WARM UP: Linear Equations multiple choice Learning Targets : TODAY IN CALCULUS… WARM UP: Linear Equations multiple choice Learning Targets : You will be able to identify the domain and range of a function using its graph or equation. You will be able to recognize even functions and odd functions using equations and graphs. You will be able to interpret and find formulas for piecewise defined functions. Independent practice

WARM UP: Which of the following is an equation of the line through (−3, 4) with slope 1 2 ? a) 𝑦−4= 1 2 (𝑥+3) b) 𝑦+3= 1 2 (𝑥−4) c) 𝑦−4=−2(𝑥+3) d) 𝑦−4=2(𝑥+3) 2. Which of the following is an equation of the vertical line through (−2, 4)? a) 𝑦=4 b) 𝑥=2 c) 𝑦=−4 d) 𝑥=−2 3. Which of the following is the x-intercept of the line 𝑦=2𝑥−5? a) 𝑥=−5 b) 𝑥=0 c) 𝑥= 5 2 d) 𝑥=− 5 2

1.2 FUNCTIONS FUNCTION: a relationship between two variables such that each independent value corresponds to exactly one dependent value No repeated x values Exactly one input paired with one output Domain: independent values Range: dependent values FUNCTION NOTATION: 𝑦=𝑓(𝑥) y is a function of x. y is dependent on x. EXAMPLE: 𝑃 𝑡 =−4.9 𝑡 2 −𝑡 P is a function of t. P is dependent on t. DOMAIN 𝒙 Independent variable input Function Machine output Dependent variable 𝒚 RANGE

EXAMPLE: Which of the equations below define y as a function of x? 1.2 FUNCTIONS EXAMPLE: Which of the equations below define y as a function of x? a. 𝑥+𝑦=1 b. 𝑥 2 + 𝑦 2 =1 c. 𝑥 2 +𝑦=1 c. 𝑥+ 𝑦 2 =1 X Y X Y − 𝑥 − 𝑥 𝒚=𝟏−𝒙 − 𝑥 2 − 𝑥 2 𝑦 2 =1− 𝑥 2 𝒚=± 𝟏− 𝒙 𝟐 −1 1 1, −1 −2 −1 1 2 3 2 1 −1 FUNCTION NOT A FUNCTION X Y −2 −1 3 , − 3 2 , − 2 1 X Y − 𝑥 2 − 𝑥 2 𝒚=𝟏− 𝒙 𝟐 − 𝑥 − 𝑥 𝑦 2 =1−𝑥 𝒚=± 𝟏−𝒙 −2 −1 1 2 −3 1 FUNCTION NOT A FUNCTION

1.2 FUNCTIONS PRACTICE: Which of the equations below define y as a function of x? a. 𝑥−𝑦=1 b. 𝑥 2 + 𝑦 2 =4 c. 𝑥 2 −𝑦=0 c. 𝑥+ 𝑦 2 =2 − 𝑥 − 𝑥 −𝑦=−𝑥+1 −1 −1 𝒚=𝒙−𝟏 − 𝑥 2 − 𝑥 2 𝑦 2 =4− 𝑥 2 𝒚=± 𝟒− 𝒙 𝟐 NOT A FUNCTION FUNCTION − 𝑥 2 − 𝑥 2 −𝑦=− 𝑥 2 −1 −1 𝒚= 𝒙 𝟐 − 𝑥 − 𝑥 𝑦 2 =2−𝑥 𝒚=± 𝟐−𝒙 NOT A FUNCTION FUNCTION

1.2 FUNCTIONS: VERTICAL LINE TEST PRACTICE: Identify which graphs are functions using the Vertical Line Test. 1. 2. 3. 4. 5. 6. FUNCTION FUNCTION NOT A FUNCTION NOT A FUNCTION FUNCTION FUNCTION

1.2 DOMAIN AND RANGE EXAMPLES: 1. 2. 3. Function Machine input output 𝒙 𝒚 DOMAIN RANGE Independent variable Dependent variable −𝟐, ∞ [𝟎, ∞) DOMAIN: RANGE: DOMAIN: RANGE: [−𝟐,𝟐] [𝟎,𝟐] DOMAIN: where the graph exists on the x-axis. RANGE: where the graph exists on the y-axis. DOMAIN: RANGE: (−∞, ∞) [−𝟏,𝟏]

1.2 DOMAIN AND RANGE EXAMPLE 1: Find the domain and range, then sketch the graph of the function. 1. 𝑓 𝑥 = 𝑥 2 −9 2. 𝑓 𝑥 =− −𝑥 3. 𝑓 𝑥 = 4 −𝑥 4. 𝑓 𝑥 =1+ 1 𝑥 2 DOMAIN: RANGE: DOMAIN: RANGE: (−∞,∞) [−𝟗, ∞) (−∞,𝟎] (−∞, 𝟎] DOMAIN: RANGE: DOMAIN: RANGE: (−∞,𝟎] [𝟎,∞) (−∞,𝟎)∪(𝟎, ∞) (𝟏, ∞)

