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Mathematics LibreTexts

1: Chapter 1 - Functions

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    • 1.1: Relations and Functions
    • 1.2: The Graph of a Function
      Descartes introduces his coordinate system, a method for representing points in the plane via pairs of real numbers. Indeed, the Cartesian plane of modern day is so named in honor of Rene Descartes, who some call the “Father of Modern Mathematics.” A Cartesian Coordinate System consists of a pair of axes, usually drawn at right angles to one another in the plane, one horizontal (labeled x) and one vertical (labeled y).
    • 1.3: Slope
      In the previous section on Linear Models, we saw that if the dependent variable was changing at a constant rate with respect to the independent variable, then the graph was a line. You may have also learned that higher rates led to steeper lines (lines that rose more quickly) and lower rates led to lines that were less steep. In this section, we will connect the intuitive concept of rate developed in the previous section with a formal definition of the slope of a line.
    • 1.4: Equations of Lines
      In this section we will develop the slope-intercept form of a line. When you have completed the work in this section, you should be able to look at the graph of a line and determine its equation in slope-intercept form.
    • 1.5: The Point-Slope Form of a Line
      In the last section, we developed the slope-intercept form of a line (y = mx + b). The slope-intercept form of a line is applicable when you’re given the slope and y-intercept of the line. However, there will be times when the y-intercept is unknown.
    • 1.6: Vertical Transformations
      In this section we study the art of transformations: scalings, reflections, and translations. We will restrict our attention to transformations in the vertical or y-direction. Our goal is to apply certain transformations to the equation of a function, then ask what effect it has on the graph of the function.
    • 1.7: Horizontal Transformations
      In the previous section, we introduced the concept of transformations. We made a change to the basic equation y = f(x), such as y = af(x), y = −f(x), y = f(x) − c, or y = f(x) + c, then studied how these changes affected the shape of the graph of y = f(x). In that section, we concentrated strictly on transformations that applied in th vertical direction. In this section, we will study transformations that will affect the shape of the graph in the horizontal direction.