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Precalculus: A Graphing Approach
Chapter Two
Linear and Quadratic Functions
Copyright © 2000 by the McGraw-Hill Companies, Inc.
Linear and Constant Functions
A function f is a linear function if
f(x) = mx + b
m0
where m and b are real numbers. The domain is the set of all real
numbers and the range is the set of all real numbers.
If m = 0, then f is called a constant function
f(x) = b
which has the set of all real numbers as its domain and the constant
b as its range.
Copyright © 2000 by the McGraw-Hill Companies, Inc.
2-1-13
Slope of a Line
m
y2  y1 Rise

x 2  x1
Run
x1  x2
y
(x , y )
2
2
y2 – y1
Rise
(x 1, y1)
x2 – x1
Run
(x 2, y 1)
x
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2-1-14
Geometric Interpretation of Slope
Line
Slope
Rising
Positive
Falling
Negative
Horizontal
Zero
Vertical
Not defined
Example
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2-1-15
Equations of a Line
Standard form
Ax + By = C
A and B not both 0
Slope-intercept form
y = mx + b
Slope: m; y intercept: b
Point-slope form
y – y1 = m(x – x1)
Slope: m; Point: (x1, y1)
Horizontal line
y=b
Slope: 0
Vertical line
x=a
Slope: Undefined
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2-1-16
Inequality Properties
An equivalent inequality will result and the sense will remain the
same if each side of the original inequality:
1. Has the same real number added to or subtracted from it; or
2. Is multiplied or divided by the same positive number.
An equivalent inequality will result and the sense will reverse if each
side of the original inequality:
3. Is multiplied or divided by the same negative number.
Note: Multiplication by 0 and division by 0 are not permitted.
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2-2-17
Completing the Square
To complete the square of the quadratic expression
x2 + bx
add the square of one-half the coefficient of x; that is, add
2
b
  or
2
b2
4
The resulting expression can be factored as a perfect square:
2
b
b

x2 + bx +   =  x  
2
2

2
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2-3-18
Properties of a Quadratic Function
Given a quadratic function f(x) = ax 2 + bx + c, a  0, and the form f(x) = a (x – h) 2 + k
obtained by completing the square:
(h, k)
(h, k)
1.
2.
3.
4.
5.
6.
The graph of f is a parabola.
Vertex: (h, k) [parabola increases on one side of vertex and decreases on the other].
Axis (of symmetry): x = h (parallel to y axis)
f(h) = k is the minimum if a > 0 and the maximum if a < 0
Domain: All real numbers
Range: (–,k] if a < 0 or [k, ) if a > 0
The graph of f is the graph of g(x) = ax2 translated horizontally h units and
vertically k units.
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2-3-19
Basic Properties of the Complex Number System
1.
Addition and multiplication of complex numbers are
commutative and associative.
2.
There is an additive identity and a multiplicative identity for
complex numbers.
3.
Every complex number has an additive inverse (that is,
a negative).
4.
Every nonzero complex number has a multiplicative inverse
(that is, a reciprocal).
5.
Multiplication distributes over addition.
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2-4-20
The Quadratic Formula
If ax2 + bx + c = 0, a  0, then
 b  b2  4ac
x 
2a
Discriminants, Roots, and Zeros
Discriminant
b2 – 4ac
Roots of
ax2 + bx + c = 0
No. of real zeros of
f(x) = ax2 + bx + c
Positive
2 distinct real roots
2
0
1 real root (double root)
1
Negative
2 imaginary roots, one the
conjugate of the other
0
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2-5-21
Power Operation on Equations
If both sides of an equation are raised to the same natural number
power, then the solution set of the original equation is a subset of
the solution set of the new equation.
Equation
Solution set
x = 3
{3}
x2 = 9
{ – 3, 3 }
Extraneous solutions may be introduced by raising both sides of an
equation to the same power. Every solution of the new equation
must be checked in the original equation to eliminate extraneous
solutions.
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2-6-22