pre-calculus lesson 1: algebra revisited formulas, definitions, and methods from algebra 1

Pre-CalculusLesson 1: Algebra RevisitedFormulas, definitions, and methods from Algebra 1

The distance d between any two points with coordinates and is given by the formula d = .

The Distance Formula

1 1( , )x y 2 2( , )x y

2 22 1 2 1( ) ( )x x y y

Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)

Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)

( 3 4)2 (2 1)2d =

Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)

( 3 4)2 (2 1)2d =

( 7)2 (1)2 49 1d =

Applying the Distance Formula Find the distance between (-3, 2) and (4, 1)

( 3 4)2 (2 1)2d =

( 7)2 (1)2 49 1d =

50 or 5 2 or 7.07d =

In the coordinate plane, the coordinates of the midpoint of a segment whose endpoints have coordinates and are .

Midpoint Formula

1 2 1 2,2 2

x x y y

1 1( , )x y 2 2( , )x y

Applying the Midpoint FormulaFind the midpoint between (-2, 5) and (6, 4)

Applying the Midpoint FormulaFind the midpoint between (-2, 5) and (6, 4)

2 62

,5 4

2

M =

Applying the Midpoint FormulaFind the midpoint between (-2, 5) and (6, 4)

42

,92

= 2,

92

M =

2 62

,5 4

2

M =

Domain and RangeDomain = the input of a functionRange = the output of a function

Vertical Line Test: If a vertical line can connect two points

on the graph of a relation, then the relation is not a function.

Examples:

http://math.tutorvista.com/calculus/vertical-line-test.html

Evaluating FunctionsTo evaluate a function, simply

replace (substitute) the function's variable with the indicated number or expression.

Evaluating FunctionsTo evaluate a function, simply

replace (substitute) the function's variable with the indicated number or expression.

f(x) = 2x + 5,  Find f(3).

Evaluating FunctionsTo evaluate a function, simply

replace (substitute) the function's variable with the indicated number or expression.

f(x) = 2x + 5,  Find f(3). f(3) = 2(3) + 5 = 11.

Evaluating Functions

Evaluating Functions

Evaluating Functions

Evaluating Functions

Slope Formula:

12

12

xx

yym

Find the slope of the line that goes through the points (-5, 3) and (2, 1).

Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS

Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS

Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

1 3

2 ( 5)m

Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS

Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

1 3

2 ( 5)m

1 3

2 5m

Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS

Find the slope of the line that goes through the points (-5, 3) and (2, 1).

m y2 y1

x2 x1

1 3

2 ( 5)m

1 3

2 5m

2

7m

Designed by Skip Tyler, Varina HS and Vicki Hiner, Godwin HS

A few things to remember about circles:

A circle is a set of points equidistant from a central point.

A circle is named using its center point.

Radius=distance from center to edge of circle.

Diameter=2 x radius

Equation of a CircleThe circle with center (0,0) and radius r is the graph of the equation

222 ryx

Equation of a CircleThe circle with center (h,k) and radius r is the graph of the equation:

222 rkyhx

Applying the Equation of a Circle Identify the coordinates of the

center and the length of the radius in the circle (x − 5)2 + (y + 2)2 = 4

Write the equation of a circle centered at (5,1) with a radius of 5

Applying the Equation of a Circle Identify the coordinates of the

center and the length of the radius in the circle (x − 5)2 + (y + 2)2 = 4

(5, -2), r = 2Write the equation of a circle

centered at (5,1) with a radius of 5

Applying the Equation of a Circle Identify the coordinates of the

center and the length of the radius in the circle (x − 5)2 + (y + 2)2 = 4

(5, -2), r = 2Write the equation of a circle

centered at (5,1) with a radius of 5 (x − 5)2 + (y – 1)2 = 25

Extension ProblemWrite the equation of the circle whose diameter extends from the point (-2,1) to the point (6,-5).