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Numbers and Their Application - Lesson 12

Cartesians, Polynomials, Quadratics

Lesson Overview

Introduction

Coordinate geometry was developed by both Descartes and Fermat. Today we use cartesian coordinates extensively which are named after the former. The relationship between two sets of numbers are often represented via a graph or an equation. For example: F = (9/5)C + 32 relates temperature in Celsius to temperature in Fahrenheit. One variable is designated the independent variable (C) and the value (F) depends on it and is thus the dependent variable. Often, it is easy to reverse these roles: C = (5/9)(F - 32). Such relationships, if plotted on a coordinate system are lines and hence termed linear.

Ordered Pairs, Quadrants, Relations, Functions, f(x), VLT

Mathematicians often speak of forming the cartesian product of several items. The cartesian product is a set operation, but results in a (potentially) bigger object which is not a member of our universal set! The cartesian coordinate system is such a cartesian product of two number lines, labelled x and y. Now instead of having points on a number line with a single number to indicate its distance from the origin (zero), we have points on a plane with two numbers to indicate position. The number lines divide the plane into four quadrants labelled I, II, III, IV counterclockwise with quadrant I having both positive x and positive y coordinates. The axes are not in any quadrant.
  II   |    I 
 III   |   IV 

These coordinates are called ordered pairs and are separated by commas and enclosed within parentheses. The first coordinate (abscissa) is x and is plotted horizontally. The second coordinate (ordinate) is y and is plotted vertically. Warning: the notation for an open intervals is identical!

Lattice points are points in the xy-plane with integer coordinates for both x and y.

A relation is a set of ordered pairs.
A function is a relation for which there is exactly one value of the dependent variable for each value of the independent variable.

Instead of writing y = x + 2, functional notation is often used: f(x) = x + 2. This does not mean to multiply f by x. It means f is the name of the function with x as the independent variable. It gives the recipe for finding f(x)=y given an x value.

The set of values of the independent variable is the domain.
The set of values of the dependent variable is the range.

The Vertical Line Test can be used to determine if a relation is a function as follows. Check if any vertical line ever crosses the relation more than once. If it does, the relation has failed the vertical line test and is not a function.

Slope, Line Equations

About half of calculus is concerned with finding the slope of any function anywhere. Slope is thus an important concept but should already be familiar.

slope = m = rise/run = dy/dx = [delta]y/ [delta]x = [change in y over change in x]

Parallel lines have equal slopes.   Perpendicular lines have slopes which are negative reciprocals.

Note: modern books tend to use an inclusive definition of parallel which allows a line to be parallel to itself. Others exclude this.

This should be well studied in Algebra, so only a quick review is presented in today's activity. In summary, if y = mx + b, then m is the slope and b is the y-intercept (i.e., the value of y when x = 0). Often linear equations are written in standard form with integer coefficients (Ax + By = C). Such relationships must be converted into slope-intercept form (y = mx + b) for easy use on the graphing calculator. In today's activity -10x + y = -5 (10x - y = 5) and y = 5 are encountered. Such systems of equations are either inconsistent (parallel lines, so have no points in common), dependent (coincident lines (same slope and y-intercept), so all points are in common), or independent (slopes are different). One other form of an equation for a line is called the point-slope form and is as follows: y - y1 = m(x - x1). The slope, m, is as defined above, x and y are our variables, and (x1, y1) is a point on the line.

Special Slopes

It is important to understand the difference between positive, negative, zero, and undefined slopes, and that is also covered in today's activity. In summary, if the slope is positive, y increases as x increases, and the function runs "uphill" (going left to right). If the slope is negative, y decreases as x increases and the function runs downhill. If the slope is zero, y does not change, thus is constant—a horizontal line. Vertical lines are problematic in that there is no change in x. Thus our formula is undefined due to division by zero. Some will term this condition infinite slope, but be aware that we can't tell if it is positive or negative infinity! Hence the rather confusing term no slope is also in common usage for this situation.

Polynomials

Polynomials are algebraic expressions involving only the operations of +, -, × of variables.

Polynomials involve no nonalgebraic operations (such as absolute value) and no operations under which the set of real numbers is not closed, such as ÷ or square root.

An expression is a collection of variables and constants connected by operation symbols (+, -, ×, ÷, etc.) which stands for a number.
A term is a part of an expression which is added or subtracted.

Quadratic functions are polynomials with degree two and will be explored below.

The degree of a polynomial is the maximum number of variables which are factors in any one term.

Polynomials (poly- means many) are named based on how many terms they have and by their degree.

