Motion Along a Line—Search Robots Again

New York State Common Core Math Geometry, Module 4, Lesson 14

Worksheets for Geometry

Student Outcomes

• Students name several points on a line given by a parametric equation and provide the point-slope equation for a line given by a parametric equation.
• Students determine whether lines given parametrically are parallel or perpendicular.

Motion Along a Line—Search Robots Again

Classwork

Exercises

a. If 𝑓(𝑡) = (𝑡, 2𝑡 − 1), find the values of 𝑓(0), 𝑓(1), and 𝑓(5), and plot them on a coordinate plane.
b. What is the image of 𝑓(𝑡)?
c. At what time does the graph of the line pass through the 𝑦-axis?
d. When does it pass through the 𝑥-axis?
e. Can you write the equation of the line you graphed in slope 𝑦-intercept form?
f. How does this equation compare with the definition of 𝑓(𝑡)?

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Example 1

Programmers want to program a robot so that it moves at a uniform speed along a straight line segment connecting two points 𝐴 and 𝐵. If 𝐴(0,−1) and 𝐵(1, 1), and the robot travels from 𝐴 to 𝐵 in 1 minute,
a. Where is the robot at 𝑡 = 0?
b. Where is the robot at 𝑡 = 1?
c. Draw a picture that shows where the robot will be at 0 ≤ 𝑡 ≤ 1.

Exercise 1

A robot is programmed to move along a straight line path through two points 𝐴 and 𝐵. It travels at a uniform speed that allows it to make the trip from 𝐴(0,−1) to 𝐵(1, 1) in 1 minute. Find the robot’s location, 𝑃, for each time 𝑡 in minutes.
a. 𝑡 = 1/4
b. 𝑡 = 0.7
c. 𝑡 = 5/4
d. 𝑡 = 2.2

Example 2

Our robot has been reprogrammed so that it moves along the same straight line path through two points 𝐴(0,−1) and 𝐵(1, 1) at a uniform rate but makes the trip in 0.6 minutes instead of 1 minute. How does this change the way we calculate the location of the robot at any time, 𝑡?
a. Find the location, 𝑃, of the robot from Example 1 if the robot were traveling at a uniform speed that allowed it to make the trip from 𝐴 to 𝐵 in 0.6 minutes. Is the robot’s speed greater or less than the robot’s speed in Example 1?
b. Find the location, 𝑃, of the robot from Example 1 if the robot were traveling at a uniform speed that allowed it to make the trip from 𝐴 to 𝐵 in 1.5 minutes. Is the robot’s speed greater or less than the robot’s speed in Example 1?

Exercise 2

Two robots are moving along straight line paths in a rectangular room. Robot 1 starts at point 𝐴(20, 10) and travels at a constant speed to point 𝐵(120, 50) in 2 minutes. Robot 2 starts at point 𝐶(90, 10) and travels at a constant speed to point 𝐷(60, 70) in 90 seconds.
a. Find the location, 𝑃, of Robot 1 after it has traveled for 𝑡 minutes along its path from 𝐴 to 𝐵.
b. Find the location, 𝑄, of Robot 2 after it has traveled for 𝑡 minutes along its path from 𝐶 to 𝐷.
c. Are the robots traveling at the same speed? If not, which robot’s speed is greater?
d. Are the straight line paths that the robots are traveling parallel, perpendicular, or neither? Explain your answer.

Example 3

A programmer wants to program a robot so that it moves at a constant speed along a straight line segment connecting the point 𝐴(30, 60) to the point 𝐵(200, 100) over the course of a minute.
At time 𝑡 = 0, the robot is at point 𝐴.
At time 𝑡 = 1, the robot is at point 𝐵.
a. Where will the robot be at time 𝑡 = 1/2?
b. Where will the robot be at time 𝑡 = 0.6?

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