Skip to main content
Mathematics LibreTexts

2.1: Lines

  • Page ID
    232
  • \( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)

    1. Lines (definitions)

    Everyone knows what a line is, but providing a rigorous definition proves to be a challenge.

    Definition: Line

    A line with slope \(m\) through a point \(P = (a,b)\) is the set of all points \((x,y)\) such that

    \[\dfrac{y-b}{x-a}= m.\]

    2. The Slope Intercept Form of the equation of a Line

    Given a point \((x_1,y_1)\) and a slope \(m\), the equation of the line is

    Definition: Slope Intercept Equation of a Line

    \[y-y_1=m(x-x_1)\]

    3. Piecewise Linear Functions

    A function is piecewise linear if it is made up of parts of lines

    Example 1

    \[f(x)=\begin{cases} x+4 & \text{if }x\leq-2 \\ 2x-1 & \text{if } -2<x<1 \\ -2x & \text{if } x\geq1\end{cases}\]

    piecewise.gif

    We graph this line by sketching the appropriate parts of each line on the same graph.

    4. Applications

    Example 2

    Suppose you own a hotel that has 150 rooms. At $80 per room, you have 140 rooms occupied and for every $5 increase in price you expect to have two additional vacancies. Come up with an equation that gives rooms occupied as a function of price.

    Solution

    Let \(x\) be the price of a room and \(y\) be the number of rooms occupied. Then we have an equation of a line that passes through the point \((80,140)\) and has slope \(-\frac{1}{5}\). Hence the equation is:

    \[y - 140 = -\dfrac{1}{5}(x - 80)\]

    or

    \[y = -\dfrac{1}{5} x + 16 + 140\]

    or

    \[y = -\dfrac{1}{5} x + 156.\]

    Exercise 1

    What should you do if your two year old daughter has a 40 degree C temperature?

    Hint: We have the two points: \((0,32)\) and \((100,212)\).

    Exercise 2

    Suppose that your company earned $30,000 five years ago and $35,000 three years ago. Assuming a linear growth model, how much will it earn this year?

    Exercise 3

    My rental was bought for $204,000 three years ago. Depreciation is set so that the house depreciates linearly to zero in twenty years from the purchase of the house. If I plan to sell the house in twelve years for $250,000 and capital gains taxes are 28% of the difference between the purchase price and the depreciated value, what will my taxes be?

    Exercise 4

    Wasabi restaurant must pay either a flat rate of $400 for rent or 5% of the revenue, whichever is larger. Come up with the equation of the function that relates rent as a function of revenue

    Larry Green (Lake Tahoe Community College)


    This page titled 2.1: Lines is shared under a not declared license and was authored, remixed, and/or curated by Larry Green.

    • Was this article helpful?