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Objectives: How do the concepts from Ch. 4 relate to the real world? What are 1st and 2nd differences, and how can you use them to identify different families of functions?

Linear and Non-linear Situations What do these 2 words mean? What kinds of situations in real life produce graphs that are straight lines?

How can you tell that the graph of Example: y How can you tell that the graph of this table would be a straight line? 20 15 +6 x y 1 10 +3 +2 1 4 +3 +3 2 7 5 +3 +3 3 10 +2 +6 +3 5 16 +1 1 2 3 4 5 6 7 8 9 10 x

Non-Linear Functions They do not make straight lines because they do not change at a constant rate. They are the main theme of Algebra 2 We’re beginning to study them right now.

Some friends form a Facebook group and start posting on each others’ walls. If everyone posts twice per day on everyone’s wall (including their own), how many total posts will be made per day? Step 1: Try it with simple examples 1 Person 2 People 3 People Writes __ posts Writes __ posts Writes 4 posts Writes __ posts Writes 6 posts Writes 6 posts 2 4 6 TOTAL: 2 TOTAL: TOTAL: (4)(2) (6)(3)

1 Person 2 People 3 People Writes __ posts Writes __ posts Writes 4 6 TOTAL: 2 TOTAL: TOTAL: (4)(2) (6)(3) Step 2: Look for a pattern in the answers. If there are 30 people If there are x people

Introduction to Quadratic Functions The equation for the Facebook example was f(x) = 2x2 Parent function is f(x) = x2 Examples of Quadratic functions: f(x) = (x – 4)2 f(x) = x2 + 3x + 2 Any function where the highest power of x is x2

Are These Quadratic Functions? f(x) = x2 f(x) = 9 – x2 + 5x f(x) = x2 + 8x3 Yes, it’s the parent function. Yes No, the highest power of x is x3.

Rates of change in the Facebook example 1 Person 2 People 3 People Writes __ posts Writes __ posts Writes 4 posts Writes __ posts Writes 6 posts Writes 6 posts 2 4 6 TOTAL: 2 TOTAL: TOTAL: (4)(2) = 8 (6)(3) = 18 ?? + 6 ?? + 10 Would the graph of this situation be a straight line? Why or why not?

1st and 2nd Differences y = 3x + 1 y = x2 x y x y 1st Diff. 1st Diff. -3 -8 -3 9 -2 -5 -2 4 -1 -2 -1 1 1 1 4 1 1 2 7 2 4 3 10 3 9

The Shape of Quadratic Functions What shape of graph do you get when you have a constant 2nd difference?

1st Difference starts at +1. 1st Difference is +1 2nd Difference is +1 1st Difference starts at +1. 1st Difference is +1 +3 +1 +1 +2 +1 +1 +1 +1

1st Difference starts at +1. 1st Difference is +1 2nd Difference is +1 1st Difference starts at +1. 1st Difference is +1 +3 +1 -2 +2 +1 -1 +1 +1 +1 +0 +1 +1

1st Difference starts at +1. 1st Difference is +1 2nd Difference is +1 1st Difference starts at +1. 1st Difference is +1 +3 +1 -2 +2 +1 -1 +1 +1 +1 +0 +1 +1 2nd Differences: 1st Differences: 1, 1, 1, 1, 1, 1 1st Differences: -2, -1, 0, 1, 2, 3

Very Common Example of Quadratic Functions Where in daily life do you see something that looks like a quadratic graph?