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Algebra 1

7.12.2 Transformations with Quadratic Functions

Algebra 17.12.2 Transformations with Quadratic Functions
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Activity

Use the graphing tool or technology outside the course. Graph f ( x ) = x 2 f ( x ) = x 2 , 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).

1.

Add different constant terms to x 2 x 2 (for example: x 2 + 5 x 2 + 5 , x 2 + 10 x 2 + 10 , x 2 − 3 x 2 − 3 ).

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2.

Multiply x 2 x 2 by different positive coefficients greater than 1 (for example: 3 x 2 3 x 2 , 7.5 x 2 7.5 x 2 ).

3.

Multiply x 2 x 2 by different negative coefficients less than or equal to -1 (for example: − x 2 − x 2 , − 4 x 2 − 4 x 2 ).

4.

Multiply x 2 x 2 by different coefficients between -1 and 1 (for example: 12 x 2 12 x 2 , − 0.25 x 2 − 0.25 x 2 ).

5.

Add different constants to x x before squaring the term (for example ( x − 1 ) 2 ( x − 1 ) 2 , ( x + 2 ) 2 ( x + 2 ) 2 , ( x − 3 ) 2 ( x − 3 ) 2 ).

6.

Multiply x x by different positive coefficients greater than 1 (for example: ( 3 x ) 2 3 x ) 2 , ( 7.5 x ) 2 ( 7.5 x ) 2 ).

7.

Multiply x x by different positive coefficients between 0 and 1 (for example: ( 0.5 x ) 2 ( 0.5 x ) 2 , ( 0.1 x ) 2 ( 0.1 x ) 2 ).

For quadratics, the parent function is f ( x ) = x 2 f ( x ) = x 2 . 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

1.

Examine the graphs of three quadratic functions.

Three parabolas are graphed on a coordinate plane that has unlabeled axes.

What can you say about the coefficient of x 2 x 2 in the expression that defines f f (in black at the top)?

2.

What can you say about the coefficient of x 2 x 2 in the expression that defines g g (in blue in the middle)?

3.

What can you say about the coefficient of x 2 x 2 in the expression that defines h h (in yellow at the bottom)?

4.

How do each of the functions compare to each other?

Self Check

How is the graph of f ( x ) = ( x − 4 ) 2 transformed from the graph of f ( x ) = x 2 ?
  1. Left 4
  2. Right 4
  3. Down 4
  4. Up 4

Additional Resources

Graphing Quadratic Functions Using Transformations

GRAPH A QUADRATIC FUNCTION OF THE FORM f ( x ) = x 2 + k f ( x ) = x 2 + k USING A VERTICAL SHIFT

The graph of f ( x ) = x 2 + k f ( x ) = x 2 + k shifts the graph of f ( x ) = x 2 f ( x ) = x 2 vertically k k units.

  • If k > 0 k > 0 , shift the parabola vertically up k k units.
  • If k < 0 k < 0 , shift the parabola vertically down | k | | k | units.
Graph of the parent quadratic function is given in blue on the coordinate plane. Functions that have been vertically translated are graphed using an oranged, dashed curve and a green, dotted curve.

For the functions graphed above, notice that the parent function, f ( x ) = x 2 f ( x ) = x 2 , is graphed using a blue, solid line.

  1. The parabola that has been shifted vertically up 3 units represents the function f ( x ) = x 2 + 3 f ( x ) = x 2 + 3 since k = 3 k = 3 and 3 > 0 3 > 0 (orange, dashed graph).

  2. The parabola that has been shifted vertically down 2 units represents the function f ( x ) = x 2 − 2 f ( x ) = x 2 − 2 since k = − 2 k = − 2 and − 2 < 0 − 2 < 0 (green, dotted graph).

GRAPH A QUADRATIC FUNCTION OF THE FORM f ( x ) = ( x − h ) 2 f ( x ) = ( x − h ) 2 USING A HORIZONTAL SHIFT

The graph of f ( x ) = ( x − h ) 2 f ( x ) = ( x − h ) 2 shifts the graph of f ( x ) = x 2 f ( x ) = x 2 horizontally h h units.

