Transcript Rene Descartes: Founder of Analytic Geometry Cartesian Coordinate System Graphing an equation Graphically finding solutions to equations.

Rene Descartes: Founder of Analytic Geometry
Cartesian Coordinate System
Graphing an equation
Graphically finding
solutions to equations
Cartesian coordinate plane is formed by two equal perpendicular lines. These lines are
called axes. The y-axis is the vertical axis while the x-axis is the horizontal axis.
y-axis
origin
The point where the axes intersect is
known as the origin.
x-axis
The Cartesian plane is divided into four quadrants.
Distances are measured from the origin along each axis.
The distances are positive moving right along the x-axis and up along the y-axis.
The distances are negative moving left along the x-axis and down along the y-axis.
The distances continue to infinity along each axis in both directions.
The position of different points on the Cartesian plane is expressed in relation to
first the horizontal distance and then the vertical distance from the origin.
The horizontal distance is known as the x-coordinate.
The vertical distance is known as the y-coordinate.
The position is expressed as ordered pairs (x,y). The x-coordinate is always
written first.
The purple point is the origin.
The green point is located 2
units right and 3 units up from
the origin.
The red point is located 3 units
left and 1 unit up from the
origin.
The blue point is located 1.5
units left and 2.5 units down
from the origin
The Cartesian Plane can be viewed as a city map.
A location within the city is indicated by street and avenue.
The avenues would be the x-values.
The streets would be the y-values.
Street
Avenue
The Cartesian plane is also used to show the relationship between two variables
x and y by plotting multiple points on the plane. The relationship between the
points on the graph below is that each time x moves one to the left, y moves
three up.
A point on a graph is located in terms of where the y-coordinate is in
relation to the x-coordinate. Thus, the y-coordinate is considered to be
dependant on the x-coordinate.
y-axis
The x-axis is the independent variable.
The y-axis is the dependent variable.
x-axis
A curve can be drawn through all the points on a graph. It is called a curve even
if the graph is in a straight line. The curve not only shows what y-coordinates
are for each integer values of x (shown on the graph below) but also for the
decimal values of x. The relationship between every value of x and y on the
curve is the same.
The relationship represented by the curve of graph satisfies a mathematical
equation. Any mathematical equation with two variables can be represented on
the Cartesian plane. The curve on the graph below satisfies the equation y=x3.
Putting any x-coordinate in the equation for x, the corresponding answer for y
will be the same as the y-coordinate on the curve.
How to graph a mathematical equation on a Cartesian Plane?
1. Get a mathematical equation.
2. Create a Table of Values. To create a table of values, take a number of create a
table with two columns with heading x and y. Put a number of integer values into
the x-column. This should include zero and number of positive and negative
numbers. Find y by putting the corresponding x values into the given equation.
X
Example
Y
Example
1. The equation is 3x-2=y
2. Create a Table of Values.
3. Select a number of positive and negative values for x, including zero.
4. Substitute these x values into the equation to get the corresponding y values.
X
Y
-3
-11
-2
-8
-1
-5
0
-2
1
1
2
4
3
7
Click on this box to
practice making Table
of Values.
Back
Graphing a mathematical equation continued
3. The different pairs of numbers in the Table of Values can now be plotted
on the Cartesian Plane.
10
9
8
7
6
5
4
3
2
1
0
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1 0
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9 10
4. Before we can connect the points with a curve, we have to know what kind of
curve it is going to be. Does it dip or curve between the points? Is it a straight
line? Is there any places where the curve is undefined?
The Highest Degree of the Equation will give the shape of the curve.
Click on the buttons to view the different graphs.
Linear
Quadratic
Cubic
Quartic
1.
2.
3.
4.
Degree: one
Shape of Curve: straight line
Simplest equation: y=x
Number x-intercepts: one
Back
1.
2.
3.
4.
Degree: two
Shape of Curve: parabola
Simplest Equation : y=x2
Number of x-intercepts: two
Back
1.
2.
3.
4.
Degree: three
Shape of Curve: shown below
Simplest Equation: y=x3
Number of x-intercepts: three
Back
1.
2.
3.
4.
Degree: four
Shape of Curve: shown below
Simplest Equation: y=x4
Number of x-intercepts: four
Back
The equation y=3x-2 is a linear equation so the curve is a straight line.
10
9
8
7
6
5
4
3
2
1
0
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-1 0
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9 10
Explore what the curves of following forms of equations will look like.
Click on the different equations.
The place where the curve crosses the x-axis is called the x-intercept.
This graph has two x-intercepts
x-intercept
x-intercept
The x-axis is the line where all the y-coordinates are zero. This
means that a x-intercept has a y-coordinate that is equal to zero.
(2,0)
(-4,0)
(-1,0)
The solutions of an equation are the values of x for when y=0.
The x-intercepts are then the solutions of an equation.
This is the graph of y=x2+3x-10
The x-intercepts are -5 and 2.
20
18
16
14
12
10
8
6
4
2
0
-2 0 1 2 3 4 5 6 7 8 9 10
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1
-4
-6
-8
-10
-12
-14
-16
-18
-20
(-5,0)
(2,0)
Solving the Equation
The solutions for y=x2+3x-10 are:
=(-5)2+3(-5)-10
=25+(-15)-10
=0
=(2)2+3(2)-10
=4+6-10
=0
The solutions of an equation can be solved graphically by finding the x-intercepts.
Sometimes x-intercepts are whole numbers and can be observed from the graph
but most times the x-intercepts falls between the whole number making it difficult
to identify. Example below shows this.
Graphing calculators have different functions that
can be used to find the x-intercept