lines & functions

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 2y = 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 ofradius 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 equationy = 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

12xxyym

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 Lines1. Ax + By + C = 02. x = a3. y = b4. y = mx + b5. 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


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

01012

)( 2 xifxxifx

xf

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 variableindependent

variabledomainrange

Summary of Functional Notation

functionfunctional notationfunctional valueimplied 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


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

xgxfx

gf

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 rangef 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


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

b 1

X Y X2 XY1 3 1 32 5 4 104 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.

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