1

PRECALCULUS I

Dr. Claude S. MooreDanville Community

College

Graphs and Lines•Intercepts, symmetry, circles

•Slope, equations, parallel, perpendicular

Equation - equality of two quantities.

Solution - (a,b) makes true statement when a and b are substituted into equation.

Point-plotting method - simplest way to graph.

x -2 -1 0 1 2

y = 2x - 3-7 -5 -3 -1 1

Graph of an Equation

The x-intercept is point where graph touches (or crosses) the x-axis.

The y-intercept is point where graph touches (or crosses) the y-axis.

1. To find x-intercepts, let y be zero and solve the equation for x.

2. To find y-intercepts, let x be zero and solve the equation for y.

Finding Interceptsof an Equation

1. The graph of an equation is symmetric with respect to the y-axis if replacing x with -x yields an equivalent equation.

2. The graph of an equation is symmetric with respect to the x-axis if replacing y with -y yields an equivalent equation.

3. The graph of an equation is symmetric with respect to the origin if replacing x with -x and y with -y yields an equivalent equation.

Tests for Symmetry

The point (x, y) lies on the circle of

radius r and center (h, k)

if and only if

(x - h)2 + (y - k)2 = r2 .

Standard Form of theEquation of a Circle

The graph of the equation

y = mx + b

is a line whose slope is m and

whose y-intercept is (0, b).

Slope-Intercept Form of the Equation of a Line

Definition: Slope of a Line

The slope m of the nonvertical line through (x1, y1) and (x2, y2) where x1 is not equal to x2 is

12

12

xx

yym

The equation of the line with slope m passing through the

point (x1, y1) isy - y1 = m(x - x1).

Point-Slope Form of the Equation of a Line

1. General form:

2. Vertical line:

3. Horizontal line:

4. Slope-intercept:

5. Point-slope:

Equations of Lines

1. Ax + By + C = 0

2. x = a

3. y = b

4. y = mx + b

5. y y1 = m(x x1)

Parallel: nonvertical l1 and l2 are parallel iff m1 = m2 and b1 b2.*Two vertical lines are parallel.

Perpendicular: l1 and l2 are perpendicular iff m1 = -1/m2 or m1 m2 = -1.

Parallel and Perpendicular Lines

11

PRECALCULUS I

Dr. Claude S. MooreDanville Community

College

Functions and Graphs•Function, domain, independent variable

•Graph, increasing/decreasing, even/odd

A function f from set A to set B is a rule of correspondence that assigns to each element x in set A exactly one element y in set B.

Set A is the domain (or set of inputs) of the function f, and set B contains range (or set of outputs).

Definition: Function

1. Each element in A (domain) must be matched with an element of B (range).

2. Each element in A is matched to not more than one element in B.

3. Some elements in B may not be matched with any element in A.

4. Two or more elements of A may be matched with the same element of B.

Characteristics of a Function

Read f(x) = 3x - 4 as “f of x equals three times x subtract 4.”

x inside parenthesis is theindependent variable.

f outside parenthesis is the dependent variable.

For the function f(x) = 3x - 4, f(5) = 3(5) - 4 = 15 - 4 = 11, and f(-2) = 3(-2) - 4 = - 6 - 4 = -10.

Functional Notation

A “piecewise function” defines the function in pieces (or parts).

In the function below, if x is less than or equal to zero, f(x) = 2x - 1; otherwise, f(x) = x2 - 1.

Piece-Wise Defined Function

01

012)( 2 xifx

xifxxf

Generally, the domain is implied to be the set of all real numbers that yield a real number functional value (in the range).

Some restrictions to domain:

1. Denominator cannot equal zero (0).

2. Radicand must be greater than or equal to zero (0).

3. Practical problems may limit domain.

Domain of a Function

dependent variable

independent variable

domain

range

Summary of Functional Notation

function

functional notation

functional value

implied domain

In addition to working problems, you should know and understand the definitions of

these words and phrases:

Vertical Line Test for a Function

A set of points in a coordinate plane is the graph of

y as a function of x

if and only if no vertical line intersects the graph at more than

one point.

On the interval containing x1 < x2,

1. f(x) is increasing if f(x1) < f(x2).Graph of f(x) goes up to the right.

2. f(x) is decreasing if f(x1) > f(x2).Graph of f(x) goes down to the right.

On any interval,

3. f(x) is constant if f(x1) = f(x2).Graph of f(x) is horizontal.

Increasing, Decreasing, and Constant Function

1. A function given by y = f(x) is even if, for each x in the domain,

f(-x) = f(x).

2. A function given by y = f(x) is odd if, for each x in the domain,

f(-x) = - f(x).

