this storm, y=\frac12x , and in the slope of the line representing this storm.

In addition to being a linear relationship between the time since the beginning of the storm and the depth of the snow, we can also call this as a proportional relationship since the depth of snow was 0 at the beginning of the storm.

A graph showing time since the beginning of a storm in hours and depth of snow in inches.There are points at (2,1) and (4,2).

During a mid-winter storm, the snow again fell at a rate of \frac12 inches per hour, but this time there was already 5 inches of snow on the ground. We can graph this storm on the same axes as the first storm by taking all the points on the graph of the first storm and translating them up 5 inches.

A graph showing time since the beginning of a storm in hours and depth of snow in inches.There are points at (2,1), (4,2), (2,6), and (4,7).

2 hours after each storm begins, 1 inch of new snow has fallen. For the first storm, this means there is now 1 inch of snow on the ground. For the second storm, this means there are now 6 inches of snow on the ground. Unlike the first storm, the second is not a proportional relationship since the line representing the second storm has a vertical intercept of 5. The equation representing the storm, y=\frac12x+5 , is of the form y=mx+b , where m is the rate of change, also the slope of the graph, and b is the initial amount, also the vertical intercept of the graph.

Lesson 8 Practice Problems

  1. Select all equations that have graphs with the same y -intercept.

    1. y=3x -8
    2. y=3x -9
    3. y=3x+8
    4. y=5x -8
    5. y=2x -8
    6. y=\frac13x -8

  2. Create a graph showing the equations y=\frac14x and y=\frac14x-5 . Explain how the graphs are the same and how they are different.

  3. A cable company charges $70 per month for cable service to existing customers.

    1. Find a linear equation representing the relationship between x , the number of months of service, and y , the total amount paid in dollars by an existing customer.
    2. For new customers, there is an additional one-time $100 service fee. Repeat the previous problem for new customers.
    3. When the two equations are graphed in the coordinate plane, how are they related to each other geometrically?

  4. Match each graph to a situation.

    Graphs A, B, C, and D showing different functions with postivie sloped linear functions
    1. The graph represents the perimeter, y , in units, for an equilateral triangle with side length of x units. The slope of the line is 3.
    2. The amount of money, y , in a cash box after x tickets are purchased for carnival games. The slope of the line is \frac14 .
    3. The number of chapters read, y , after x days. The slope of the line is \frac54 .
    4. The graph shows the cost in dollars, y , of a muffin delivery and the number of muffins, x , ordered. The slope of the line is 2.

  5. A mountain road is 5 miles long and gains elevation at a constant rate. After 2 miles, the elevation is 5500 feet above sea level. After 4 miles, the elevation is 6200 feet above sea level.

    1. Find the elevation of the road at the point where the road begins.
    2. Describe where you would see the point in part (a) on a graph where y represents the elevation in feet and x represents the distance along the road in miles.