1.2 DOMAIN AND RANGE EXAMPLE 2: Use a graphing utility to graph the function. Then determine the domain and range of the function. 1. 𝑓 𝑥 = 3𝑥+2 𝑥<0 2−𝑥 𝑥≥0 2. 𝑓 𝑥 = 𝑥 2 1−𝑥 3. 𝑓 𝑥 =5 𝑥 3 +6 𝑥 2 −1 4. 𝑓 𝑥 = 9− 𝑥 2 DOMAIN: RANGE: DOMAIN: RANGE: (−∞,∞) (−∞,𝟐] (−∞, 𝟏)∪(𝟏, ∞) −∞,−𝟒 ∪[𝟎, ∞) DOMAIN: RANGE: DOMAIN: RANGE: (−∞,∞) [−𝟑,𝟑] [𝟎,𝟑]

1.2 SYMMETRY-EVEN AND ODD FUNCTIONS A function 𝑦=𝑓(𝑥) is an… EVEN FUNCTION if 𝑓 −𝑥 =𝑓(𝑥) ODD FUNCTION if 𝑓 −𝑥 =−𝑓(𝑥) (−𝑥, 𝑦) (𝑥, 𝑦) (𝑥, 𝑦) (−𝑥, −𝑦) Symmetry: about the y-axis Symmetry: about the origin

1.2 SYMMETRY-EVEN AND ODD FUNCTIONS EXAMPLE: Determine whether the function is even, odd, or neither. 1. 𝑓 𝑥 = 𝑥 2 −1 2. 𝑓 𝑥 = 𝑥 5 − 𝑥 3 −𝑥 3. 𝑓 𝑥 = 𝑥 2 −2𝑥−1 4. 𝑓 𝑥 =1+ 1 𝑥 2 EVEN ODD EVEN NEITHER

1.2 SYMMETRY-EVEN AND ODD FUNCTIONS EXAMPLE 2: Determine whether the function is even, odd, or neither without graphing. 1. 𝑓 𝑥 = 𝑥 2 −1 2. 𝑓 𝑥 = 𝑥 5 − 𝑥 3 −𝑥 3. 𝑓 𝑥 = 𝑥 2 −2𝑥−1 4. 𝑓 𝑥 =1+ 1 𝑥 2 𝑓 −𝑥 = (−𝑥) 2 −1 𝑓 −𝑥 = 𝑥 2 −1 𝑓 −𝑥 =𝑓(𝑥) 𝑓 −𝑥 = (−𝑥) 5 − −𝑥 3 −(−𝑥) 𝑓 −𝑥 =− 𝑥 5 + 𝑥 3 +𝑥 𝑓 −𝑥 =−𝑓(𝑥) EVEN ODD 𝑓 −𝑥 = (−𝑥) 2 −2 −𝑥 −1 𝑓 −𝑥 = 𝑥 2 +2𝑥−1 𝑓 −𝑥 =1+ 1 (−𝑥) 2 𝑓 −𝑥 =1+ 1 𝑥 2 𝑓 −𝑥 =𝑓(𝑥) NEITHER EVEN

1.2 PIECEWISE FUNCTIONS Different functions defined in certain domains. EXAMPLE: 𝑓 𝑥 = −𝑥, 𝑥<0 𝑥 2 , 0≤𝑥≤1 1, 𝑥>1

1.2 SYMMETRY-EVEN AND ODD FUNCTIONS EXAMPLE 2: Write a piecewise formula for the function. 𝑓 𝑥 = −𝑥+2, 0<𝑥≤2 − 1 3 𝑥+ 5 3 , 2<𝑥≤5 (2, 1) 2, 1 5, 0 𝑦=𝑚𝑥+𝑏 𝑚= −1 3 0=− 1 3 5 +𝑏 0=− 5 3 +𝑏 𝑏= 5 3 𝒚=− 𝟏 𝟑 𝒙+ 𝟓 𝟑 0, 2 2, 0 𝑦=𝑚𝑥+𝑏 𝑚= −2 2 𝑏=2 𝒚=−𝒙+𝟐 =−1

HOMEWORK #2: Pg.19: QR: 1-11odd E: 1-19odd, 21-30, 31-35odd, 37-40, 44-45 If finished, work on other assignments: HW #1: Pg.9: QR:1-9odd E:1-23odd, 27-33odd, 37, 41, 53