Graph of function g on a coordinate plane. X axis from negative 3 to 5. Y axis from negative 4 to 4. From left to right, the function begins around negative 3 comma 2, moves downward and to the right to about negative 1 comma negative 2, stays level until about 2 comma negative 2, moves upward and to the right to about 3 comma 2, moves downward and to the right to 4 comma negative 1, then stays level until 5 comma negative 1.
Graph of function h on a coordinate plane. X axis from negative 5 to 3. Y axis from negative 4 to 4. From left to right, the function begins around negative 5 comma 1, moves to the right to about negative 4 comma 1, moves downward and to the right to about negative 3 comma negative 2, moves upward and to the right to about negative 2 comma 2, stays level until about 1 comma 2, then moves downward and to the right until about 3 comma negative 2.
4.2
Activity
Reflecting Across
Here is the graph of function and a table of values.
A graph of function f on a coordinate plane. X axis from negative 4 to 4, by 2’s. Y axis from negative 4 to 4, by 2’s. From left to right, the function begins in the second quadrant, moves downward and to the right, crossing the x axis at point negative 3 comma 0, continues downward reaching point negative 1 point 5 comma negative 4 point 3. It then curves up and to the right passing through points negative 1 comma negative 4 and 0 comma negative 1 point 8 then crossing the x axis at 0 point 6. It moves upward and to the right until it reaches point 2 point 6 comma 3 point 9. Then it moves downward and to the right through point 4 comma 0 and ends in the fourth quadrant.
-3
0
-1.5
-4.3
-1
-4
0
-1.8
0.6
0
2.6
3.9
4
0
Let be the function defined by . Complete the table.
Sketch the graph of on the same axes as the graph of but in a different color.
Describe how to transform the graph of into the graph of . Explain how the equation produces this transformation.
4.3
Activity
Reflecting Across a Different Way
Here is another copy of the graph of from the earlier activity. This time, let be the function defined by .
A graph of function f on a coordinate plane. X axis from negative 4 to 4, by 2’s. Y axis from negative 4 to 4, by 2’s. From left to right, the function begins in the second quadrant, moves downward and to the right, crossing the x axis at point negative 3 comma 0, continues downward reaching point negative 1 point 5 comma negative 4 point 3. It then curves up and to the right passing through points negative 1 comma negative 4 and 0 comma negative 1 point 8 then crossing the x axis at 0 point 6. It moves upward and to the right until it reaches point 2 point 6 comma 3 point 9. Then it moves downward and to the right through point 4 comma 0 and ends in the fourth quadrant.
Use the definition of to find . Does your answer agree with your prediction?
What does your prediction tell you about ? Does your answer agree with the definition of ?
Complete the tables. The values for will not be the same for the two tables.
-3
0
-1.5
-4.3
-1
-4
0
-1.8
0.6
0
2.6
3.9
4
0
Sketch the graph of on the same axes as the graph of but in a different color.
Describe what happened to the graph of to transform it into the graph of . Explain how the equation produces this transformation.
Student Lesson Summary
Here are graphs of the functions , , and , where and . How do these equations match the transformation we see from to and from to ?
A graph of function f on a coordinate plane. X axis from negative 3 to 3, by 1’s. Y axis from negative 3 to 3, by 1’s. From left to right, the function begins in the third quadrant, moves upward and to the right, crossing the x axis at approximatelynegative 1, continues upward reaching about 0 comma 1. It then curves back downward slightly then back up and to the right, and continues to move upward and to the right, ending in the first quadrant.
A graph of function g on a coordinate plane. X axis from negative 3 to 3, by 1’s. Y axis from negative 3 to 3, by 1’s. From left to right, the function begins in the second quadrant, moves down and to the right crossing the x axis at about negative 1. It moves downward and to the right crossing the y axis at about negative 1 then moves downward and to the right passing through about 1 point 5 comma negative 2 point 5, then ending in the fourth quadrant.
A graph of function h on a coordinate plane. X axis from negative 3 to 3, by 1’s. Y axis from negative 3 to 3, by 1’s. From left to right, the function begins in the second quadrant, moves down and to the right to about negative 1 comma 1. It crosses the y axis at about 1 then moves downward and to the right passing through the x axis at about 1, then ending in the fourth quadrant.
Considering first the equation , we know that for the same input , the value of will be the opposite of the value of . For example, since , we know that . We can see this relationship in the graphs where is the reflection of across the -axis.
Looking at , this equation tells us that the two functions have the same output for opposite inputs. For example, 1 and -1 are opposites, so (and is also true!). We can see this relationship in the graphs where is the reflection of across the -axis.