Linear Functions Section 1-1 Points Lines Objective: To find the intersection of two lines and to find the length and the coordinates of the midpoint of a segment.

Each point in a plane can be associated with the ordered pair of numbers called the coordinates of the point. Also each ordered can be associated with a point in the plane. The association of points and ordered pairs is the basis of Coordinate Geometry, a branch of mathematics that connects Geometric and Algebraic ideas.

To set up a coordinate system we use two number lines perpendicular to each other these are called the x axis and y axis. We designate the intersection as the origin. The axes separate the plane into 4 quadrants. We see several points on the plane. Each point is described by the x-coordinate and the y-coordinate.

Linear Equations A solution of the equation 2x – 3y = 12 is the set of ordered pair of numbers that make the equation true. For example (0,-4) is a solution. Because 2(0) – 3(-4) = 12 Several solutions are shown. (0,-4) is called the y-intercept and (6,0) is called the x-intercept. Note: That the with the y-intercept in the ordered pair x=0 and in the y- intercept the y=0.

Example 1 Sketch the graph of 2x + 6y = 12 One way is to find the x and y intercepts.

Graphing an equation using the point plotting method Once you've picked x-values, you have to compute the corresponding y- values: you can pick whatever values you like, but it's often best to "space them out" a bit. For instance, picking x = 1, 2, 3 might not give you as good a picture of your line as picking x = –3, 0, 3. That's not a rule, but it's often a helpful method. x y = 2x + 3 Ordered Pair -3 y = 2( -3) + 3 = = 0 ( -3, 0 ) -2 y = 2( -2) + 3 = = - 1 (-2, -1) y = 2( -1) + 3 = = 1 (-1, 1)

Graphing using x and y intercepts An x – intercept is where the line crosses the x axis and y=0 and the y – intercept is where the line crosses the y axis and x=0 first we create a T- chart. Graph 2x – 6y =12 x y

Using The slope- intercept form We can use the slope intercept form y=mx + b By first calculating the y – intercept and then plotting that point and use the rise and run to locate a second point. If we need a third point we can use the rise and run again to locate that point

Example 3 First plot the y-intercept (0,4) then from the y- intercept (0,4) then run- 3 and rise 2 to locate the second point (3,2). To find the third point, run -3 and rise 2 and plot (6, 0) Now we can graph the line.

Using a graphing utility to graph linear equation A graphing utility such as a TI-83 or computer software such as microsoft excel. A graphing utility is a powerful tool that quickly generates the graph of an equation in two variables. We start by converting an equation in the form of Ax + By = C into slope intercept form y=mx + b

Setting the Viewing rectangle The next step is setting the viewing rectangle First type the WINDOW button and you will see X min= Xmax = X sci = Y Min = Y Max = Y scl = X res =

The viewing Rectangle X min stands for the minimum value or leftmost value on the x - axis, and X max stands for the maximum value or rightmost value on the x – axis. The same goes for Y min, Y max, and Y scl.

Intersections of Lines You can determine where two lines intersect by drawing their graphs or by solving their equations simultaneously. Consider the following pair of linear equations. 2x + 5y = 10 3x + 4y = 12

Intersection of Lines We could solve graphically but that method is not accurate, The other method is Elimination Follow the steps and solve for x and y.

When two lines are drawn with the same slope and different y- intercepts they are called parallel lines m 1 =m 2 and b 1 ≠ b 2

When two linear equations have infinitely many solutions they have the same line which means they have the same slope and the same y-intercept. m 1 =m 2 and b 1 = b 2

We denote the line segment with endpoints A and B as AB and the length as AB. You can use the formula below to find the distance of AB and the midpoint of segment AB. The Distance and Midpoint formula Let A = (x 1,y 1 ) and B = (x 2,y 2 ) and M is the midpoint of AB then

Example 2 Calculate the distance between the points (2.-6) and (5,3)

Example 3: If A= (1,0) and B= (7,8) find the midpoint of AB

Practice Exercises Find the length and the midpoint coordinate of CD. 1.) C(0,0) D(8,6) ______ (2.) C(4,2),D(6,6) ___ 3.) C(-2,-1),D(4,9)_____(4.) C(-8,-3),D(7,5)_____ No Calculators Please

Homework 3,7,10,11,17,19- 22,25,27,32,33,35 pp. 5-7