Download Slide 1 - Elgin Local Schools

Pre-Calculus
Lesson 1.1
Linear Functions
Points and Lines
Point A position in space. Has no size
Coordinate
An ordered pair of numbers which describe a point’s
position in the x-y plane (x,y)
2nd
1st
X-axis
The horizontal axis.
Quadrant
Quadrant
Y-axis
The vertical axis
3rd
quadrant
4th
Quadrant
Origin
The point of intersection of the
x- and y axis. Identified as (0,0)
Quadrants
One of 4 areas the x-y axes
divide a coordinate plane into.
Linear Equations
• General Form Ax + By = C
• Solution Any ordered pair (x,y) that
makes the equation true
Example 1 Sketch the graph of the equation
3x + 2y = 18.
Method 1 Find the x- and y-intercepts of the graph.
(To find the x-intercept, let y = 0. To find the y-intercept, let x = 0)
Substituting 0 in for y yields:
Substituting 0 in for x yields:
3x + 2(0) = 18
3(0) + 2y = 18
3x = 18
2y = 18
x=6
y=9
Now plot the points (6,0) and (0,9) and draw your line.
The graph of 3x + 2y = 18:
Slope-intercept method: y = (m)x + (b)
Here the equation is solved for y. Once the equation is solved for
Y, (m) -- the coefficient of x -- will always identify the slope of the
line. (b) – the constant term will always identify the point where the
line crosses the y-axis (y-intercept)
Graph the equation: 3x - 2y = 6
1st : solve for y
-3x
- 3x
- 2y = - 3x + 6
-2
-2
y = (3/2)x - 3
Since the equation is solved for y: y = (3/2)x – 3
(we align under the equation y = (m)x + (b)
So we can identify values for m = (3/2), & b = - 3
(Knowing m = slope  rise & b  y-intercept
run
We go to our graph and place a point at:
- 3 on the y – axis
Then from there we move:
Up 3 spaces and right 2 spaces
Special cases from the General form: Ax + By = C
a) If C = 0, the line will always pass
through the origin. 3x + 2y = 0
(blue line)
b) If A = 0, (no x-term) The line will
always be horizontal: 0x + 2y = 6
or
2y = 6
or
y=3
(red line)
c) If B = 0, (no y term) the line will
always be vertical: 3x + 2(0) = 6
or
3x = 6
or
x=2
When working with 2 lines at the same time (called a
system of equations) one of ‘3’ things can happen:
a) Parallel lines (no solutions occur)
y = (2/3) x - 3
2x – 3y = 9
a) Intersecting lines (one solution occurs)
y = (-2/3) x + 2
5x – 4y = 8
a) Same line (Concurrent,
Or coincident lines)
(infinite number of solutions)
y = (-5/2)x + 4
5x + 2y = 8
Solving a system of equations:
1st : Remember there are three different methods:
i) Graphing
ii) substitution
iii) Addition-subtraction
(Elimination method)
Example 2 Solve this system: 3x – y = 9
7x – 5y = 25
(Grapher’s can be used to check the algebra process only!!!!!!)
(I expect to see pencil/paper detailed processes at all times!!!!!!)
Use your method of choice 
(Check the solution process for example 2 in the book)
Two synonymous terms are: Length and Distance
To find the length of a line segment we need to calculate
The distance between two points: (x1,y1), (x2,y2)
Remember the Distance Formula---Oh you better! 
To find the ‘midpoint’ of a line segment, we find the
‘average’ between the two endpoints!
Remember the Midpoint Formula-- it is so suite!
Example 3:
If A = (-1,9) and B = (4,-3), find:
a) The length of AB
(check the solution process in the book)
b) The coordinates of the
midpoint of AB
(check the solution process in the book)