P.1 Graphs and Models
Example 1a
Sketch the graph of y = x2 – 2
Example 1b
Sketch the graph of y = x3 – x2 – 25
Intercepts
To find the x-intercept(s) of a graph let y = 0 and solve for x.
To find the y-intercept(s) of a graph let x = 0 and solve for y.
Example 2
Find the x- and y-intercepts of the graph y = x3 – 4x
Tests for Symmetry
x-axis: Replace y by –y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the x-axis.
Ex.: x = y2
y-axis: Replace x by –x in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the y-axis.
Ex: y = x2
Origin: Replace x by –x and y by –y in the equation. If an equivalent equation results, the graph of the equation is symmetric with respect to the origin.
Ex: y = x3
Example 3
Show that the graph of y = 2x3 – x is symmetric with respect to the origin.
Example 4
Use intercepts and symmetry to graph x – y2 = 1
A point of intersection of the graphs of two equations is a point that satisfies both equations. You can find the points of intersection of two graphs by solving their equations simultaneously.
Example 5
Find all points of intersection of the graphs of x2 – y = 3 and x – y = 1
P.2 Linear Models and Rates of Change
Slope:
Slope is the average rate of change of y with respect to x
Example 1
Find the slope of the line joining the points (1 , 2) and (5 , -3).
When the slope a line is:
Positive, then the line slants upward from left to right.
Negative, then the line slants downward from left to right.
Zero, then the line is horizontal.
Undefined, then the line is vertical.
Vertical Line: x = a , where a is the x-intercept
Example: x = 3
Horizontal Line: y = b , where b is the y-intercept
Example: y = -2
Point-Slope Form of an Equation of a Line: An equation of a nonvertical line of slope m that passes through the point ( x1 , y1) is
y – y1 = m(x – x1)
Example 2
Find an equation of the line that has a slope of 3 and passes through the point (1, -2).
General Form of an Equation of a Line
Ax + By + C = 0
Slope – Intercept Form of an Equation of Line
y = mx + b
Example 3
Sketch the graph of each equation.
a. y = 2x + 1 b. y = 2 c. 3y + x – 6 = 0
Parallel and Perpendicular Lines
Two distinct nonvertical lines are parallel if and only if their slopes are equal.
Two nonvertical lines are perpendicular if and only if the product of their slopes is –1 (the slopes are negative reciprocals).
Example 4
Find the equations of the lines that passes through the point (2, -1) and are:
a. parallel to the line 2x – 3y = 5
b. perpendicular to the line 2x – 3y = 5
P.3 Functions and Their Graphs
Definition of a Function: Let X and Y be two nonempty sets of real numbers. A function from X into Y is a rule of correspondence that associates with each element of X a unique element of Y.
domain: x-values range: y-values
Implicit Form Explicit Form Function Notation
3x + y = 5 y = -3x + 5 f(x) = -3x + 5
Example 1
Evaluate each expression where f(x) = x2 + 7
a. f(2) b. f(3a) c. f(b – 1)
Example 2
Find the domain and range of
a. f(x) = b. f(x) = 1 – x if x < 1
if x 1
Vertical Line Test: A set of points in the xy plane is the graph of a function if and only if a vertical line intersects the graph in at most one point.
Library of Functions
Identity Function: f(x) = x Squaring Function: f(x) = x2
Cubing Function: f(x) = x3 Square Root Function: f(x) =
Rational Function: f(x) = 1/x Absolute Value Function: f(x) = | x |
Sine Function f(x) = sin x Cosine Function f(x) = cos x
Transformations of Functions
Basic types of transformations ( c > 0 )
Original graph: y = f(x)
Right Shift: y = f(x – c)
Left Shift: y = f(x + c)
Downward Shift: y = f(x) – c
Upward Shift: y = f(x) + c
Reflection about x-axis: y = -f(x)
Reflection about y-axis: y = f(-x)
Reflection about origin: y = -f(-x)
Example 3
Graph:
a. y = x2 b. y = x2 – 2
c. y = (x + 3)2 d. y = -x2
e. y = (-x)2 f. y = -(x – 1)2 + 2
Sketching Polynomial Graphs (End Behaviors)
f(x) = anxn + a n-1 x n-1 + … + a1x + a0
Even Degree
Leading Coefficient Positive Leading Coefficient Negative
Odd Degree
Leading Coefficient Positive Leading Coefficient Negative
Composite Functions
Given two functions f and g, the composite function, denoted f o g (read as “f composed with g”), is defined by
(f o g)(x) = f(g(x))
The domain of f o g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.
Example 4
Given that f(x) = 2x – 3 and g(x) = cos x , find each composite function.
a. f o g b. g o f
An x-intercept of a function is called a zero.
To find the zeros of a function solve f(x) = 0
Test for Even and Odd Functions
The function y = f(x) is even if f(-x) = f(x)
The function y = f(x) is odd if f(-x) = -f(x)
Example 5
Determine whether each function is even, odd, or neither. Then find the zeros of the function.
a. f(x) = x3 – x b. g(x) = 1 + cos x