Bellwork: Graph each line: 1. 3x – y = 6 2. Y = -1/2 x + 3 Y = -2 If f(x) = 3x2 – 9, find each. 4. f(-2) 5. f(3) 6. f(4a) Algebra II

Evaluating, Graphing, and Writing Piecewise Functions Algebra II

Piecewise Functions A combination of equations each corresponding to a given domain. Algebra II

Example 1 f(x)= { 2x2 – 2 if x < 1 x + 4 if x ≥ 1 Evaluate each. = (3) + 4 = 7 = 2(-4)2 – 2 = 2(16) – 2 = 30 = (1) + 4 = 5 Algebra II

Example 2 x − 1 if x ≤ -1 f(x)= { (x − 3)2 if x > -1 Evaluate each. = (-3) − 1 = -4 = (-1) − 1 = -2 = ((2) – 3)2 = (-1)2 = 1 Algebra II

Example 3 g(x)= { 3x – 2 if x < -3 4 if -3 ≤ x < 5 Evaluate each. 1. g(5) 2. g(-2) 3. g(-8) = 2(5)2 – 3 = 2(25) – 3 = 47 = 4 = 3(-8) – 2 = -24 – 2 = -26 Algebra II

Find each. f(½) f(2) f(-5) -3/2 9 -10 4. g(½) 5. g(0) 6. g(-1) 3 2 2x if x < -2 x – 2 if -2 ≤ x < 2 2x2 + 1 if x ≥ 2 f(x)= { x + 3 if x < ½ 2x – 1 if x ≥ ½ g(x)= { Evaluate each piecewise function for the given values. f(½) f(2) f(-5) -3/2 9 -10 4. g(½) 5. g(0) 6. g(-1) 3 2 Algebra II

Graphing: Example 1 f(x)= { x – 3 if x < 2 -½x + 1 if x ≥ 2 Graph the piecewise function: f(x)= { x – 3 if x < 2 -½x + 1 if x ≥ 2 x – 3 if x < 2 -½x + 1 if x ≥ 2 Algebra II

Graphing: Example 2 f(x)= { 3x + 1 if x ≤ -1 x + 2 if x > -1 Graph the piecewise function: f(x)= { 3x + 1 if x ≤ -1 x + 2 if x > -1 3x + 1 if x ≤ -1 x + 2 if x > -1 Algebra II

Graphing: Example 3 f(x)= { -2x if x < -2 ⅔x – 1 if x ≥ -2 Graph the piecewise function: f(x)= { -2x if x < -2 ⅔x – 1 if x ≥ -2 -2x if x < -2 ⅔x – 1 if x ≥ -2 Algebra II

Graphing: Example 4 f(x)= { -3 if x < 0 -x – 1 if x ≥ 0 Graph the piecewise function: f(x)= { -3 if x < 0 -x – 1 if x ≥ 0 -3 if x < 0 -x – 1 if x ≥ 0 Algebra II

Graphing: Example 5 f(x)= { Graph the piecewise function: 2x – 1 if x < -2 3 if -2 ≤ x ≤ 2 -¼x if x > 2 2x – 1 if x < -2 3 if -2 ≤ x ≤ 2 -¼x if x > 2 Algebra II

Find each. Graph: Evaluate each given the piece-wise function: f(x)={ 1/3x + 1 if x < -3 3x if -3 ≤ x < 2 2 if x > 2 4x2 + 5 if x < -3 3x – 2 if -3 ≤ x < 5 –3 if x ≥ 5 f(x)= { Algebra II

Writing a Piecewise Function Write the equation for each piece of the function Write the domain for each piece of the function use inequality notation to represent the domain in each piece Algebra II

function that is graphed. Example 1 Write the piecewise function that is graphed. (3 – 1) = -2 = -2 (-3 + 2) 1 y – 1 = -2(x + 2) y = 2x – 3 (-5 + 4) = -1 (2 + 1) 3 y + 4 = -⅓(x + 1) y = -⅓x – 13/3 f(x) = { 2x – 3 if x ≤ -1 -⅓x – 13/3 if x > -1 Algebra II

function that is graphed. Example 2 Write the piecewise function that is graphed. (6 – 5) = 1 (1 + 4) 5 y – 6 = ⅕(x – 1) y = ⅕x + 29/5 (0 + 1) = 1 = 1 (3 – 2) 1 y – 0 = 1(x + 3) y = x – 3 f(x) = { ⅕x + 29/5 if x < 1 x – 3 if x ≥ 1 Algebra II

function that is graphed. Example 3 Write the piecewise function that is graphed. (2 + 1) = 3 = -1 (-3 - 0) -3 y + 1 = -1(x + 0) y = -x – 1 (2 – 0) = 2 = 2 (1 – 0) 1 y – 0 = 2(x + 0) y = 2x f(x) = { -x – 1 if x ≤ 0 2x if x > 0 Algebra II

function that is graphed. Example 4 Write the piecewise function that is graphed. (5 – 3) = 2 = 1 (-3 + 5) 2 y – 5 = 1(x + 3) y = x + 8 Slope is 0 horizontal line (1 – 0) = 1 = -½ (1 – 3) -2 y = 3 y – 0 = -½(x – 3) y = -½x + 3/2 f(x) = { x + 8 if x < -3 3 if -3 ≤ x < 1 -½x + 3/2 if x ≥ 1 Algebra II

Closure: Graph the piece-wise function: Write a piece-wise function for this graph: f(x) = { x + 8 if x < -3 3 if -3 ≤ x < 1 -½x + 3/2 if x ≥ 1 Algebra II