Overview of Chapter 3

• • • • • • Slope Y=mx+b Line of best fit Barbie Bungee Point-slope equation Systems of Equations  Graphing  Elimination  Substitution

Recursive Explicit Linear Equations

3.1

Goal

• Given one form if a linear equation, convert it to one of the other forms.

Remember when….?

• What does the graph of an arithmetic sequence look like?

• We know there is another way calculate linear equations other than knowing the previous term right?

• Recursions are ONE type of equation. We will learn the other EQUIVALENT forms.

• 𝑈 𝑛 = 𝑈 𝑛−1 + 𝑑

Recursive

• Find the next term by looking at the previous

Explicit

• 𝑈 𝑛 = 𝑎 ∙ 𝑛 + 𝑏 • • b = Y-intercept. The initial value ( 𝑈 𝑜 ) recursion. in the a= Slope (d in the recursion) • Nice because you do not have to know the previous term to find the next.

• y=mx+b • m=slope • b=y-intercept • Linear uses x and y.

Linear

So…

You will be given one of the three types just discussed, and will be asked to write it in a different way.

Example 1

• Given the recursion 𝑈 0 = 2, 𝑈 𝑛 = 𝑈 𝑛−1 + 6 1. Find the explicit formula 2. Find 𝑈 22 using the explicit 3. Find n such that 𝑈 𝑛 = 86

Example 1: answers

1.

𝑈 𝑛 = 6𝑛 + 2 slope initial value 2. 𝑈 22 𝑈 22 = 6 22 + 2 = 134 3. 86=6n+2 n=14

You try!

• Given the recursion 𝑈 0 = 5, 𝑈 𝑛 = 𝑈 𝑛−1 − 2 1. Find the explicit formula 2. Find 𝑈 8 using the explicit 3. Find n such that 𝑈 𝑛 = −35

Example 2

• You spend $2 a day on lunch and have $17 left after today.

Write a recursive and explicit formula modeling this situation. So: Recursive: 𝑈 1 = 17 𝑎𝑛𝑑 Explicit: 𝑈 𝑛 = −2𝑛 + 17 𝑈 𝑛 = 𝑈 𝑛−1 − 2

Example 3

• Write an equation in the form y=a +bx of the line the passes through the points of an arithmetic sequence with 𝑈 0 = 20 and a common difference of -5.7. • Answer: 𝑈 0 = 𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 = 𝑎 = 20 -5.7=slope=b

y=20-5.7x

• • 3.1

Problems: 1,4,5

Homework