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##### An Application of Slope

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**An Application of Slope**Mike Haley**Objective**• Utilize our graphing skills to make decisions • Understand the equation y = mx + b, where m is slope and b is the y-intercept**Abe and Brian**• Two twin brothers (Abe and Brian) are trying to make decisions about when they should retire. • How many years must Brian (who retires at 66) wait until his Social Security compensation is greater than what his brother Abe (who retires at 62) receives?**Social Security Benefits**Your guess:_________****Graph Abe’s Compensation**• Age 62 • Slope is a rate! • The slope is dollars per month • x-axis is number of months • y-axis is compensation in dollars**Graph Brian’s Compensation**• Age 66 • Slope is still a rate! • x-axis is number of months • y-axis is compensation in dollars**Graphing Abe and Brian’s Compensation**• Comment on these two different slopes:** • A slope of 1200 is steeper than a slope of 900. • The problem that we face is more complex than who makes more per month. We are asking about the total Social Security compensation the brothers receive.**Let’s Graph This!**• How much compensation has Abe received by Brian’s 66th birthday? ** • $43200 (48 months * 900 $/month) • Graph Brian’s age 66 retirement. (Origin) • Graph Abe’s age 62 retirement. • What does their intersection signify?**The Final Answer**• How old are the brothers before Brian’s compensation is greater than Abe’s?** • 144 months or 12 years!**Using a Different Origin**• What if we use Abe’s retirement at 62 as the origin? • Do the equations of the lines change? • What would the new equations be? • Does the time we have to wait for the optimal compensation from the later retirement change? • In the equation y = mx + b, how does b influence the function?****Adjusting Slope (m)**• What if Abe’s Social Security check had stopped coming on his 66th birthday? What would the compensation graph look like?** • What does a horizontal slope mean?** • What would a vertical slope mean? Is it possible?****Looking Ahead**• y = mx + b • where m is slope, and b is the y-intercept • Beyond linear equations….