Math 131 Notes Section 1 2 Catalog of Functions Linear functions are of the form f x mx b m is the slope and b is the y intercept The y intercept is the y value assigned to x 0 and is the y value where the graph crosses the y axis If m 0 the line is f x b and is a constant function the graph is a horizontal line Any two points determine a line The slope m is the rise run that is m y 2 y1 for any two points on the line The pt x 2 x1 slope form of the line is y y1 m x x1 This is then simplified to the slope intercept form Because m is constant and does not depend on which two points are used the graph is a straight line m rise run where the rise is the change in y and the run is the change in x If we know m and the change in x we can find the change in y rise m run In other words change in y m change in x Example Given 2 5 and 1 11 are on the line find a the slope intercept form b the change in y corresponding to a decrease of 7 in x Solution a m 11 5 1 2 2 y 5 2 x 2 y 5 2 x 4 y 2 x 9 is the slope int form b If x decreases by 7 the change in y is 2 7 14 so y increases by 14 Linear Regression in the Calculator Please see the Linear Regression Instructions on my 131 webpage www math tamu edu jlewis then Link to Math 131 The linear regression line is the line which is closest to the data points according to a specific rule for closeness The regression line is also called the least squares line or the best fit line We will do problems from the link on my webpage Regression Problems Polynomials A polynomial is a sum of terms which are constants multiplied by nonnegative whole powers of x Some examples of polynomials are f x 4 x 3 2 x 2 1 3 x 15 g x 5 x 4 2 x The highest power of x is called the degree of the polynomial The term with the highest power of x is called the leading term f x has degree 3 and leading term 4 x 3 g x has degree 4 and leading term 5x 4 A root of a polynomial p x is an x value r for which p r 0 If the degree of a polynomial is 2 it is called a quadratic A quadratic can have 0 1 or 2 roots To find the roots of the quadratic ax 2 bx c solve the quadratic formula The roots are b b 2 4 ac 2a If the expression under the radical is negative the quadratic has no roots if it is 0 the quadratic has one root and if it is positive the quadratic has two roots A polynomial of degree 3 is a cubic of degree 4 is a quartic A rational function is of the form f x A root function is of the form f x n p x q x where p and q are polynomials x x1 n An algebraic function is a function which only involves addition subtraction multiplication division and taking roots For example f x x 2 1 x 2 1 3 5x 2 or g x x3 7 x2 4 An exponential function is of the form f x a x where a is a constant Exponential functions behave very differently than power functions which have the form x a Exponential functions will be explored in section 1 5 Logarithmic functions will be explored in section 1 6
View Full Document
Unlocking...