Precalculus Functions
Precalculus Functions is a course that is taken during highschool to prepare you for a college class of calculus. Precalculus Functions is the study of all different kinds of functions. There are many different functions however, in this class there are only five functions that we focus on. These five functions are the Quadratic, Polynomial, Rational, Exponential, and Log functions.
Quadratic Functions
Quadratic functions are U-shaped parabolas. Each parabola has a domain and range. The domain of a function is the set of all possible input values (usually x), which allows the function formula to work; in the domain the function is always all real numbers. The range is the set of all possible output values (usually y), which result from using the function formula. Quadratics have x-intercepts which are points that intercept the x-axis. The axis of symmetry splits the parabola in half so one side mirrors the opposite side. There are minimum and maximum values in a parabola that determine which way the parabola will point, upward or downward. These are also called vertex's.
Different Forms
Standard Form: y = ax²+bx+c
Vertex Form: y = a(x-h)²+k
Example:
Different Forms
Standard Form: y = ax²+bx+c
Vertex Form: y = a(x-h)²+k
Example:
Polynomial Functions
Polynomials are four different shaped graphs. The odd exponent with a positive number will start off in the lower left corner of the graph going into negative infinite to the upper right corner of the graph into positive infinite. An odd exponent with a negative number will start off in the upper left corner of the graph into positive infinite to the lower right corner of the graph into negative infinite. The even exponential graphs stay pointing up into positive infinite with a positive number or pointing down into negative infinite with a negative number. If you find yourself with the wrong shape of graph then double check if there is a x in front of the first parentheses and count it as having one exponent. Do not exclude the first x or else it will mess up your whole entire graph.
Different Forms:
F(x) = c
F(x) = mx+b
F(x) = ax²+bx+c
Example:
Different Forms:
F(x) = c
F(x) = mx+b
F(x) = ax²+bx+c
Example:
Rational functions
A rational function is formed when a polynomial is divided by another polynomial. So, all rational functions are polynomials. In rational functions there are x-intercepts, y-intercepts, vertical asymptotes, horizontal asymptotes, and sometimes holes. X-intercepts are points on the graph that are located on the x-axis while y-intercepts are points on the graph that are located on the y-axis. Vertical asymptotes are dotted lines that go through a point in the x-axis that goes on forever. Horizontal asymptotes are dotted lines that go through a point in the y-axis that goes on forever. Holes are points that both the numerator and the denomenator in the equation has such as x=3 so that spot would not have a line going through it but instead a circle, or hole, that the line connects to. When there are 2 or more vertical or horizontal asymptotes you will need to use a chart to check to see what way the function points, positive infinite or negative infinite.
Example:
Example:
Exponentials and Logarithms
Exponential and logarithmic functions are inverses of each other. There are always a base, a exponential, and an outcome. You can solve logarithmmic and exponential functions just as easily as any other function. It may be easier!