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Reflecting in the line y = x (#2)

Author:Callum Marshall

Part A

When you take a point such as (3, 5) or (-2, 8) or any other such point and reflect in the line with equation y = x, what happens..? In the diagram below there are four points A, B, C and D. Using the applet reflect these points in the line shown, the line y = x.

Question 1

What do you notice about the coordinates of the IMAGES of A, B, C and D under reflection in the line y = x?

Part B

In the applet below the point A lies on the graph of the function , and the black line is the line y = x. With Trace Off change the value of the slider to move the position of point A.

Question 2

What is the relationship between points A and A' ?

Question 3

With Trace On, again using the slider, change the position of point A. i. What do you notice about the line traced out by A' ? ii. How could you use the relationship between the coordinates of A and A' to help you find the equation of the line traced out by A' as point A moves along the graph of the function.

Question 4A

In the space below, write the equation of the line traced out by A' as point A moves.

Question 4B

Which of the following sets represents the domain of the trace function?

Tick all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Part C - linear functions

The applet below shows the line y = x (the blue broken line), and the graph of the function . The point A lines on the line with equation .

Question 5

Find the equation of the red line in the form

Question 6A

Turn Trace on and drag the point A. Which of the equations below represent the line traced out by A' ?

Tick all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 6B

Which of the following represent the range of the trace function?

Tick all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 7A

Consider the graph shown below, which of the following represent the domain and range of the red line?    

Tick all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 7B

In the space below write the equation for both: i. the red line and, ii. the trace line

Question 7C

In the space below write the domain and range of the trace function.

Part D - Rational functions

Questions 9 and 10 refer to the graph shown in the applet below

Question 8

The graph above has an equation in the form for . Find the values for both a and b. 

Question 9

Turn on the trace and move point A. i. Find the equation of the line which is traced out by point A'. ii. Write the domain and range for the trace function.

Question 10

The graph below shows the graph of the function for . i. Given that a = 3, find the correct values for b and c. ii. Write the equation for the line traced out by A' as A is moved along the line.

Part E - Quadratic functions

Question 11 refers to the applet shown below.

Question 11 A

The function with equation for is shown in the applet above. i. Find the correct values for a and b. ii. Hence, find the equation traced out by A' as point A moves along the line.

Question 11 B

Which of the options shown below represent the domain and range for the trace function.

Tick all that apply
  • A
  • B
  • C
  • D
Check my answer (3)

Question 12

The graph shown below illustrates the function where . Using the graph, find the value of b.

Question 13

i. Find the equation of the line traced out by A' as the point A moves along the graph of . ii. State the domain and the range of the trace function.

Part F - Summarising what you have done.

Having worked through the problems above in the space below explain: i. how to find the equation of a function when it is reflected in the line y = x. ii. the relationship between the domain and range of the original function and the domain and range of the reflected function.

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