Honors Calculus I

Chapter P: Prerequisites

Section P.1: Lines in the Plane

Intercepts of a Graph

The x-intercept is the point at which the graph crosses the x-axis. (a, 0) Let y = 0, and solve for x.

The y-intercept is the point at which the graph crosses the y-axis. (0, b) Let x = 0, and solve for y.

Symmetry of a Graph

A graph is symmetric with respect to the y-axis if, whenever (x, y) is a point on the graph, (-x, y) is also a point on the graph.

A graph is symmetric with respect to the x-axis if, whenever (x, -y) is a point on the graph, (-x, y) is also a point on the graph.

A graph is symmetric with respect to the origin if, whenever (x, y) is a point on the graph, (-x, -y) is also a point on the graph.

Tests for Symmetry

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

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

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

Points of Intersection

The points of intersection of the graphs of two equations is a point that satisfies both equations.

Think substitution or elimination.

Honors Calculus I

Chapter P: Prerequisites

Section P.2: Linear Models and Rate of Changes

Slope of a Line

Slope =

Delta, , means “change in” Given two points in the plane:

rise

run

y

x

m y2 y1

x2 x1

x1,y1 x2, y2

Point Slope Form of a Linear Equation

Given two points in the plane:

Point-Slope Form:

m y y1

x x1

m x x1 y y1

y y1 m x x1

x1,y1 x,y Find Slope & Cross-multiply

Now, switch sides

Slope-Intercept from of a Linear Equation

Slope-intercept form:

m is the slope of the given line b is the y-intercept of the given line

the point (0, b) is on the graph

y mx b

Equations of special lines

Vertical lines intersect the x-axis, therefore the equation of a vertical line is x = a Where a is the x-intercept x = 3 intersects the x-axis at 3

Horizontal lines intersect the y-axis, therefore the equation of a horizontal line is y = b Where b is the y-intercept y = 3 intersects the y-axis at 3 (& has a slope of 0)

Parallel and Perpendicular Lines

Parallel lines never intersect, therefore they have the SAME SLOPE

Perpendicular lines intersect at right angles, therefore they have OPPOSITE INVERSE SLOPES

m1 m2

m1 1

m2

Honors Calculus I

Section P.3: Functions and

Their Graphs

Function and Function Notation

A relation is a set of ordered pairs (x, y). A function is a relation in which each x value

is paired with exactly one y value. A function f(x) is read “f of x ” The independent variable: x The domain is the set of all x The dependent variable: y The range is the set of all y

Equations

An explicit form of an equation is solved for y or f(x)

An implicit form of an equation is when the equation is not solved for (or cannot be solved for) y. It is implied.

y 2

3x 5

x 2 y 2 16

Domain of Function

The implied domain is the set of all real numbers for which the function is defined.

Two considerations: The expression under an even root must be

non-negative (positive or zero). The expression in the denominator cannot

equal zero.

Domain of a Function

For those two considerations: Set the expression under an even root ≥ 0. Set the expression in the denominator equal

to zero to find out what the variable CANNOT be. The domain is everything else.

Use interval notation to designate domain.

Range of a Function

Think of the graph of the function and the intervals of y values related to the domain.

The Graph of a Function

Identity Function Quadratic Function Cubic Function Square Root Function Absolute Value Function Rational Function Sine Function Cosine Function

Transformations of Functions

Horizontal Shift to the Right: y = f(x – c) Horizontal Shift to the Left: y = f(x + c) Vertical Shift Up: y = f(x ) + c Vertical Shift Down: y = f(x ) – c Reflection about the x-axis: y = – f(x ) Reflection about the y-axis: y = f(– x ) Reflection about the origin: y = – f(– x )

Classifications of Functions

Algebraic Functions: Polynomial Functions: expressed as a finite

number of operations of xn

Rational Functions: expressed as a fraction Radical Functions: expressed with a root

Transcendental Functions: Trigonometric Functions: sine, cosine, tangent,

etc.

Composite Functions

Combination of functions such that the range of one function is the domain of the other.

(f ° g)(x) = f(g(x)) (g ° f)(x) = g(f(x))

Even and Odd Functions

An even function is symmetric with respect to the y- axis

Test: substitute -x for x and get back the original function.

An odd function is symmetric with respect to the origin.

Test: substitute -x for x AND -y for y and get back the original function.