2. Preliminaries 0.1 THE REAL NUMBERS AND THE CARTESIAN PLANE 0.2
0.1
THE REAL NUMBERS AND THE CARTESIAN PLANE
0.2
LINES AND FUNCTIONS
0.3
GRAPHING CALCULATORS AND COMPUTER ALGEBRA SYSTEMS
0.4
TRIGONOMETRIC FUNCTIONS
0.5
TRANSFORMATIONS OF FUNCTIONS
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3. 0.2 LINES AND FUNCTIONS 2.1 For x1 ≠ x2, the slope of
The straight line through
the points (x1, y1) and (x2, y2)
is the number
When x1 = x2 and y1 ≠ y2,
the line through (x1, y1)
and (x2, y2) is vertical and the slope is undefined.
Notice that a line is horizontal if and only if its slope is zero.
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4. 0.2
LINES AND FUNCTIONS
2.2
Using Slope to Determine If Points Are Colinear
Use slope to determine whether the points (1, 2), (3, 10) and (4, 14) are colinear.
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5. 0.2
LINES AND FUNCTIONS
2.2
Using Slope to Determine If Points Are Colinear
Since the slopes are the same, the points must be colinear.
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6. POINT-SLOPE FORM OF A LINE
0.2
LINES AND FUNCTIONS
POINT-SLOPE FORM OF A LINE
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7. 0.2
LINES AND FUNCTIONS
2.4
Finding the Equation of a Line Given Two Points
Find an equation of the line through the points (3, 1) and (4, −1) and graph the line.
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8. 0.2
LINES AND FUNCTIONS
2.4
Finding the Equation of a Line Given Two Points
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9. 0.2
LINES AND FUNCTIONS
2.1
Two (nonvertical) lines are parallel if they have the same slope.
Further, any two vertical lines are parallel.
Two (nonvertical) lines of slope m1 and m2 are
perpendicular whenever the product of their slopes is −1 (i.e., m1 · m2 = −1).
Also, any vertical line and any horizontal line are perpendicular.
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10. 0.2
LINES AND FUNCTIONS
2.6
Finding the Equation of a Perpendicular Line
Find an equation of the line perpendicular to y = −2x + 4 and intersecting the line at the point (1, 2).
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11. 0.2
LINES AND FUNCTIONS
2.6
Finding the Equation of a Perpendicular Line
The slope of y = −2x + 4 is −2.
The slope of the perpendicular
line is then −1/(−2) = ½.
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12. 0.2 LINES AND FUNCTIONS 2.2 A function f is a rule that
assigns exactly one element
y in a set B to each element
x in a set A. In this case, we write y = f (x).
We call the set A the domain of f. The set of all values f (x) in B is called the range of f , written {y|y = f (x), for some x in set A}.
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13. 0.2
LINES AND FUNCTIONS
2.2
Unless explicitly stated otherwise, whenever a function f is given by a particular expression, the domain of f
is the largest set of real numbers for which the expression is defined.
We refer to x as the independent variable and to y as the dependent variable.
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14. 0.2 LINES AND FUNCTIONS Functions
By the graph of a function f, we mean the graph of the equation y = f (x).
That is, the graph consists of all points (x, y), where x is in the domain of f and where
y = f (x).
Notice that not every curve is the graph of a function, since for a function, only one y-value can correspond to a given value of x.
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15. 0.2 LINES AND FUNCTIONS Functions
We can graphically determine whether a curve is the graph of a function by using the vertical line test:
if any vertical line intersects the graph in more than one point, the curve is not the graph of a function, since in this case, there are two y-values for a given value of x.
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16. 0.2 LINES AND FUNCTIONS 2.8 Using the Vertical Line Test
Determine which of the curves correspond to functions.
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17. 0.2 LINES AND FUNCTIONS 2.8 Using the Vertical Line Test Slide 17
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18. 0.2
LINES AND FUNCTIONS
2.3
A polynomial is any function that can be written in the form
where a0, a1, a2, , an are real numbers (the coefficients of the polynomial) with an ≠ 0 and n ≥ 0 is an integer (the degree of the polynomial).
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19. 0.2
LINES AND FUNCTIONS
2.4
Any function that can be written in the form
where p and q are polynomials, is called a rational function.
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20. 0.2 LINES AND FUNCTIONS 2.10 A Sample Rational Function
Find the domain of the function
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21. 0.2 LINES AND FUNCTIONS 2.10 A Sample Rational Function Slide 21
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22. 0.2 LINES AND FUNCTIONS Functions
The square root function is defined in the usual way. When we write we mean that y is that number for which y2 = x and y ≥ 0.
In particular, Be careful not to write erroneous statements such as In particular, be careful to write
Similarly, for any integer n ≥ 2, whenever yn = x, where for n even, x ≥ 0 and y ≥ 0.
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23. 0.2
LINES AND FUNCTIONS
2.11
Finding the Domain of a Function Involving a Square Root or a Cube Root
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24. 0.2
LINES AND FUNCTIONS
2.11
Finding the Domain of a Function Involving a Square Root or a Cube Root
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25. 0.2 LINES AND FUNCTIONS Functions
A solution of the equation f (x) = 0 is called a zero of the function f or a root of the equation f (x) = 0.
Notice that a zero of the function f corresponds to an x-intercept of the graph of y = f (x).
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26. 0.2 LINES AND FUNCTIONS 2.12 Finding Zeros by Factoring
Find all x- and y-intercepts of f (x) = x2 − 4x + 3.
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27. 0.2 LINES AND FUNCTIONS 2.12 Finding Zeros by Factoring y-intercept
x-intercepts
For the quadratic equation ax2 + bx + c = 0 (for a ≠ 0), the solution(s) are also given by the familiar quadratic formula:
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28. 0.2 LINES AND FUNCTIONS 2.2 Slide 28
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29. 0.2
LINES AND FUNCTIONS
2.2
Polynomials may also have complex zeros. For instance,
f (x) = x2 + 1 has only the complex zeros x = ±i, where i
is the imaginary number defined by
We confine our attention in this text to real zeros.
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30. 0.2 LINES AND FUNCTIONS 2.3 (Factor Theorem) Slide 30
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31. 0.2 LINES AND FUNCTIONS 2.14 Finding the Zeros of a Cubic Polynomial
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32. 0.2 LINES AND FUNCTIONS 2.14 Finding the Zeros of a Cubic Polynomial
By calculating f (1), we can see that one zero of this function is x = 1.
Factor:
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