CHAP 15: PRECALCULUS in .NET framework

Encoding Data Matrix ECC200 in .NET framework CHAP 15: PRECALCULUS

CHAP 15: PRECALCULUS
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Substituting this value for x into the rst equation yields: 4 12 12 5 7 y = 3 1 = 1 = = 5 5 5 5 5 The coordinates of the point of intersection are 4 , 7 5 5
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Function A function is a mathematical relationship between elements of a given set X = {x}, referred to as the domain, and another set Y = {y}, referred to as the range A function describes a mathematical relationship between a dependent variable y and an independent variable x, which can represent any number within a speci ed interval Polynomial functions are represented by the form y = f(x) = a 0 + a1x + a2x 2 + a3x 3 + + anxn, where a0, a1, a2, a3, an are coef cients (constants) and n is a positive integer or zero that describes the degree or order of the polynomial Special cases A rst-order polynomial function (linear function) is given by y = f(x) = a 0 + a1x y = f ( x ) = a 0 + a1x + a2x2 If the quadratic equation is written in the form ax 2 + bx + c = 0, then the solution x can be determined from the following formula: x= Power Functions Power functions are de ned by y = axn, where a is a constant and n is a real number, representing the mathematical power of the function If p and q are any numbers and a > 0 but a 1: a p aq = a p + q a p + aq a p + q ap aq = ap q b b2 4 ac 2a a 1 0 a1,a 2 0
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A second-order polynomial function (quadratic equation) is given by
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1 an
a1/n = n a
a0 = 1
Graphs of Functions
Graphs, a means for illustrating information from scienti c observations, can reveal relationships and trends between the graphed variables and can allow the scientist to draw relevant conclusions about the variables and the underlying principles involved in the scienti c question being answered
Important Features of a Graph
1 All graphs should have a title that describes the experiment and variables being investigated 2 The independent variable (the variable that is free to move) is displayed on the horizontal x-axis, while the dependent variable (the variable that depends upon the value of the independent variable) is displayed on the vertical y-axis 3 The scale and range for each variable should be noted in a legend on the graph 4 Looking at the data points in the graph, you should note: Is there a general trend exhibited by the data and, if so, is the relationship between the two variables linear, inverse, quadratic or something else If the trend is linear, is the relationship between the two variables constant (represented by a straight horizontal line), increasing (represented by an upward slanted line), or decreasing (represented by a downward slanted line) If the relationship between the two variables is constant, what is the numerical value of the dependent variable that holds throughout the entire experiment If the relationship between the two variables is increasing or decreasing, by how much is the trend increasing or decreasing (ie, what is the slope of the curve ) and where does the function begin on the y-axis (ie, what is the y-intercept of the curve )
Calculating the Slope and y-Intercept
For the function describing a line, y = mx + b, m is the slope and b is the y-intercept of the function The y-intercept is the point at which the line intercepts or passes through the y-axis The slope de nes the rate of change of the line in the y-axis with respect to the corresponding change in the x axis (also known as the rise over the run) and is given as
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