3.4 Graphs and Transformations

1. Define parent functions.2. Transform graphs of parent functions.

Parent functions are used to illustrate the basic shape and characteristics of various functions.

The rules of transforming these functions can be applied to ANY function.

Parent Functions

Parent Functions

1 2 3 4 5–1–2–3–4–5 x

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5

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–5

y

1 2 3 4 5–1–2–3–4–5 x

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5

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–2

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–5

y

1 2 3 4 5–1–2–3–4–5 x

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5

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–2

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y

constant function

f(x) 1

identity (linear) function

absolute-value function

f(x) x f(x) x

Parent Functions

greatest integer function

cubic function

f(x) x f(x) x 2

quadratic function

f(x) x 3

1 2 3 4 5–1–2–3–4–5 x

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5

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–5

y

1 2 3 4 5–1–2–3–4–5 x

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5

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–2

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y

1 2 3 4 5–1–2–3–4–5 x

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–2

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y

Parent Functions

reciprocal function

cube root function

square root function

f(x) 1x

f(x) x f(x) x3

1 2 3 4 5–1–2–3–4–5 x

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–5

y

1 2 3 4 5–1–2–3–4–5 x

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–2

–3

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–5

y

1 2 3 4 5–1–2–3–4–5 x

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–1

–2

–3

–4

–5

y

Vertical Shifts

g(x) f(x) c

g(x) f(x) c

Vertical shift upward c units.

Vertical shift downward c units.

Example #1Shifting a Graph Vertically

g(x) x 2 3Vertical shift 3 units up.

2 4 6 8 10 12–2–4–6–8–10–12 x

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–4

y

2 4 6 8 10 12–2–4–6–8–10–12 x

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y

h(x) x 5Vertical shift 5 units down.

Horizontal Shifts

Horizontal shift left c units.

Horizontal shift right c units.

g(x) f(x c) or g(x) f(x c)

g(x) f(x c)

Example #2Shifting a Graph Horizontally

Horizontal shift 2 units right. Horizontal shift 4 units left.

g(x) 1x 2

h(x) x 4

2 4 6 8 10 12–2–4–6–8–10–12 x

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2 4 6 8 10 12–2–4–6–8–10–12 x

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y

Reflections

Reflection over the x-axis.

Reflection over the y-axis.

g(x) f(x)

g(x) f( x)

Example #3Reflecting a Graph

Reflection over the x-axis. Reflection over the y-axis.

g(x) x h(x) x

2 4 6 8 10 12–2–4–6–8–10–12 x

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y

2 4 6 8 10 12–2–4–6–8–10–12 x

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y

Vertical Stretches & Compressions

g(x) c f(x)Given a function with the transformation:

If c > 1, the graph of f is stretched vertically, away from the x-axis, by a factor of c.

If c < 1, the graph of f is compressed vertically, toward the x-axis, by a factor of c.

Every point of the function is changed by cyx,

Example #4Vertical Stretches & Compressions

Vertical stretch by a factor of 2.

g(x) 2 x

Vertical compression by a factor of .14

h(x) 14

x 3

2 4 6 8 10 12–2–4–6–8–10–12 x

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2 4 6 8 10 12–2–4–6–8–10–12 x

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y

Horizontal Stretches & Compressions

Given a function with the transformation:

g(x) f(c x)

If c > 1, the graph of f is compressed horizontally, toward

the y-axis, by a factor of .

If c < 1, the graph of f is stretched horizontally, away from

the y-axis, by a factor of .

1c

1c

Every point of the function is changed by

yx

c,

1

Example #5Horizontal Stretches & Compressions

Horizontal stretch by a factor of 5 .

g(x) 3x3

2 4 6 8 10 12–2–4–6–8–10–12 x

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2 4 6 8 10 12–2–4–6–8–10–12 x

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y

Horizontal compression by a factor of .13

2

5

1)(

xxh

Combining Transformations

g(x) c f(a(x b)) d

1. If a < 0, reflect over the y-axis.

2. Stretch or compress horizontally by a factor of .

3. Shift the graph horizontally b units left or right.

4. If c < 0, reflect over the x-axis.

5. Stretch or compress vertically by a factor of .

6. Shift the graph vertically d units up or down.

1a

c

Example #6Combining TransformationsDescribe the transformations on the following functions, then graph.

A.) g(x) (3x 12)2 2

1. Horizontal compression by a factor of 1/3.

2. Shift 4 units right.3. Reflect over x-axis.4. Shift 2 units up.

Apply transformations using the order of operations.

2 4 6 8 10 12–2–4–6–8–10–12 x

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y

Example #6Combining TransformationsDescribe the transformations on the following functions, then graph.

1. Reflection over the y-axis.2. Horizontal stretch by a factor of 4.3. Shift 4 units left.4. Vertical stretch by a factor of 2.5. Shift 4 units down.

Apply transformations using the order of operations.

2 4 6 8 10 12–2–4–6–8–10–12 x

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4

6

8

10

12

–2

–4

–6

–8

–10

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y414

12)(

3

xxhB.)