transformations of functions · transformations theorem (translations) given a function f(x), it...

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Section2.6

Transformations of Functions

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Parent Functions

Linear Function y = x

Quadratic Function y = x2

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Parent Functions

Cubic Function y = x3

Square Root Function y =√x

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Parent Functions

Absolute Value Function y = |x |

Constant Function y = b for some constant b

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Transformations

Theorem (Translations)

Given a function f (x), it can be shifted around the graph usingthe following modifications, called translations:

f (x) + a is a vertical shift of a units upwards

f (x)− b is a vertical shift of b units downwards

f (x + c) is a horizontal shift c units left

f (x − d) is a horizontal shift d units right

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Explain how the graph of g is obtained from the graph of f.f (x) = x2, g(x) = (x − 7)2

Answer: Shift right 7 unitsf (x) =

√(x) g(x) =

√(x) + 6

Answer: Shift up 6 units

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Explain how the graph of g is obtained from the graph of f.f (x) = x2, g(x) = (x − 7)2

Answer: Shift right 7 units

f (x) =√

(x) g(x) =√

(x) + 6Answer: Shift up 6 units

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Explain how the graph of g is obtained from the graph of f.f (x) = x2, g(x) = (x − 7)2

Answer: Shift right 7 unitsf (x) =

√(x) g(x) =

√(x) + 6

Answer: Shift up 6 units

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Explain how the graph of g is obtained from the graph of f.f (x) = x2, g(x) = (x − 7)2

Answer: Shift right 7 unitsf (x) =

√(x) g(x) =

√(x) + 6

Answer: Shift up 6 units

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Graph f (x) = |x | − 4

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Graph f (x) = |x | − 4

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Theorem (Reflections over x and y axis)

Given a function h(x), a reflection is obtained by flipping thefunction over the x or y axis.

f (−x) reflects the function over the y-axis

−f (x) reflects the function over the x-axis.

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Explain how the graph of j is obtained from the graph of h.h(x) = x2, j(x) = −x2

Answer: Reflection over the x-axis.h(x) = x3, j(x) = (−x)3 Answer: Reflection over y-axis.

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Explain how the graph of j is obtained from the graph of h.h(x) = x2, j(x) = −x2Answer: Reflection over the x-axis.

h(x) = x3, j(x) = (−x)3 Answer: Reflection over y-axis.

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Explain how the graph of j is obtained from the graph of h.h(x) = x2, j(x) = −x2Answer: Reflection over the x-axis.h(x) = x3, j(x) = (−x)3

Answer: Reflection over y-axis.

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Explain how the graph of j is obtained from the graph of h.h(x) = x2, j(x) = −x2Answer: Reflection over the x-axis.h(x) = x3, j(x) = (−x)3 Answer: Reflection over y-axis.

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Graph −x2

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Graph −x2

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Theorem (Stretches and Compressions)

A function f (x) can be stretched or compressed vertically orhorizontally by the coefficient.

f (ax) is a horizontal compressions when a > 1 and astretch when a < 1

bf (x) is a vertical stretch when b > 1 and a compressionwhen b < 1

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Explain how the graph of g is obtained from the graph of f.f (x) =

√x . g(x) = 3

√x

Answer: A vertical stretchf (x) = x2, g(x) = 1

4x2

Answer: A vertical compression

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Explain how the graph of g is obtained from the graph of f.f (x) =

√x . g(x) = 3

√x

Answer: A vertical stretch

f (x) = x2, g(x) = 14x

2

Answer: A vertical compression

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Explain how the graph of g is obtained from the graph of f.f (x) =

√x . g(x) = 3

√x

Answer: A vertical stretchf (x) = x2, g(x) = 1

4x2

Answer: A vertical compression

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Explain how the graph of g is obtained from the graph of f.f (x) =

√x . g(x) = 3

√x

Answer: A vertical stretchf (x) = x2, g(x) = 1

4x2

Answer: A vertical compression

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Given the parent graph, f (x) = x2 below, graph h(x) = 3x2

and g(x) = 13x

2

The cyan graph is h(x) and the gray graph is g(x)

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Given the parent graph, f (x) = x2 below, graph h(x) = 3x2

and g(x) = 13x

2

The cyan graph is h(x) and the gray graph is g(x)

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Given the parent graph, f (x) = x2 below, graph h(x) = 3x2

and g(x) = 13x

2

The cyan graph is h(x) and the gray graph is g(x)

