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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….