Unit 1Pre-Calculus Review

Section 1Graphs and Models

VocabularySolution point: a point that satisfies the equation when _________ _____________________________Table of values: a table with several x-values and the _______________________________________Graph: the set of all _____________________ to an equationParabola: a graph that opens either up or downIntercept: point at which the graph intersects the x- or y-axisx-intercept: the point ____________To find- let y=0 and solve for xy-intercept: the point ____________To find- let x=0 and solve for yPoint of intersection: a point that satisfies two or more equationsMathematical models: real-life applications of mathematics

Example 1.1Sketch the graph of y = x2 – 2

Exploration

Example 1.2Find the x- and y-intercepts of y=x3−4 x

Symmetry1. 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. This means that the portion of the graph to the left of the y-axis is a mirror image of the portion to the right of the y-axis.

2. 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. This means that the portion of the graph above the x-axis is a mirror image of the portion below the x-axis.

3. 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. This means the graph is unchanged by a rotation of 180º about the origin.

Tests for symmetry1. Symmetric with respect to the y-axis

2. Symmetric with respect to the x-axis

3. Symmetric with respect to the origin

Example 1.3Test the graph of y=2x3−x for symmetry with respect to (a) the y-axis and (b) the origin.

Example 1.4Use intercepts and symmetry to sketch the graph of x - y2 = 1

Example 1.5Find all points of intersection of x2 - y = 3 and x - y = 1

Mathematical ModelsUsed for:

Goals:

Example 1.6The Mauna Loa Observatory in Hawaii records the carbon dioxide concentration y (in parts per million) in Earth’s atmosphere. The January readings for various years are shown below.

In the July 1990 issue of Scientific American, these data were used to predict the carbon dioxide level in Earth’s atmosphere in the year 2035, using the quadratic model y=0.018t 2+0.70 t+316.2 where t=0 represents 1960 as shown in part (a) above. The data shown in part (b) above represent the years 1980 through 2010 and can be modeled by y=1.68 t+303.5 where t=0 represents 1960. What was the prediction given in the Scientific American article in 1990? Given the new data for 1990 through 2010, does this prediction for the year 2035 seem accurate?

Section 2Linear Models and Rates of Change

SlopeThe slope of a nonvertical line is a measure of the number of units the line rises (or falls) vertically for each unit of horizontal change from left to right.Picture:

Slope Definition:

Positive slope:

Slope of zero:

Negative slope:

“Undefined” slope:

Point-Slope Form

Example 2.1Find the equation of a line that has a slope of 3 and passes through the point (1, -2).

Ratios and Rates of ChangeRatio: if the x- and y-axes have the same unit of measure and the slope has no unitsRate of change: if the x- and y-axes have different units of measure

Example 2.2The maximum recommended slope of a wheelchair ramp is 1/12. A business installs a wheelchair ramp that rises to a height of 22 inches over a length of 24 feet. Is the ramp steeper than recommended?

Example 2.3The population of Colorado was about 4,302,000 in 2000 and about 5,029,000 in 2010. Find the average rate of change of the population over this 10-year period. What will the population of Colorado be in 2020?

Slope-Intercept Form

Summary of Equations of Lines1. General Form:

2. Vertical Line:

3. Horizontal Line:

4. Slope-Intercept Form:

5. Point-Slope Form:

Parallel and PerpendicularDrawings:

Parallel: two lines with equal slopesPerpendicular: two lines with slopes that are negative reciprocals of each other

Example 2.4Find the general forms of the equations of the lines that pass through the point (2,-1) and are

(a)parallel to the line 2x-3y=5(b) perpendicular to the line 2x-3y=5

Section 3Functions and Their Graphs

VocabularyRelation: between two sets X and Y, a set of ordered pairs ______ where x is a member of X and y is a member of YFunction: a relation between X and Y that has the property that any two ordered pairs with the same x-value have the __________Independent Variable: ____Dependent Variable: ____Real-Valued Function of a Real Variable x: a correspondence that assigns to each number x in X exactly one number _________Domain: ___________Image: the number ______, denoted by _______Value of f at x: f(x)Range: a ___________ and consists of all images of numbers in XEvaluate (a function): find the y-value that corresponds to a given x-valueImplicit: not expressed directly in terms of the independent variable (ex: 3x-2y=7)Explicit: expressed directly in terms of x (ex: y=5x-2)

Example 3.1For the function f defined by f ( x )=x2+7, evaluate each expression(a) f (3a ) (b ) f (b−1 )(c)

f ( x+∆ x )−f (x)∆ x

Example 3.2

Find the domain and range of the functions:a. f(x)=√x-1

b. f(x)=tan(x)

Example 3.3Find the domain and range of:

More VocabularyOne-to-one: a function which assigns to each y-value in the range exactly one ______________________Onto: a function whose range consists of all of _________Vertical Line Test: a vertical line can intersect the graph of a function of x ___________________

Basic 8 Functions

TransformationsTransformation: vertical shifts, horizontal shifts, and reflections of a graph

Basic Types of TransformationsOriginal graph:

Horizontal shift c units to the right:

Horizontal shift c units to the left:

Vertical shift c units downward:

Vertical shift c units upward:

Reflection (about the x-axis):

Reflection (about the y-axis):

Reflection (about the origin):

Elementary Functions: a collection of functions that can create mathematical models to represent many real-world phenomena

Three Categories:1. Algebraic Functions- polynomial, radical, rational2. Trigonometric Functions- sine, cosine, tangent3. Exponential and Logarithmic Functions

VocabularyPolynomial Function: most common type of algebraic function

where n is a nonnegative integerCoefficients: the numbers _________Leading coefficient: _________Constant Term: ________Degree: ______ if an≠0

Some basic polynomial functions:

Leading Coefficient Test: the function's degree (even or odd) and the leading coefficient an can be used to determine a function's right and left behavior of the graph- but tells nothing about the middle

Rational Function: written as the quotient of two polynomials

Algebraic Functions: functions that can be expressed as a finite number of sums, differences, multiples, quotients, and radicals involving xn

Transcendental: functions that are not algebraicComposition: a way of combining functions by substituting one into the other for x

Example 3.4For f ( x )=2x−3 and g ( x )=cos (x) find each composite function:

(a) f ∘ g(b) g∘ f

More Vocabulary

Zero (of a function): a solution of the equation ______________Even (function): graph is symmetric with respect to the _________Odd (function): graph is symmetric with respect to the _________

Example 3.5Determine 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)

Exploration

Section 4

Fitting Models to Data

Example 4.1

Position Functions ( t )=1

2g t 2+v0 t+s0

Example 4.2

Example 4.3