Section 3.5 Piecewise 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  xy y = 3 – x xy

Identify the following: a) Domain: ____________ b) Range: _______________ c) x-intercept: ____________ d) y-intercept: ____________ e) Increasing: ____________ f) Decreasing: ____________ g) Point of Discontinuity: ______________ All reals (-1,0) and (3, 0) (0, 1) -1< x ≤ 2 x ≤ -1 OR x > 2 x = 2

y = x 2 + 2x y = 4 xy xy

(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

(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   = 15 5 – (-6) = 11

(4). Blue Ray: m = 3/2; y-int = 1 Domain: x  0 x  0 Red Ray: m = 0 Domain: x < 0 x < 0

(5). Red Segment: m = 0; y-int = -2 Equation: y = -2 Domain: 0  x < 3 Blue Segment: m = 0 Equation: y = 0 Domain: 3  x < 5 Green Segment: m = 0 Equation: y = 2 Domain: 5  x  8 0  x <  x <  x  8 2

6. Find the intervals over which the functions above are constant: a. f(x): b. g(x): x < 0 0  x < 33  x < 5 5  x  8

(7). Evaluate each of the following for y = g(x) in problem #5. a.g(2) = _______ b. g(3) = _______ c. g(6) = _______ 0  x <  x <  x 

To write an absolute value function as a piecewise function, following the steps below: Step 1: Identify the vertex of the function Step 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 y = -1(x + 4) y = -x – 4 (-4, 0) Up 1

8. f(x) =  x + 4  Right: x > _____ Left: x < _____ y = 1(x + 4) y = x + 4 (-4, 0) - 4 y = -1(x + 4) y = -x – 4 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) y = -2(x – 3) + 7 y = -2x y = -2x + 13 y = 2(x – 3) + 7 y = 2x – y = 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) + 7 y = -2x y = -2x + 13 y = 2(x – 3) + 7 y = 2x – y = 2x x + 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 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 4x – 5 if x  0 -4x – 5 if x < 0