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Transformations with Quadratic Functions

7.12.2 Transformations with Quadratic Functions

7.12.2 • Transformations with Quadratic Functions

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).

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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 ).

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.

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 ).

Compare your answer:

Multiplying x 2 x 2 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.

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 ).

Compare your answer:

Multiplying x 2 x 2 by a negative number flipped the graph (or reflected the graph) over the x x -axis so that it opens downward rather than upward.

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 ).

Compare your answer:

Multiplying x 2 x 2 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.

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 ).

Compare your answer:

When a constant is added to the x x in x 2 x 2 , the graph moves horizontally to the left. When a constant is subtracted from the x x in x 2 x 2 , the graph moves horizontally to the right.

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 ).

Compare your answer:

Multiplying x x 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.

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 ).

Compare your answer:

Multiplying x x 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 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)?

Compare your answer:

It is positive because the graph of f f opens upward.

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)?

Compare your answer:

It is positive because the graph of g g opens upward.

Or

The coefficient is less than the coefficient for the graph of f f (but still greater than 0) because g g is wider.

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)?

Compare your answer:

It is negative because the graph of h h opens downward.

Or

The absolute value of the coefficient of h h is less than the absolute value of the coefficients for both f f and g g because h h is wider.

4.

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 f f is greater than that of g g because the graph of f f is narrower, indicating that it is growing faster than the graph of g g . The absolute value of the coefficient of h h is less than that of g g because the graph of h h is wider, indicating that it is not growing as quickly.

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

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 ?

Here is how to determine shifts in graphs:

The function f ( x ) = x 2 f ( x ) = x 2 is the parent function for quadratics. For f ( x ) = x 2 + 3 f ( x ) = x 2 + 3 , when 3 is added to f ( x ) = x 2 f ( x ) = x 2 , the graph shifts up 3 units. This shift occurs because the value of 3 is being added to the output of the x 2 x 2 function. Thus, the shift is seen vertically because it affects the output values.