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
Standards Alignment
Building On
Addressing
F-BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
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
Standards Alignment
Building On
Addressing
F-BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.
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.
Glossary
None
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Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.