Equations of Lines

  1. What is an equation for the line that passes through the point \((2,3)\) and is parallel to the line given by the equation \(y = -\frac{1}{2}x+7?\) Protip: it might help the sketch the initial line, the point, then the new line.
  2. What is an equation for the line that passes through the point \((-3,5)\) and is perpendicular to the line given by the equation \(y = -\frac{1}{2}x+7?\) Protip: it might help the sketch the initial line, the point, then the new line.
  3. Without appealing to technology, explain, as if convincing a skeptical peer in the class, how you can tell that the three points \((-1,3)\) and \((1,-1)\) and \((5,-9)\) all lie on the same line.
  4. Suppose a line intersects the \(x\)-axis at the number \(12,\) and the area of the triangle formed by this line and the positive \(x\)- and \(y\)-axis is \(30\) What must the slope of this line be?
  5. The price of Johnson & Johnson (NYSE:JNJ) stock closed at $156.74 per share on 29 Dec 2023, but closed at $137.49 per share on 29 Nov 2019. What was the average monthly change in share price between these two dates? What was the average annual change in share price between these two dates?
  6. According to the US Energy Information Administration (EIA), the price of mid-grade conventional retail gasoline in Colorado was $1.99 at the beginning of May 2020, but was $3.02 at the beginning of January 2024.
    1. What was the average monthly change in gas price between these two dates?
    2. Assuming the price of gas changed at about this same average rate prior to May 2020, approximately what was the price of gas at the beginning of January 2020?
    3. Using these two figures, write down an equation \(p(t)\) of a linear function \(p\) that models the price of gas as a function of \(t\) months since January 2020.
    4. According to this model \(p\) what was the price of gas at the beginning of August 2022?
    5. We built this model using only two data points, so maybe its not a very accurate model. Plot the graph \(y = p(t)\) over a scatterplot of the following data from EIA.
      2020
      Month Price
      January $2.95
      March $2.65
      May $1.99
      July $2.77
      September $2.64
      November $2.47
      2021
      Month Price
      January $2.53
      March $2.96
      May $3.27
      July $3.77
      September $3.94
      November $3.85
      2022
      Month Price
      January $3.62
      March $4.08
      May $4.30
      July $5.21
      September $3.93
      November $3.77
      2023
      Month Price
      January $3.19
      March $4.33
      May $3.75
      July $4.20
      September $4.20
      November $3.65
      2024
      Month Price
      January $3.02
      Visually, does \(p\) appears to be an accurate model? If not, try to adjust the parameters of the model slightly, to make it better fit the data.
    6. Do you think it’s a reasonable idea to use \(p\) as a model for gas prices in the future? I.e. do you think \(p\) can accurately predict gas prices in the year 2025? or 2030?