Module 1 Day 1 Evaluating Functions

Evaluating functions: Plug the value in for the variable Evaluate the function given the value Example 𝑓(𝑥) = 𝑥 2 +5 find 𝑓(3) 𝑓(3)= 14

Evaluating functions: Example: 𝑓 𝑥 = −3𝑥 2 −2𝑥 find 𝑓( 𝑥 2 ) 𝑓 𝑥 2 =−3 𝑥 4 −2 𝑥 2 Example: g 𝑥 =12 𝑥 2 +6 find g(−2𝑥) 𝑓 −2𝑥 =48 𝑥 2 +6 Example: h 𝑥 =3𝑥+5 find h(2𝑥+1) ℎ(2𝑥+1)=6𝑥+11

Arithmetic Sequences Adding or subtracting to get the next term The pattern is called the common difference.

Geometric Sequences Multiplying to get the next term The pattern is called the common ratio

Explicit Formula: Arithmetic Explicit is used to find any term in the sequence 𝑦 = 𝑚𝑥 + 𝑏 or 𝑓(𝑥)= 𝑚𝑥 + 𝑏 𝑚 = common difference 𝑏 = zero term Plug in values to equation Example: Write the explicit equation: 2, −4, −10, −16… 𝑓(𝑥)= −6𝑥 +8

Explicit Formula: Geometric Explicit is used to find any term in the sequence 𝑓(𝑥)= 𝑎∙ 𝑏 𝑥 𝑎 = zero term 𝑏= common ratio Example: Write the explicit equation: 6, -24, 96, -384… 𝑓 𝑥 = −3 2 −4 𝑥

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Explicit Formula: Geometric Example: Write the explicit equation given: 2, -4, 8, -16….. And f(1)= 2

Explicit Formula: Geometric Example: Write the explicit equation given 𝑓(1) = 2, common ratio is −4 𝑓 𝑥 = −1 2 ∙ −4 𝑥 𝑜𝑟 𝑓 𝑥 =2∙ −4 𝑥−1 Find 𝑓(5) 𝑓(5)= 512

Check For Understanding: If f(x)= 2x2 -3x – 4, find f(-4) f(-4)= 40 Find the common difference: 8, 2, -4, -10… CD= -6 Write the explicit equation from the sequence above. y= -6x +14 What is the 10th term? -46 Write the recursive formula: f(x)= f(x-1) - 6

Check For Understanding: Find the common ratio: -3, 6, -12, 24… CR= -2 Write the explicit equation from the sequence above. 𝑓 𝑥 =−3∙ −2 𝑥−1 𝑜𝑟 𝑓 𝑥 = 3 2 ∙ (−2) 𝑥 What is the 5th term? -48 Write the recursive formula: f(x)= f(x-1) ∙ -2

Recursive Formulas: Arithmetic and Geometric Recursive is used to find the next term from the previous term The previous term is represented by: f(x-1) or any other variable 𝑓(𝑛) = 𝑓( 𝑛 – 1 ) +/− Common Difference or 𝑓(𝑛) = 𝑓( 𝑛 – 1 ) ∙ Common Ratio 𝑛 = the term number Example: Determine the 2nd term if 𝑓(1) = 2 and the recursive formula given is 𝑓(𝑛) = 𝑓(𝑛−1) − 6 Answer: 𝑓(2)= −4