Monomials have one term.    Binomials have two terms.    Trinomials have three terms.

Linear functions are a special class of polynomials with degree one. A constant function has degree zero.

If only one variable is present, such as x, we have a polynomial in x. The coefficient of the term with highest degree is called the leading coefficient. There may also be a constant coefficient which has no x multiplier.

Quadratic Functions

The general equation for a quadratic function is
y = ax2 + bx + c,
where a, b, and c are constants, and a [not equal] 0. (If a = 0, then the function is linear.)

Learn the Quadratic Formula (its derivation is given below):

ax2 + bx + c = 0 Given: the general quadratic equation
ax2 + bx = -c Move constant to other side, by subtracting c from both sides.
ax2  + bx =  -c
aa a
 Remove coefficients from quadratic term (x2) by dividing everything by the coefficient.
x2 + bx/a + (b/2a)2 = -c/a + (b/2a)2 To have perfect square trinomial (that's why method is called Completing the Square), need to take half of "b", square it, and add that to both sides.
(x + b/2a)(x + b/2a) = -c/a + (b/2a)2Factor left side since it is now a perfect square.
(x + b/2a)2 = -c/a + (b2/4a2)Rewrote into exponential form (x×x = x2).
(x + b/2a)2 = -4ac +   b2
4a24a2
 On the right side, rewrote fractions to have common denominator, 4a2.
x + b/2a = ±(-4ac + b2)1/2
2a
 Took square root of both sides (As you do to one side, do to the other.) When adding fractions with a common denominator, add the numerators.
x =  -b ± (b2 - 4ac)1/2
2a
 Isolate the variable by subtracting b/2a from both sides.

The shape of the graph of a quadratic equation is called a parabola. On both sides of the vertex (the maximum or minimum point on the graph), the graph of the equation either increases or decreases. The vertex lies on the axis of symmetry. Thus the graph on one side of the line (axis) of symmetry is a reflection of the graph on the other side. Several examples of parabolas are explored in today's activity. Where the graph crosses the x-axis are points called x-intercepts where y = 0. The general equation then degenerates into ax2 + bx + c = 0. To solve for x, the quadratic formula method must be mastered. It involved fractions and radicals. Quadratic Relations will be explored in Algebra II, Precalculus, and Calculus BC. They will allow the full nature of conic sections to be explored.

To obtain the solution to a quadratic equation, Completing the Square is sometimes used. Using the completing-the-square method, as outlined above in the derivation of the quadratic formula, on the general equation (ax2 + bx + c = 0) will find the solutions to any equation.

Discriminant

If ax2 + bx + c = 0, then the quantity D = b2 - 4ac is called the discriminant.

Gauss's Fundamental Theorem of Algebra state that the number of solutions to any equation cannot exceed its degree. In fact, if we carefully count repeated (see Activity 12) and complex roots (see Lesson 15), we will find equality. So, a quadratic equation may have up to two solutions. To determine quickly how many and what type of solutions a quadratic equation has, analyze the discriminant.

Given: ax2 + bx + c = 0, where a, b, and c are real numbers.
If b2 - 4ac < 0 The equation has no real-number solutions. The solutions, involving non-real complex numbers, will be discussed in lesson 15.
If b2 - 4ac > 0 The equation has two different real-number solutions. If D is a perfect [rational] square, the solutions are rational.
If b2 - 4ac = 0 Then the equation has a repeated real-number solution with the vertex on the x-axis. If a and b are rational, then the solution will also be rational.

An example is x2 - 6x + 8 = 0 where a = 1, b = -6, and c = 8. So the discriminant becomes (-6)2 - 4(1)(8) = 36 - 32 = 4. Since 4 is a positive number, the equation will yield two real-number solutions. These answers are (6+2)/2 and (6-2)/2, which reduce to 4 and 2. These are related to the original equations as follows: x2 - 6x + 8 = (x - 4) (x - 2) = 0.

Solutions, Roots, Zeroes, and x-intercepts

The four terms solutions, roots, zeroes, and x-intercepts are often used somewhat interchangeably to refer to the values of x where an equation is zero.

Cubic, Quartic, Quintic

Polynomials with degree three are referred to as cubic functions. Degree four polynomials are quartic functions and degree five polynomials are quintic functions. There are ways to solve cubic functions and quartic functions, but the general quintic function ax5 + bx4 + cx3 + dx2 + ex + f = 0 is not solveable algebraically—only numerical approximation can be obtained. Functions with only even or odd exponents are termed even or odd and relate to its symmetry when graphed.

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