Section 3.5Piecewise Functions Day 2

Standard: MM2A1 ab

Essential Question: How do I graph piecewise functions and given a graph, can I write the rule?

y = x + 1

x y

2

1

0

-1

-2

-3

3

2

1

01

2

y = 3 – x

x y

2

3

4

10

-1

2,3

2,1)(.1

xifx

xifxxg

2,3

2,1)(.1

xifx

xifxxg

Identify the following:

a) Domain: ____________

b) Range: _______________

c) x-intercept: ____________

d) y-intercept: ____________

e) Increasing: ____________

f) Decreasing: ____________

g) Point of Discontinuity: ______________

All reals

All reals

(-1,0) and (3, 0)

(0, 1)

-1< x ≤ 2

x ≤ -1 OR x > 2

x = 2

3,4

23,2)(.2

2

xif

xifxxxh

y = x2 + 2x y = 4

x y

-3

-2

-1

0

1

2

x y

-3

-4

-5

3

0

-1

03

8

44

4

3,4

23,2)(.2

2

xif

xifxxxh

(a). Domain: _______________

(b). Range: ________________

(c). x-intercept: ______________

(d). y-intercept: ______________

(e). Increasing: ______________

(f). Decreasing: ______________

(g). Points of discontinuity: ___________

(h). Interval over which the function is constant: ___________

x < 2

-1 ≤ y < 8(0, 0) and (-2, 0)

(0, 0)-1 ≤ x < 2-3 ≤ x ≤ -1

x = -3

x < -3

5,6

54,2

4,5

)( 2

xifx

xifx

xifx

xp

(3). Find each of the following:

a. p(4) = b. p(-4) =

c. p(9) =

d. p(-6) =

2 + (4)2 = 18

5 – (-4) = 9

9 + 6 = 15

5 – (-6) = 11

____________,__________

___________,__________)(

if

ifxf(4).

12

3 xy

Blue Ray: m = 3/2; y-int = 1

12

3x

Domain: x 0

x 0

2y

Red Ray: m = 0

Domain: x < 0

2 x < 0

____________,__________

____________,__________

___________,__________

)(

if

if

if

xg(5).

Red Segment: m = 0; y-int = -2Equation: y = -2Domain: 0 x < 3

Blue Segment: m = 0Equation: y = 0Domain: 3 x < 5

Green Segment: m = 0Equation: y = 2Domain: 5 x 8

0 x < 3-23 x < 505 x 82

6. Find the intervals over which the functions above are constant:

a. f(x):

b. g(x):

x < 0

0 x < 3 3 x < 5 5 x 8

____________,__________

____________,__________

___________,__________

)(

if

if

if

xg(7).

Evaluate each of thefollowing for y = g(x) in problem #5.

a. g(2) = _______

b. g(3) = _______

c. g(6) = _______

0 x < 3-23 x < 505 x 8

2

-2

0

2

To write an absolute value function as a piecewise function, following the steps below:

Step 1: Identify the vertex of the functionStep 2: Determine if the function opens upward or downward.Step 3: Identify the rate of change to the right and left of the vertex.Step 4: Use the rate of change for each portion of the graph to write a linear equation.Step 5: Simplify each linear equation and write your solution as a piecewise defined function.

8. f(x) = x + 4

Vertex is at __________

Graph opens _________

Slope to the right is ____, slope to the left is ____.

Right: x > _____ Left: x < _____

y = 1(x + 4)y = x + 4

(-4, 0)

- 4- 4

y = -1(x + 4)y = -x – 4

(-4, 0)

Up

1-1

8. f(x) = x + 4

Right: x > _____ Left: x < _____

y = 1(x + 4)y = x + 4

(-4, 0)

- 4- 4

y = -1(x + 4)y = -x – 4

)(xf

x + 4 if x -4

-x – 4 if x < -4

9. Write the function g(x) = -2|x – 3 | + 7 as a piecewise function.

Vertex is at __________

Graph opens _________

Slope to the right is ____,slope to the left is ____.

Right: x > _____ Left: x < _____

(3, 7)

Down

(3, 7)

-2

2

3 3

y = -2(x – 3) + 7y = -2x + 6 + 7y = -2x + 13

y = 2(x – 3) + 7 y = 2x – 6 + 7y = 2x + 1

9. Write the function g(x) = -2|x – 3 | + 7 as a piecewise function.

Right: x > _____ Left: x < _____

(3, 7)

3 3

y = -2(x – 3) + 7y = -2x + 6 + 7y = -2x + 13

y = 2(x – 3) + 7 y = 2x – 6 + 7y = 2x + 1

)(xf

-2x + 13 if x 3

2x + 1 if x < 3

10. Write the function h(x) = 4|x| – 5 as a piecewise function.

Vertex is at __________

Graph opens _________

Slope to the right is ____,slope to the left is ____.

Right: x > _____ Left: x < _____

(0, -5)

4

y = 4(x) – 5 y = 4x – 5

(0, -5)

Up

-4

0 0

y = -4(x) – 5 y = -4x – 5

10. Write the function h(x) = 4|x| – 5 as a piecewise function.

Right: x > _____ Left: x < _____

y = 4(x) – 5 y = 4x – 5

(0, -5)

0 0

y = -4(x) – 5 y = -4x – 5

)(xf

4x – 5 if x 0

-4x – 5 if x < 0