Activity
Use the graphing tool or technology outside the course. Graph , and then experiment with each of the changes to the function in questions 1 – 7. Record your observations of what you noticed or wondered about each set of changes to the function (include sketches, if helpful).
Add different constant terms to (for example: , , ).
Compare your answer:
Changing the constant term either raised the graph (when the constant is positive) or lowered the graph (when the constant is negative). Adding the constant term shifted the graph vertically, but the overall shape of the graph did not change.
Multiply by different positive coefficients greater than 1 (for example: , ).
Compare your answer:
Multiplying by different positive numbers changed the shape of the graph, making it "steeper" when the constant is larger and less steep when the constant is smaller. The size of the coefficient determined how much of a vertical stretch was applied to the graph.
Multiply by different negative coefficients less than or equal to -1 (for example: , ).
Compare your answer:
Multiplying by a negative number flipped the graph (or reflected the graph) over the -axis so that it opens downward rather than upward.
Multiply by different coefficients between -1 and 1 (for example: , ).
Compare your answer:
Multiplying by numbers between -1 and 1 changed the opening of the graph as well as the direction of the graph. Shrinking the coefficient from 1 toward 0 made the graph "shallower" and wider. This change vertically compressed the graph. When the coefficient is 0, the graph becomes a horizontal line. Raising the coefficient from -1 to 0 had the same effect of vertically compressing the graph, except that the graph started out opening downward.
Add different constants to before squaring the term (for example , , ).
Compare your answer:
When a constant is added to the in , the graph moves horizontally to the left. When a constant is subtracted from the in , the graph moves horizontally to the right.
Multiply by different positive coefficients greater than 1 (for example: ( , ).
Compare your answer:
Multiplying by different positive numbers changed the shape of the graph, making it more narrow when the constant is larger and less narrow when the constant is smaller.
Multiply by different positive coefficients between 0 and 1 (for example: , ).
Compare your answer:
Multiplying by different positive numbers between 0 and 1 changed the shape of the graph, making it wider as the decimal gets closer to zero.
For quadratics, the parent function is . From there, any moves created by making changes to the function like in questions 1 – 7 above are called transformations.
Are you ready for more?
Extending Your Thinking
Examine the graphs of three quadratic functions.
What can you say about the coefficient of in the expression that defines (in black at the top)?
Compare your answer:
It is positive because the graph of opens upward.
What can you say about the coefficient of in the expression that defines (in blue in the middle)?
Compare your answer:
It is positive because the graph of opens upward.
Or
The coefficient is less than the coefficient for the graph of (but still greater than 0) because is wider.
What can you say about the coefficient of in the expression that defines (in yellow at the bottom)?
Compare your answer:
It is negative because the graph of opens downward.
Or
The absolute value of the coefficient of is less than the absolute value of the coefficients for both and because is wider.
How do each of the functions compare to each other?
Compare your answer:
The coefficients cannot be identified exactly without knowing the coordinates of some points on the graph. The coefficient of the squared term of is greater than that of because the graph of is narrower, indicating that it is growing faster than the graph of . The absolute value of the coefficient of is less than that of because the graph of is wider, indicating that it is not growing as quickly.
Self Check
Additional Resources
Graphing Quadratic Functions Using Transformations
GRAPH A QUADRATIC FUNCTION OF THE FORM USING A VERTICAL SHIFT
The graph of shifts the graph of vertically units.
- If , shift the parabola vertically up units.
- If , shift the parabola vertically down units.
For the functions graphed above, notice that the parent function, , is graphed using a blue, solid line.
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The parabola that has been shifted vertically up 3 units represents the function since and (orange, dashed graph).
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The parabola that has been shifted vertically down 2 units represents the function since and (green, dotted graph).
GRAPH A QUADRATIC FUNCTION OF THE FORM USING A HORIZONTAL SHIFT
The graph of shifts the graph of horizontally units.
- If , shift the parabola horizontally right units.
- If , shift the parabola horizontally left units.
Again, for the functions graphed above, the parent function, , is graphed using a blue, solid line.
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The parabola that has been shifted horizontally right 3 units represents the function since and (orange, dashed graph).
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The parabola that has been shifted horizontally left 2 units represents the function since and (green, dotted graph).
GRAPH OF A QUADRATIC FUNCTION OF THE FORM
The coefficient a in the function affects the graph of by stretching or compressing it.
- If , the graph of will be “wider” than the graph of .
- If , the graph of will be “skinnier” than the graph of .
Again, the parent function, , is graphed using a blue, solid line.
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The parabola that has been vertically compressed by a scale factor of 2 represents the function since and (green, dotted graph).
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The parabola that has been vertically stretched by a scale factor of 3 represents the function since and (orange, dashed graph).
GRAPH OF A QUADRATIC FUNCTION OF THE FORM
The coefficient in the function affects the graph of by stretching or compressing it.
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If , the graph of will appear “wider” than the graph of .
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If , the graph of will appear “skinnier” than the graph of .
Again, the parent function, , is graphed using a blue, solid line.
-
The parabola that has been horizontally compressed by a scale factor of 3 represents the function since and (orange, dashed graph).
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The parabola that has been horizontally stretched by a scale factor of 2 represents the function since and (green, dotted graph).
Try it
Try It: Graphing Quadratic Functions Using Transformations
How does the graph of change from ?
Here is how to determine shifts in graphs:
The function is the parent function for quadratics. For , when 3 is added to , the graph shifts up 3 units. This shift occurs because the value of 3 is being added to the output of the function. Thus, the shift is seen vertically because it affects the output values.