## Overview

• Section 2.1 in the textbook:

– Cartesian Plane

– Distance & Midpoint Formulas

– Graphing Equations

– Intercepts

– Circles

**Cartesian Plane**

• The** Cartesian Plane** consists of two number

lines – one horizontal and one vertical

– The point of intersection of the two lines is known as

the origin and has coordinates (0, 0)

– A coordinate (x, y) consists of two numbers :

• x re presents the horizontal (left to right) direction from the

origin

• y represents the vertical (top to bottom) from the origin

• We plot points by starting at the origin and then

using the x and y parts of the point

– Split into four sections called** quadrants**

based on the sign of x and y of the coordinate

(x, y)

## Cartesian Plane (Example)

**Ex 1:** Plot the fol lowing points on the

Cartesian Plane and tell the quadrant in

which each lies: (-5, 3), (0,2), (1, 1),

(-4, -4), (-3, 0), (2, -1)

**Distance & Midpoint Formulas**

## Distance Formula

• Some times we want to know the distance between two

points on a graph that may not necessarily lie on a

straight line

• Use the Pythagorean

Theorem

• From this, we get the

distance formula for

any two points

(x_{1}, y_{1}) and (x_{2}, y_{2}):

## Midpoint Formula

• Sometimes we need to know the midpoint of the line

segment between (x_{1}, y_{1}) and (x_{2}, y_{2})

• Thus, the midpoint of the

line segment connecting

(x_{1}, y_{1}) and (x_{2}, y_{2}) is:

## Distance & Midpoint Formulas

(Example)

**Ex 2:** Find the distance and midpoint of

(-6, 2) and (-1, -1). For the distance, give

both the exact and approximate answers.

**Ex 3:** Find the distance and midpoint of

(4, 5) and (2, -6). For the distance, give

both the exact and approximate answers.

**Graphing Equations**

• Given an equation, we can pick values for

one of the variables and solve for the other

– Ex: Given y = -x^{4} -> when x = -2, y = -16

• Thus, (-2, -16) lies on the graph of y = -x^{4}

• By repeating the process a few times, we

obtain a graph of the equation

– Usually 3 to 4 points are satis factory

– Pick both positive and negative values

## Graphing Equations (Example)

**Ex 4:** Graph using a table of values:

y = |x – 3| + 1

**Ex 5:** Graph using a table of values:

**Intercepts**

**• x-intercept:** where the graph of an

equation crosses the x-axis

–Written in coordinate form as (x, 0)

– To find, set y = 0, and solve for x:

• May entail solving a linear or quadratic equation

**• y-intercept:** where the graph of an

equation crosses the y-axis

–Written in coordinate form as (0, y)

– To find, set x = 0, and solve for y

## Intercepts (Example)

**Ex 6:** Find the x and y intercepts of

**Ex 7: **Find the x and y intercepts of

**Circles**

## Standard Equation of a Circle

• **Circle:** the set of all points r units away, where
r

is the **radius**, from a point (h, k) called the

**center**

• Given the radius and the center, we can

construct the **standard equation of a circle:**

where:

(h, k) is the center

r is the radius

## Standard Equation of a Circle

(Example)

**Ex 8:** Find the standard equation of a circle

with center (-2, 1) and contains the point

(0, 3)

**Ex 9:** Find the standard equation of a circle

with a diameter of (-1, -2) and (5, -6)

## General Equation of a Circle

• There is another commonly used form of

the equation of a circle

• Ex: Start with the standard equation

(x – 3)^2 + (y + 2)^2 = 16

– Square the binomials :

x^2 – 6x + 9 + y^2 + 4y + 4 = 16

– Collect like terms :

x^2 – 6x + y^2 + 4y + 13 = 16

– Set the right side to 0 and rearrange the left:

x^2 + y^2 – 6x + 4y – 3 = 0

– This last equation can be written in the form

x^2 + y^2 + Ax + By + C = 0 (A, B, and C are

constants) and is known as the **general**

equation of a circle

• Notice that the right side of the general equation

is set to 0

**• Complete the square** to convert the

general equation to the standard equation

– Complete the square on x

– Complete the square on y

## General Equation of a Circle

(Example)

**Ex 10: **Find the center and radius of

x^2 + y^2 – 4x + y – 7 = 0

• After studying these slides, you should know how to do

the following:

– Graph points on a Cartesian Plane and be able to identify the

quadrant in which it lies

– Apply the distance and midpoint formulas

– Graph an equation by picking points

– Identify the intercepts of an equation

– Extract the center and radius of a circle from either the standard

or general equations

– Construct equations of circles

• Additional Practice

– See the list of suggested problems for 2.1

• Next lesson

– Introduction to Functions (Section 2.2)