Section 1.4Shifting, Reflecting, and Stretching

Graphs

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Do NowGraph the following functions

a. f(x) = c ; c represents a constantb. f(x) = xc. f(x) = |x|d. f(x) = √xe. f(x) = x2

f. f(x) = x3

We will call these parent functions.

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Do Nowa. f(x) = c

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Do Nowb. f(x) = x

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Do Nowc. f(x) = |x|

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Do Nowd. f(x) = √x

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Do Nowe. f(x) = x2

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Do Nowf. f(x) = x3

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Activity 1Complete Activity 1

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Activity 11. f(x) = x2 2. h(x) = x2 + 2 3. g(x) = (x - 2)2

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Vertical and Horizontal ShiftsActivity 2

Graph the following functions.

a. f(x) = |x| + 2b. f(x) = x2 - 3c. f(x) = |x - 4|d. f(x) = (x - 3)2

e. f(x) = √(x + 2)f. f(x) = |x | - 1

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Vertical and Horizontal ShiftsActivity 2

Graph the following functions.

a. f(x) = |x| + 2

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Vertical and Horizontal ShiftsActivity 2

Graph the following functions.

b. f(x) = x2 - 3

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Vertical and Horizontal ShiftsActivity 2

Graph the following functions.

c. f(x) = |x - 4|

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Vertical and Horizontal ShiftsActivity 2

Graph the following functions.

d. f(x) = (x - 3)2

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Vertical and Horizontal ShiftsActivity 2

Graph the following functions.

e. f(x) = √(x + 2)

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Vertical and Horizontal ShiftsActivity 2

Graph the following functions.

f. f(x) = |x| - 1

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Vertical and Horizontal ShiftsLet c be a positive real number. Vertical and Horizontal shifts in the graph of y = f(x) and represented as follows.

1. Vertical shifts c units upward: h(x) = f(x) + c2. Vertical shifts c units downward: h(x) = f(x) - c3. Horizontal shifts c units to the right: h(x) = f(x-c)4. Horizontal shifts c units to the left: h(x) = f(x+c)

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Exampleg(x) = (x+2)3 - 1

The parent function is:

f(x) = x3

The transformation is:

Left 2 units,

Down 1 unit

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1. g(x) = (x-1)3 + 2

The parent function is:

f(x) = x3

The transformation is:

Right 1 unit,

Up 2 units

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2. g(x) = √(x-3) + 1

The parent function is:

f(x) = √x

The transformation is:

Right 3 units,

Up 1 unit

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3. g(x) = (x-2)2 - 3

The parent function is:

f(x) = x2

The transformation is:

Right 2 units,

Down 3 units

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4. g(x) = |x+1| - 2

The parent function is:

f(x) = |x|

The transformation is:

Left 1 unit,

Down 2 units

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Activity 3

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1. f(x) = x2

g(x) = -x2

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2. f(x) = √x

g(x) = -√x

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3. f(x) = √x

g(x) = √-x

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4. f(x) = x3

g(x) = (-x)3

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Reflection in the Coordinate Axes:Reflections in the coordinate axes of the graph of y = f(x) and represented as follows.

1. Reflections in the x-axis: h(x) = -f(x)2. Reflections in the y-axis: h(x) = f(-x)

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PracticeIdentify the parent function and describe the transformation.

1. g(x) = -(x+6)2 + 72. g(x) = -√(x-1) + 83. g(x) = √(3-x)

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Practice1. g(x) = -(x+6)2 + 7

Parent: f(x) = x2

Transformation:

-Reflection over x-axis

-Horizontal shift left 6 units

-Vertical shift upward 7 units

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Practice2. g(x) = -√(x-1) + 8

Parent: f(x) = √x

Transformation:

-Reflection over x-axis

-Horizontal shift right 1 unit

-Vertical shift upward 8 units

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Practice3. g(x) = -|3 - x| - 2

Parent: f(x) = |x|

Transformation:

-Reflection over x-axis

-Horizontal shift right 3 units

-Vertical shift downward 2 units

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Activity 4Graph each of the functions on the same graph. Include x- and y-intercepts.

f(x) = x2 - 3

g(x) = 16x2 - 3

h(x) = ¼ x2 - 3

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Activity 4Graph each of the functions on the same graph. Include x- and y-intercepts.

f(x) = x2 - 3

g(x) = 16x2 - 3

h(x) = ¼ x2 - 3

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Horizontal Stretch and Shrink● y-intercept does not change

y = f(cx)

1. Horizontal shrink if c >1 (narrower)2. Horizontal stretch if 0 < c < 1 (wider)

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Activity 5Graph each of the functions on the same graph. Include x- and y-intercepts.

f(x) = x2 - 2

g(x) = ⅓ (x2 - 2)

h(x) = 2 (x2 - 2)

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Activity 5Graph each of the functions on the same graph. Include x- and y-intercepts.

f(x) = x2 - 2

g(x) = ⅓ (x2 - 2)

h(x) = 2 (x2 - 2)

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Vertical Stretch and Shrink● x-intercepts do not change

y = cf(x)

1. Vertical shrink if c >1 (narrower)2. Vertical stretch if 0 < c < 1 (wider)

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Practicea. Identify the parent function, f(x)b. Use the function to write g in terms of fc. Describe the sequence of transformations from f to g.

1. g(x) = 2 - (x + 5)2

2. g(x) = 3(x - 2)3

3. g(x) = -2|x - 1| - 44. g(x) = - ½ √(x+3) - 1

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