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Given the parent graph, f (x) = x2 below, graph h(x) = 3x2

and g(x) = 13x

2

The cyan graph is h(x) and the gray graph is g(x)

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = x , g(x) = 2x − 4

Answer: Stretched vertically (orcompressed horizontally) by a factor of 2, then shifted down 4unitsf (x) = x3, g(x) = −(x − 4)3 + 1 Answer: Reflected over they-axis downward, and shifted 4 units right and 1 unit up

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = x , g(x) = 2x − 4 Answer: Stretched vertically (orcompressed horizontally) by a factor of 2, then shifted down 4units

f (x) = x3, g(x) = −(x − 4)3 + 1 Answer: Reflected over they-axis downward, and shifted 4 units right and 1 unit up

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = x , g(x) = 2x − 4 Answer: Stretched vertically (orcompressed horizontally) by a factor of 2, then shifted down 4unitsf (x) = x3, g(x) = −(x − 4)3 + 1

Answer: Reflected over they-axis downward, and shifted 4 units right and 1 unit up

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = x , g(x) = 2x − 4 Answer: Stretched vertically (orcompressed horizontally) by a factor of 2, then shifted down 4unitsf (x) = x3, g(x) = −(x − 4)3 + 1 Answer: Reflected over they-axis downward, and shifted 4 units right and 1 unit up

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = |x |, g(x) = −|12x + 1| − 2

Answer: Reflected (flipped)over the y-axis, stretched horizontally by a factor of 1/2, andshifted left 1 and down 2.f (x) =

√x , g(x) = −

√2x − 4 + 1 Answer: Reflected over

y-axis, compressed horizontally by a factor of 2, and shiftedright 4 and up 1.

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = |x |, g(x) = −|12x + 1| − 2 Answer: Reflected (flipped)over the y-axis, stretched horizontally by a factor of 1/2, andshifted left 1 and down 2.

f (x) =√x , g(x) = −

√2x − 4 + 1 Answer: Reflected over

y-axis, compressed horizontally by a factor of 2, and shiftedright 4 and up 1.

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = |x |, g(x) = −|12x + 1| − 2 Answer: Reflected (flipped)over the y-axis, stretched horizontally by a factor of 1/2, andshifted left 1 and down 2.f (x) =

√x , g(x) = −

√2x − 4 + 1

Answer: Reflected overy-axis, compressed horizontally by a factor of 2, and shiftedright 4 and up 1.

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Combining TransformationsExplain how the graph of g(x) is obtained from f(x).f (x) = |x |, g(x) = −|12x + 1| − 2 Answer: Reflected (flipped)over the y-axis, stretched horizontally by a factor of 1/2, andshifted left 1 and down 2.f (x) =

√x , g(x) = −

√2x − 4 + 1 Answer: Reflected over

y-axis, compressed horizontally by a factor of 2, and shiftedright 4 and up 1.

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Graph g(x) =√−x − 1

The red graph is the parent graph. G(x) is shown in blue.

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Graph g(x) =√−x − 1

The red graph is the parent graph. G(x) is shown in blue.

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Graoh f (x) = −x3 − 1

The parent graph is shown in red. The graph of f(x) is shown

in magenta.

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Graoh f (x) = −x3 − 1The parent graph is shown in red. The graph of f(x) is shown

in magenta.

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Graph f (x) = 35 |x − 1|+ 2

The parent function is shown in red. The graph of f(x) isshown in purple.

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Graph f (x) = 35 |x − 1|+ 2

The parent function is shown in red. The graph of f(x) isshown in purple.

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Even and Odd Functions

Theorem (Even and Odd Functions)

An even function is a function f (x) such thatf (−x) = f (x) for all x.

An odd function is a function f (x) such thatf (−x) = −f (x) for all x.

Graphically, even functions are symmetric with respect tothe y-axis.

Graphically, odd functions are symmetric with respect tothe origin.

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Decide whether each function is even, odd, or neither.

y = x4

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Decide whether each function is even, odd, or neither.

y = 4x3 − 2x

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Decide whether the function is even, odd, or neither:

y = x3 − x2

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The Bottom Line

Numbers inside the function affect x; numbers outside thefunction affect y.

Translations shift a function around the coordinate plane.

Reflections flip a function over the x-axis or y-axis.

Coefficients stretch or compress functions in the x- ory-direction.

Even and odd functions can make it easier to graph themif you know the symmetry involved.