Non-Linear Relationships

Exponential Functions

• Exponential functions take the form $y=a^{x}$. $a$ is a constant,such as 2, 50, π, $\frac{1}{3}$, 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 $y=2^x$, 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 $y=2^x$ has a horizontal asymptote at $y=0$, meaning that the graph will approach zero, but never reach it.

Quadratic Functions

• Quadratic Functions take the form $y=ax^{2}+bx+c$, 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 $(h)$: $x=\frac{-b}{2a}$, 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:

    $\text{When }ax^2+bx+c=0, x=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

    • Note the $\pm$ 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 $y=\frac{a}{x}$ 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 $x=\frac{a}{y}$, and we get the same issue.
  • This is usually how we end up with asymptotes: when we need to divide by 0.