Non-Linear Relationships

Table of Contents

This post includes graphs and lots of formulae. If page-loading is slow, that’s to be expected on older computers (and definitely on phones).

Exponential Functions

  • Exponential functions take the form y=ax. a is a constant,such as 2, 50, π, 13, etc.
  • a can also be a negative value:
  • Note the green line in both graphs: this is called an asymptote.
    • Asymptotes are lines which a graph will approach, but never reach.
    • For example, no matter how small the value of x, for the graph y=2x, y will never equal zero (Try it for yourself if you want).
    • There are three kinds of asymptotes, but only two we care about right now: vertical and horizontal.
      • Horizontal asymptotes are parallel to the x-axis, and are represented as y=b, where b is some constant
      • Vertical asymptotes are parallel to the y-axis, and are represented as x=b, where b is some constant.
      • So the graph y=2x has a horizontal asymptote at y=0, meaning that the graph will approach zero, but never reach it.

A slightly more technical definition:

An asymptote of a curve is a line, such that the distance between the curve and the line approaches zero as either the x or y coordinates (or both) tend to infinity.

  • Exponential growth occurs when x is greater than 1, while exponential decay occurs when x is less than 1.

Quadratic Functions

  • Quadratic Functions take the form y=ax2+bx+c, where a, b, and c are constants.
  • When graphed, quadratic functions are referred to as “parabolas”.
  • If a is positive, the parabola has a maximum turning point. If a is negative, the parabola has a minimum turning point.

  • Quadratics have several properties, which you might be asked about:

    • Vertex/Turning point (h): x=b2a, substitute the value of x into the original equation to get the y value.

    • Axis of Symmetry: x=h (this should be a vertical line)

    • Roots/Zeroes: These are just the x-intercepts, and can be calculated using the Quadratic Formula:

      When ax2+bx+c=0,x=b±b24ac2a

      • Note the ± in the formula: there can be 0, 1, or 2 roots of a quadratic, depending on the value of the bit under the square root.
      • If the bit under the square root is negative, you’ll have no roots, meaning the graph never crosses the x-axis.
      • If the bit under the square root is 0, you have 1 root, meaning the vertex is on the x-axis.
      • If the bit under the square root is positive, you have 2 roots, meaning the graph cuts the x-axis twice.

Hyperbolas

  • Hyperbolas (also known as reciprocal functions) are used to express when two values (x and y) are inversely proportional
  • Inversely proportional means that as one value increases, the other decreases, and vice versa.
  • Hyperbolas take the form y=ax where a is a constant.
  • Hyperbolas have 2 asymptotes: x=0 and y=0.
    • Take a look at the formula, and you’ll see that if x=0, you’d need to divide by 0, which isn’t a thing.
    • The formula can be rearranged to xy=a, then to x=ay, and we get the same issue.
    • This is usually how we end up with asymptotes: when we need to divide by 0.
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