Pre-Calculus (Advanced Algebra and Geometry)

We are going to begin our work with linear and quadratic functions

More specifically, their intersections are very interesting to us…

The slope of a straight line measures the steepness of that line (km measure distance Kg measure mass…) Linear:

Write the equation of the line that passes through the points A (1,4) and B (4, -2) y = mx + b First find the slope: m = y 2 – y 1 x 2 – x 1

A( 1, 4 ) B( 4, -2) x 1, y 1 x 2, y 2 m = -2 – (+4) m = -6 3 m = -2 y = -2x + b 4 = -2(1) + b 4 = -2 + b b = 6 y = -2x + 6

Graph the following quadratic y = x 2 + 3x + 5 xy FD D Draw in sketchpad…. quadratic:

To factor an expression means to: re-write it as a product Why factor? Once an expression has been factored, equivalent factors can divide to one Mechanics:

Remember factor trees? 9 33X 91X 9 = 3 X 3

Notice: 4(x + 2) = 4x + 8 Expanding Factoring

3x sub = 2 Direct = 3(2) = 12 3 = 4 Factoring 3x = 3(x + 2) = x + 2 = 4

Crossing out 3x = x + 6 = 8

Trinomials: ax 2 + bx + c Simple (a = 1): Add to the middle Multiply to the last

Factor x 2 + 5x + 6 =(x )(x ) simple Add: 5 Multiply: 6

Complex: (a > 1) Decomposition Add to the middle, multiply to (first)(last) Common Factor twice

Factor 6x 2 – 1x – 2 =6x 2 – 4x + 3x – 2 = 2x(3x – 2) + 1(3x – 2) = (3x – 2)(2x + 1) You may also guess and check complex Add: -1 Multiply: -12 CF the first pair, then CF the second pair

Difference of Squares x 2 – y 2 = (x – y)(x + y) x 2 – 25 =(x – 5)(x + 5)

Rationalize the denominator: X

Expand and Simplify: X-2

Simplify

For the function f(x) = 3x + 12, determine f(-2) (-2,6)

For the function f(x) = 6x – 2, determine f(3 + h) in simplified form 6h + 16

For the function, determine in simplified form f(x) = 6x 6

See sheet 2,3,4,6,8,9, 10,11,12