section 1.1 introduction to graphing copyright ©2013, 2009, 2006, 2001 pearson education, inc
TRANSCRIPT
Section 1.1
Introduction to Graphing
Copyright ©2013, 2009, 2006, 2001 Pearson Education, Inc.
Objectives
Plot points. Determine whether an ordered pair is a solution of an equation. Find the x-and y-intercepts of an equation of the form Ax + By = C. Graph equations. Find the distance between two points in the plane and find the midpoint of a segment. Find an equation of a circle with a given center and radius, and given an equation of a circle in standard form, find the center and the radius. Graph equations of circles.
Cartesian Coordinate System
Example
To graph or plot a point, the first coordinate tells us to move left or right from the origin. The second coordinate tells us to move up or down.
Plot (3, 5).Move 3 units left.Next, we move 5 units up.Plot the point.
(–3, 5)
Solutions of Equations
Equations in two variables have solutions (x, y) that are ordered pairs.
•Example: 2x + 3y = 18
When an ordered pair is substituted into the equation, the result is a true equation. The ordered pair has to be a solution of the equation to receive a true statement.
Examples
a.Determine whether the ordered pair (5, 7) is a solution of 2x + 3y = 18.
2(5) + 3(7) ? 1810 + 21 ? 18 11 = 18 FALSE
(5, 7) is not a solution.
b.Determine whether the ordered pair (3, 4) is a solution of 2x + 3y = 18.
2(3) + 3(4) ? 18 6 + 12 ? 18 18 = 18 TRUE (3, 4) is a solution.
Graphs of Equations
To graph an equation is to make a drawing that represents the solutions of that equation.
x-Intercept
The point at which the graph crosses the x-axis.
An x-intercept is a point (a, 0). To find a, let y = 0 and solve for x.
Example
Find the x-intercept of 2x + 3y = 18.
2x + 3(0) = 18 2x = 18
x = 9
The x-intercept is (9, 0).
y-Intercept
The point at which the graph crosses the y-axis.
A y-intercept is a point (0, b). To find b, let x = 0 and solve for y.
Example
Find the y-intercept of 2x + 3y = 18.
2(0) + 3y = 18 3y = 18
y = 6
The y-intercept is (0, 6).
Example
Graph 2x + 3y = 18.
We already found the x-intercept: (9, 0)We already found the y-intercept: (0, 6)We find a third solution as a check. If x is replaced with 5, then
253y18103y18
3y8
y8
3Thus, is a solution.5,
8
3
Example (continued)
Graph: 2x + 3y = 18.x-intercept: (9, 0)y-intercept: (0, 6)Third point:
5,8
3
Example
Graph y = x2 – 9x – 12 .
(12, 24)2412
–2
32
32
26
12
–2
24
y
(10, –2)10
(5, 32)5
(4, 32)4
(2, 26)2
(0, 12)0
(1, –2)1(3, 24)3(x, y)x
Make a table of values.
The Distance Formula
The distance d between any two points (x1, y1) and (x2, y2) is given by
d (x2 x1)2 (y2 y1)2 .
Example
d (3 2 )2 ( 6 2)2
d 52 ( 8)2 2564
d 89 9.4
Find the distance between the points (–2, 2) and (3, 6).
Midpoint Formula
If the endpoints of a segment are (x1, y1) and (x2, y2), then the coordinates of the midpoint are
x1 x2
2,y1 y2
2
.
Example
Find the midpoint of a segment whose endpoints are (4, 2) and (2, 5).
4 2
2,
25
2
2
2,
3
2
1, 3
2
Circles
A circle is the set of all points in a plane that are a fixed distance r from a center (h, k).
The equation of a circle with center (h, k) and radius r, in standard form, is
(x h)2 + (y k)2 = r2.
Example
Find an equation of a circle having radius 5 and center (3, 7).
Using the standard form, we have
(x h)2 + (y k)2 = r2
[x 3]2 + [y (7)]2 = 52
(x 3)2 + (y + 7)2 = 25.