  • If h > 0 h > 0 , shift the parabola horizontally right h h units.
  • If h < 0 h < 0 , shift the parabola horizontally left | h | | h | units.
Graph of the parent quadratic function is given in blue on the coordinate plane. Functions that have been horizontally translated are graphed using an oranged, dashed curve and a green, dotted curve.

Again, for the functions graphed above, the parent function, f ( x ) = x 2 f ( x ) = x 2 , is graphed using a blue, solid line.

  1. The parabola that has been shifted horizontally right 3 units represents the function f ( x ) = ( x − 3 ) 2 f ( x ) = ( x − 3 ) 2 since h = 3 h = 3 and 3 > 0 3 > 0 (orange, dashed graph).

  2. The parabola that has been shifted horizontally left 2 units represents the function f ( x ) = ( x + 2 ) 2 f ( x ) = ( x + 2 ) 2 since h = − 2 h = − 2 and − 2 < 0 − 2 < 0 (green, dotted graph).

GRAPH OF A QUADRATIC FUNCTION OF THE FORM f ( x ) = a x 2 f ( x ) = a x 2

The coefficient a in the function f ( x ) = a x 2 f ( x ) = a x 2 affects the graph of f ( x ) = x 2 f ( x ) = x 2 by stretching or compressing it.

  • If 0 < | a | < 1 0 < | a | < 1 , the graph of f ( x ) = a x 2 f ( x ) = a x 2 will be “wider” than the graph of f ( x ) = x 2 f ( x ) = x 2 .
  • If | a | > 1 | a | > 1 , the graph of f ( x ) = a x 2 f ( x ) = a x 2 will be “skinnier” than the graph of f ( x ) = x 2 f ( x ) = x 2 .
Graph of the parent quadratic function is given in blue on the coordinate plane. Functions that have been vertically dilated are graphed using an oranged, dashed curve and a green, dotted curve.

Again, the parent function, f ( x ) = x 2 f ( x ) = x 2 , is graphed using a blue, solid line.

  1. The parabola that has been vertically compressed by a scale factor of 2 represents the function f ( x ) = 1 2 x 2 f ( x ) = 1 2 x 2 since a = 1 2 a = 1 2 and 0 < 1 2 < 1 0 < 1 2 < 1 (green, dotted graph).

  2. The parabola that has been vertically stretched by a scale factor of 3 represents the function f ( x ) = 3 x 2 f ( x ) = 3 x 2 since a = 3 a = 3 and 3 > 1 3 > 1 (orange, dashed graph).

GRAPH OF A QUADRATIC FUNCTION OF THE FORM f ( x ) = ( b x ) 2 f ( x ) = ( b x ) 2

The coefficient b b in the function f ( x ) = ( b x ) 2 f ( x ) = ( b x ) 2 affects the graph of f ( x ) = x 2 f ( x ) = x 2 by stretching or compressing it.

  1. If 0 < | b | < 1 0 < | b | < 1 , the graph of f ( x ) = ( b x ) 2 f ( x ) = ( b x ) 2 will appear “wider” than the graph of f ( x ) = x 2 f ( x ) = x 2 .

  2. If | b | > 1 | b | > 1 , the graph of f ( x ) = ( b x ) 2 f ( x ) = ( b x ) 2 will appear “skinnier” than the graph of f ( x ) = x 2 f ( x ) = x 2 .

Graph of the parent quadratic function is given in blue on the coordinate plane. Functions that have been horizontally dilated are graphed using an oranged, dashed curve and a green, dotted curve.

Again, the parent function, f ( x ) = x 2 f ( x ) = x 2 , is graphed using a blue, solid line.

  1. The parabola that has been horizontally compressed by a scale factor of 3 represents the function f ( x ) = ( 3 x ) 2 f ( x ) = ( 3 x ) 2 since b = 3 b = 3 and 3 > 1 3 > 1 (orange, dashed graph).

  2. The parabola that has been horizontally stretched by a scale factor of 2 represents the function f ( x ) = ( 1 2 x ) 2 f ( x ) = ( 1 2 x ) 2 since b = 1 2 b = 1 2 and 0 < 1 2 < 1 0 < 1 2 < 1 (green, dotted graph).

Try it

Try It: Graphing Quadratic Functions Using Transformations

How does the graph of f ( x ) = x 2 + 3 f ( x ) = x 2 + 3 change from f ( x ) = x 2 f ( x ) = x 2 ?

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