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 = y/ x =
Parallel lines have equal slopes. Perpendicular lines have slopes which are negative reciprocals.
Note: modern books tend to use an inclusive definition of parallelwhich 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 havea polynomial in x. The coefficient of the term withhighest degree is called the leading coefficient.There may also be a constant coefficient which has nox multiplier.
Quadratic Functions
The general equation for a quadratic function is y = ax2 + bx + c, where a, b, and c are constants, and a 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
a
a
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)2
Factor 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
4a2
4a2
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 Squareis sometimes used. Using the completing-the-square method,as outlined above in the derivation of the quadratic formula, on the generalequation (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.
![[delta]](deltabig.gif){kind=small}
y/ ![[change in y over change in x]](slopefor.gif)
![[not equal]](ne.gif){kind=small}
0.(If a = 0, then the function is linear.)