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Math 1000 Stuart Jones Section2.6 Transformations of Functions
Math 1000
Stuart Jones
Section2.6
Transformations of Functions
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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.
Math 1000
Stuart Jones
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.
Math 1000
Stuart Jones
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.
Math 1000
Stuart Jones
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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Stuart Jones
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
Math 1000
Stuart Jones
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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Stuart Jones
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
Math 1000
Stuart Jones
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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Stuart Jones
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)
Math 1000
Stuart Jones
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)
Math 1000
Stuart Jones
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)
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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
Math 1000
Stuart Jones
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.
Math 1000
Stuart Jones
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.
Math 1000
Stuart Jones
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.
Math 1000
Stuart Jones
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.
Math 1000
Stuart Jones
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.
Math 1000
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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.
