**Chapters P and One**

Chapter P - Finding Intercepts

Example!

Find any intercepts for the following equation: y=(x-3)/(x-4)

To find the y-intercept, set all x values to 0

x = 0, y = (0-3)/(0-4)

y = 3/4

(0, 3/4) y-intercept

To find the x-intercept, set all y values to 0

y = 0, 0 = (x-3)/(x-4)

0 = x-3

x = 3

(3, 0) x-intercept

Chapter P - Testing for Symmetry

Tests for Symmetry Rules

1. the graph of an equation in x and y is

symmetric with respect to the y-axis

when

replacing x by -x

yields an equivalent equation

2. the graph of an equation in x and y is

symmetric with respect to the x-axis

when

replacing

y by -y

yields an equivalent equation

3. the graph of an equation in x and y is

symmetric with respect to the origin

when replacing

x by -x and y by -y

yields an equivalent equation

Using Intercepts and Symmetry to Sketch a Graph

If you know the intercepts and the symmetry of an equation, you can use this information to sketch a graph.

Example!

Test for symmetry with respect to each axis and to the origin

**Jack Kiriazes and Donald Phan**

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y² = x² - 5

with respect to the y-axis

y² = (-x)² - 5

y² = x² - 5

with respect to the x-axis

(-y)² = x² - 5

y² = x² - 5

with respect to the origin

(-y)² = (-x)² - 5

y² = x² - 5

y² = x² - 5 is symmetric with respect to

both axes and the origin

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Example!

y= -½x + 3

y-intercept: y = -½(0) + 3 = 3

(0, 3)

x-intercept: -½x + 3 = 0

-½x = -3

x = 6

(6, 0)

Symmetry: None

Finding Points of Intersection

Given two equations, make sure that

they are

both set equal to y

.

Once they are both equal to y,

set both

equations equal to each other.

Solve for x

, and

plug back in

to equation

to get y value.

Example

Given: 5x + 3y = -1

x - y = -5

5x + 3y = -1

y = 1/3(-5x -1)

x - y = -5

y = x + 5

1/3(-5x -1) = x + 5

-5x - 1 = 3x + 15

-16 = 8x

-2 = x

When x = -2, y = 3

Point of Intersection is: (-2, 3)

Finding the Slope of a Line

Slope Formula

Example!

Find the slope of the line passing through the following points

(-7, 8) and (-1, 6)

(6-8)

(-1-(-7))

_________

=

____

-2

6

-1

3

____

Slope =

Finding an Equation of a Line

Point: (-3, 0)

Slope: m = -2/3

Point-Slope Form

y - 0 = -2/3 (x - (-3))

y = -2/3 (x + 3)

Slope-Intercept Form

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**Rate of Change**

The purchase of a new machine is $12,500, and its value will decrease by $850 a year. Use this information to write a linear equation that gives the value V of the machine t years after it is purchased. Find its value at the end of 3 years.

Slope = -850

V(t) = -850t + 12,500

V(3) = -850(3) + 12,500 = $9950

The value of the machine after 3 years will be $9950.

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Evaluating a Function

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Finding Domain and Range of a Function

The Domain of a function is the set of all real numbers for which the equation is defined. (the x-values)

The Range of a function is the set of all real numbers that the function can equal. (the y-values)

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Find the Domain and Range of:

Domain: (-infinity, infinity)

Range: (-infinity, 0]

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Example!

Important Graphs

to Know!

Transformations!

Composite Functions

Even and Odd Functions

The function y = f(x) is

even

when

f(-x) = f(x)

The function y = f(x) is

odd

when

f(-x) = -f(x)

Example!

f(x) = 1 + cos(x)

f(-x) = 1 + cos(-x)

= 1 + cos(x)

this function is even

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Chapter 1- The Tangent Line Problem

Chapter 1- The Area Problem

As point Q approaches point P, the slope of the secant approaches the slope of the tangent. Thus, the slope of the tangent is the limit of the slopes of the secant lines.

Slope of the secant line

Pre-Calculus

Spoiler Alert!

Pre-Calculus

One can estimate the are under the curve by putting rectangles underneath it, and then finding the area of the rectangles. The more rectangles that are used, the better the estimation becomes. The final goal is to determine the limit of the sum of the areas of the rectangles as the number of rectangles increases without bounds.

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Finding the Limit Numerically and Graphically

Numerically

Graphically

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Evaluating Limits with an Indeterminate

By Factoring Example!

By Rationalizing Example!

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More Fun with Limits

Piecewise Limit

No Limit

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Finding the Limit of Polynomials

Just Plug It In

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Limits of Trig Functions

Squeeze Theorem

Continuity

Graphs That Aren't Continuous

Removable

Removable

Non-Removable

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Greatest Integer Function

Intermediate Value Theorem

Infinite Limits

As f(x) approaches 2 the function shoots up from the left and down from the right. Thus the limits are - infinity from the left and + infinity form the right.

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Continuity Example

f(x)= x^3+3x^2+1

@ (2,21)

1) f(2)= 21

2) lim x^3+3x^+1 = 21

x approaches 2

3)

lim x^3+3x^+1 = f(2)

x approaches 2

Continuous

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