Essential Question: In the equationg(x) = c[a(x-b)] + d

what do each of the letters do to the graph?

3.4: Graphs and TransformationsParent function: A function with a certain shape

that has the simplest rule for that shape.For example, f(x) = x2 is the simplest rule for a

parabolaAny parabola is a transformation of that parent

functionAll of the following parent functions are on page

173 in your books… there is no need to copy them now.

It’s most important that you get down the words in blue. Everything else is predominately mathematical definition.

3.4: Graphs and TransformationsIdentify the parent function.

( ) 4 3g x x

( ) 2 4h x x

3( ) 2 5f x x

x

x

3 x

3.4: Graphs and TransformationsConstant function Identity Function

f(x) = 1 f(x) = x

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

3.4: Graphs and TransformationsAbsolute-value function Greatest Integer

Functionf(x) = |x| f(x) = [x]

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

3.4: Graphs and TransformationsQuadratic function Cubic

Functionf(x) = x2 f(x) = x3

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

3.4: Graphs and TransformationsReciprocal function Square Root Function

f(x) = 1/x f(x) = x

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

3.4: Graphs and TransformationsCube root function

f(x) = . 3 x

(0, 0)

(1, 1)

1 2 3 4–1–2–3–4 x

1

2

3

4

–1

–2

–3

–4

y

3.4: Graphs and TransformationsVertical shifts

When a value is added tof(x), the effect is to add thevalue to the y-coordinateof each point, effectivelyshifting the graph up anddown.

If c is a positive number, then:

The graph g(x) = f(x) + c is the graph of f shifted up c units The graph g(x) = f(x) – c is the graph of f shifted down c units

Numbers adjusted after the parent function affect the graph vertically, as one would expect (+ up, – down)

| x| + 2

| x| - 3

| x|

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

3.4: Graphs and TransformationsHorizontal shifts

When a value is added tothe x of a function, the effect is to readjust the graph,effectively shifting the graph left and right.

If c is a positive number, then:

The graph g(x) = f(x+c) is the graph of f shifted c units to the left

The graph g(x) = f(x-c) is the graph of f shifted c units to the right

Numbers adjusted to the x of the parent function [inside a parenthesis] affect the graph horizontally, in reverse of expected values (+ left, – right)

(x+3)² (x-1)²

x²

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

3.4: Graphs and TransformationsReflections

Adding a negative sign beforea function reflects the graph about the x-axis. Adding a negative sign before the x in the function reflects the graph aboutthe y-axis.

A negative sign before the function flips up & down(vertically, also called “reflected across the x-axis”)

A negative sign before the x flips left & right (horizontally, also called “reflected across the y-axis”)

sqrt(-y)

-sqrt(y)

sqrt(y)

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

3.4: Graphs and TransformationsStretches & Compressions (Vertical)

If a function is multiplied by a number, it will stretch or compress the parent function vertically

If c > 1, then the graph g(x) = c • f(x) is the graph of f stretched vertically (away from the x-axis) by a factor of c

If c < 1, then the graph g(x) = c • f(x) is the graph of f compressed vertically (towards the x-axis) by a factor of c

Multiplying the entire function will stretch or compress a function (proportionally) towards or away from the x-axis, as expected (large numbers stretch, small numbers compress)

2(x+1)³

¼(x+1)³

(x+1)³

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

3.4: Graphs and TransformationsStretches & Compressions (Horizontal)

If the x of a function is multiplied by a number, it will stretch or compress the parent function horizontally

If c > 1, then the graph g(x) = f(c • x) is the graph of f compressed horizontally (towardsthe y-axis) by a factor of 1/c

If c < 1, then the graph g(x) = f(c • x) is the graph of f stretched horizontally (away from the y-axis) by a factor of 1/c

Multiplying the x of a function will stretch or compress a function (inversely) away from or towards the y-axis, opposite as expected (large numbers compress by the reciprocal, small numbers stretch by the reciprocal)

1 2 3 4 5–1–2–3–4–5 x

1

2

3

4

5

–1

–2

–3

–4

–5

y

(x+1)3

(2(x+1))3

(¼(x+1))3

3.4: Graphs and FunctionsAssignment

Page 1821-21, odd problems

Essential Question: In the equationg(x) = c[a(x-b)] + d

what do each of the letters do to the graph?

3.4: Graphs and TransformationsWe have our grand equation:

g(x) = c[a(x-b)] + d

Addition on the outside shifts the graph vertically (d)(as expected: positive == up, negative == down)

A negative on the outside flips the function vertically (c)

Multiplication on the outside stretches/compresses the graph vertically (c)(as expected: large numbers == stretch, small numbers == compress)

Addition on the inside of the parenthesis shifts the graph horizontally (b)(opposite as expected: positive == left, negative == right)

A negative on the inside of the parenthesis flips the graph horizontally (a)

Multiplication on the inside stretches/compresses the graph horizontally (a)(opposite: large numbers == compress by reciprocal,

small numbers == stretch by reciprocal)

3.4: Graphs and TransformationsWe have our grand equation:

g(x) = c[a(x-b)] + dOrder of application

a (horizontal reflection)a (horizontal stretch/compression)b (horizontal shift)c (vertical reflection)c (vertical stretch/compression)d (vertical shift)

3.4: Graphs and TransformationsExample set #1 – write a rule for the function

whose graph can be obtained from the given parent function by performing the given transformation.

Parent function: f(x) = x2

Transformations: shift 5 units left and up 4 units

Parent function: f(x) = Transformations: shift 2 units right, stretched vertically by a factor of 2, and shift up 2 units

g(x) = (x + 5)2 + 4 1

x

2( ) 2

( 2)

g x

x

3.4: Graphs and TransformationsExample set #2 – describe a sequence of

transformations that transform the graph of the parent function f into the graph of the function g. Do not graph the function.

a = -1/2, b = 6 (b is always its opposite), c = -1, d = 01) Horizontal reflection2) Horizontal stretch by a factor of 2 (horizontal stretches

are inverses)3) Horizontal shift right 6 units4) Vertical reflection

1( ) ( ) 3

2f x x g x x 1

Rewrite ( ) ( 6)2

g x x

3.4: Graphs and TransformationsExample set #2 – describe a sequence of

transformations that transform the graph of the parent function f into the graph of the function g. Do not graph the function.

1 3( ) ( )

4 2f x g x

x x

a = -2, b = 2, c = 3, d = 01) Horizontal reflection2) Horizontal compression by a factor

of 0.53) Horizontal shift right 2 units4) Vertical stretch by a factor of 3

3 3 1Rewrite ( )

2( 2) 2 ( 2)g x

x x

3.4: Graphs and FunctionsAssignment

Page 18223-41, odd problems

35 – 41: Ignore the directions Instead, identify the parent function and the

transformations that occurred to get to the transformed function.