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 4th

quadrant 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 forY, (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 = 61st : 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 – axisThen 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) substitutioniii) 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 calculateThe 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)