Even and Odd Functions

21

PRECALCULUS I

Dr. Claude S. MooreDanville Community

College

Composite and Inverse Functions•Translation, combination, composite

•Inverse, vertical/horizontal line test

For a positive real number c, vertical shifts of y = f(x) are:

1. Vertical shift c units upward:h(x) = y + c = f(x) + c

2. Vertical shift c units downward:h(x) = y c = f(x) c

Vertical Shifts(rigid transformation)

For a positive real number c, horizontal shifts of y = f(x) are:

1. Horizontal shift c units to right: h(x) = f(x c) ; x c = 0, x = c

2. Vertical shift c units to left: h(x) = f(x c) ; x + c = 0, x = -c

Horizontal Shifts (rigid transformation)

Reflections in the coordinate axes of the graph of y = f(x) are represented as follows.

1. Reflection in the x-axis: h(x) = f(x)(symmetric to x-axis)

2. Reflection in the y-axis: h(x) = f(x)(symmetric to y-axis)

Reflections in the Axes

Let x be in the common domain of f and g.

1. Sum: (f + g)(x) = f(x) + g(x)

2. Difference: (f g)(x) = f(x) g(x)

Product: (f g) = f(x)g(x)

4. Quotient:

Arithmetic Combinations

0)(,)(

)()(

xg

xg

xfx

g

f

The domain of the composite function f(g(x)) is the set of all x in the domain of g such that g(x) is in the domain of f.

The composition of the function f with the function g is defined by

(f⃘g)(x) = f(g(x)).Two step process to find y = f(g(x)):

1. Find h = g(x).

2. Find y = f(h) = f(g(x))

Composite Functions

One-to-One Function

For y = f(x) to be a 1-1 function, each x corresponds to exactly one y, and each y corresponds to exactly one x.

A 1-1 function f passes both the vertical and horizontal line tests.

VERTICAL LINE TEST for a Function

A set of points in a coordinate plane is the graph of

y as a function of x

if and only if no vertical line intersects the graph at more than

one point.

HORIZONTAL LINE TEST for a 1-1 Function

The function y = f(x) is a one-to-one (1-1) function if no horizontal line intersects

the graph of f at more than one point.

A function, f, has an inverse function, g, if and only if (iff) the

function f is a one-to-one (1-1) function.

Existence of an Inverse Function

A function, f, has an inverse function, g, if and only if f(g(x)) = x and g(f(x)) = x,for every x in domain of gand in the domain of f.

Definition of an Inverse Function

If the function f has an inverse function g, then

domain range

f x yg x y

Relationship between Domains and Ranges of f and g

1. Given the function y = f(x).

2. Interchange x and y.

3. Solve the result of Step 2 for y = g(x).

4. If y = g(x) is a function, then g(x) = f-1(x).

Finding the Inverse of a Function

34

PRECALCULUS I

Dr. Claude S. MooreDanville Community

College

Mathematical Modeling•Direct, inverse, joint variations;

Least squares regression

1. y varies directly as x.

2. y is directly proportional to x.

3. y = mx for some nonzero constant m.

NOTE: m is the constant of variation or the constant of proportionality.

Example: If y = 3 when x = 2, find m.y = mx yields 3 = m(2) or m = 1.5.

Thus, y = 1.5x.

Direct Variation Statements

1. y varies directly as the nth power of x.

2. y is directly proportional to the nth power of x.

3. y = kxn for some nonzero constant k.

NOTE: k is the constant of variation or constant of proportionality.

Direct Variation as nth Power

1. y varies inversely as x.

2. y is inversely proportional to x.

3. y = k / x for some nonzero constant k.

NOTE: k is the constant of variation or the constant of proportionality.

Example: If y = 3 when x = 2, find k.y = k / x yields 3 = k / 2 or k = 6.

Thus, y = 6 / x.

Inverse Variation Statements

1. z varies jointly as x and y.

2. z is jointly proportional to x and y.

3. y = kxy for some nonzero constant k.

NOTE: k is the constant of variation. Example: If z = 15 when x = 2 and y = 3,find k.y = kxy yields 15 = k(2)(3) or k = 15/6 = 2.5.

Thus, y = 2.5xy.

Joint Variation Statements

This method is used to find the “best fit” straight line

y = ax + b for a set of points, (x,y),

in the x-y coordinate plane.

Least Squares Regression

The “best fit” straight line, y = ax + b, for a set of points, (x,y), in the x-y coordinate plane.

Least Squares Regression Line

22 xxn

yxxyna

xayn

b1

X Y X2 XY

1 3 1 3

2 5 4 10

4 5 16 20

7 13 21 33

Least Squares Regression Line

22 xxn

yxxyna xay

nb

1

Solving for a = 0.57 and b = 3, yields y = 0.57x + 3.