\[ \large\begin{array}{rcl} x_0&=& {(x +m y) - m b\over m^2 + 1}\\ & & \\ y_0 &=& {m(x + m y)+b\over m^2 + 1}\\ \end{array} \]

1. Determine the perpendicular line.
2. Solve for \( x_0 \) of the pivot point common to both the line and the perpendicular.
3. Solve for \( \large y_0 \) of the pivot point

\[ \large\begin{array}{rcl} y &=& 2x + 2\\ & & \\ y &=& - {1\over 2}x + b_p \\ 5 &=& -{8\over 2} + b_p\\ 9 &=& b_p\\ & & \\ 2 x_0 + 2 &=& -{1\over 2} x_0 + 9\\ {5\over 2} x_0 &=& 7\\ x_0 &=&{14\over 5} = 2.8\\ & & \\ y_0 &=& 2 \times 2.8 + 2 = 7.6\\ \end{array} \]

\[ \large\begin{array}{rcl} Dist & = &\sqrt{(x_0 - x)^2 + (y_0 - y)^2 }\\ & & \\ Dist & = &\sqrt{(2.8 - 8)^2 + (7.6 - 5)^2 }\\ &=& \sqrt{5.2^2 + 2.6^2} = \sqrt{33.8}\\ &=& 5.8 \\ \end{array} \]

• Growth of bamboo, hair, nails
• Rate of production
• Distance travelled
• Burn rate
• Conversions
• Exchange rates

\[ \large Revenue = Cost \]

The Federal Helium Reserve held about \( \large 16 \times 10^9 \) cubic feet of helium in 2010 and is being depleted by about \( \large 2.1 \times 10^9 \) cubic feet each year.

• Give a linear equation for the remaining federal helium reserves, R, in terms of t, the number of years since 2010.

• What will the helium reserves be in 2015?

• When will the Federal Helium Reserve be depleted if the trend continues?

Housing prices

• In which state have home values increased at a higher rate?
• If these trends continued, what was the median home value in Indiana in 2010?
• In which year would we expect the two states’ median house values to be the same ?

In 2012, a guest house had 1,001 visitors. By 2016, the number of visitors grew to 1,697. Assume linear growth.

• How much did the number of visitors grow between 2012 and 2016?
• How long did it take for the number to grow from 1,001 to 1,697 visitors?
• What is the average growth rate per year?
• What was the number of visitors in the year 2000?
• Find the equation for the annual visitors, V, for t years after 2000.
• Predict the number of visitors in 2020.

• Find the linear function y, where y depends on t, the number of years since 1980.
• Find and interpret the y-intercept.
• Find and interpret the x-intercept.
• Find and interpret the slope.

Use these data to answer the following questions

• What was the population in 2003? Is this calculation interpolation or extrapolation?
• Estimate when the population will reach 15,000. Is this calculation interpolation or extrapolation?

• What is the slope of the line?
• What is the intercept?
• What is the distance between Point A and the line?
• What is the slope of the shortest path between A and the line?

The nearest road is the \( y= .33x + 20 \)

The village is located at \( (10,30) \)

• Find the point on the road closest to the village.
• Determine how far the village is from the road.

chirps=c(44,35,
  20.4,33,31,35,
  18.5,37,26)
temp =c(80.5,70.5,
  57,66,68,72,52,
  73.5,53)
plot(chirps,temp,
  pch=19,col="blue")
ln = 
  lm(temp ~ chirps)
abline(ln,col="red")

Cricket chirping

\[ \large\begin{array}{rcl} \overline x &=& {\sum x \over n}\\ \\ \\ m &=& {\sum x y - \left(\overline x \sum y\right) \over \sum x^2 - n\left(\overline x\right)^2}\\ \\ \\ \\ b &=& \overline y - m \overline x\\ \end{array} \]

\[ \large\begin{array}{rcl} \overline x &=& {279.9 \over 9}\\ &=& 31.1\\ \\ m &=& {19037.8 - \left(31.1\times 592.5\right) \over 39778.8 - 9\left(31.1\right)^2}\\ &=& {19037.8 - 18426.8\over 9239.4 -8704.8} = {611.0 \over 534.2}\\ &=& 1.144\\ \\ b &=& 65.8 - 1.144\times 31.1\\ &=& 65.8 - 35.6\\ &=& 30.2\\ \end{array} \]

You are measuring the time required to weave silk cloth.

• How long does it take to start up the weaving process?
• What is the average weaving rate once the process is started?
• How linear is the process?

Te U.S. Census tracks the percentage of persons 25 years or older who are college graduates.

• Determine if the trend is linear.

• What line best describes the trend?

• If the trend continues, in what year will the percentage exceed 35%?

• 3.1 Complex Numbers
• 3.2 Quadratic Functions
• 3.3 Power Functions and Polynomial Functions
• 3.4 Graphs of Polynomial Functions
• 3.5 Dividing Polynomials

• 3.6 Zeros of Polynomial Functions
• 3.7 Rational Functions
• 3.8 Inverses and Radical Functions
• 3.9 Modeling Using Variation