## MAT-261: Pre-Calculus

**MAT 261**

**Section 1.2 – Visualizing & Graphing Data**

Cartesian Coordinate System

Ordered pairs= (x, y)

Domain: set of all possible input values

Range: set of all possible output values

Pythagorean Theorem

log²+log²=hypotenuse²

Distance Formula

d= √ ( ( x2−x1)

2

+( y2−y1)

2

Circles

The set of all points in a plane are an equal distance from the center point

That distance is the radius (r)

When centered at (0,0) use: x²+y²=r²

When centered at (h,k) use: (xh)²+(yk)²=r²

o Square root of a negative

When finding the range:

o Use a graph!

Section 1.4: Types of Functions & Their Rates of Change

Slope

Slope represents the average rate of change of y with respect to x

M= rise/run

Intercepts

The x intercept (horizontal intercept) is the point where the graph crosses the xaxis

The y intercept (vertical intercept) is the point where the graph crosses the yaxis

Slopeintercept form is y=mx+b

M is slope

(0, b) is yintercept

Difference Quotients

Average rate of change (similar to slope of a line)

f ( x+h )− f (x )

h

Section 2.1: Equations of Lines

Two equations for a line

o Slope intercept form: y=mx+b

o Point slope form: y−y1=m(x−x1

)

o M is slope in both equation

WEEK 2:

**Section 3.1: Quadratic Functions & Models**

The simplest quadratic function

o F(x)=x²

o Vertex at (0,0)

o Symmetry at x=0

o Decreasing on (-∞,0)

o Increasing on (0, ∞)

Vertex form of a quadratic

o F(x)= a (x-h) ²+k

o Vertex at (h, k)

o If a>0, the vertex is the minimum

o If a<0, the vertex is the maximum

o Compare to expanded form f(x)= ax²+bx+c

Domain and Range

o Domain is the all the x values that work

Domain of any quadratic is (-∞, ∞)

o Range is all the y values that the function hits

If a>0, start at the vertex and go up forever.

If a<0, start down at -∞ and go up to the vertex

Section 3.2: Quadratic Equations & Problem Solving

The Zero Product Property

o If a × b =0, then a=0, b=0, or both

A second degree or quadratic equation is an equation that can be written in

the form ax²+bx+c, where a, b, and c are real numbers and a is not zero.

Y-intercept (vertical intercept)

o Crosses the y-axis at (0, c)

X-intercepts (horizontal intercept)

o For any function solve f(x)=0

o For a quadratic function

Factor

Complete the square

Use the quadratic formula

The Discriminant

o b²-4ac

o Determines number and type of solutions

Section 3.3: Complex Numbers

Imaginary i

o √-1= i

o i² = -1

o The standard form for an imaginary number is z=a+bi

o To add or subtract imaginary numbers, you combine like term

MAT-261: Pre-Calculus

**week 3**

Section 4.2:** Polynomials Functions and Models**

Names of polynomial functions

o Degree 2

o Degree 3

o Degree 4

Number of turning points

o The number of turning points of the graph of a polynomial function of

degree n ≥ 1 is at most n-1

o If a polynomial has t > 0 turning points, its degree must be at least t+1

End Behavior

o Highest degree term “wins” for large x and negative x

Zeros of a function

o For real numbers, these are all the same:

X-intercepts of a graph

Zeros of a function

Roots of a polynomial function

Solutions of p(x)= 0

o If the solution/roots/zeros have a nonzero, imaginary part, there are no

x-intercepts

Even and Odd Functions

o Even function: f(-x) = f(x)

Symmetric about the y-axis

If (x, y) is on the graph, then (-x, y) is too

o Odd function: f(-x) = -f(x)

Symmetric about the origin

If (x, y) is on the graph, then (-x,-y) is too

o Even functions

Polynomials with only even powers

Cos(x)

Sec(x)

o Odd functions

Polynomial with only odd powers

Sin(x)

Tan(x)

Csc(x)

Cot(x)

o Both even and odd f(x)=0

High and Low points

o Extremum: maximum or minimum

o Extrema: minima or maxima

o Local or relative extrema

Highest or lowest point in the neighborhood

Occurs at the turning point of polynomials

o Absolute maximum

**week 4: quiz**