MATH 30P - Lecture 5 - Catalog of Functions PDF

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MATH 30P – Lecture 5 – Catalog of Functions I.

Linear Functions a. f(x) = mx + b b. graph  non-vertical lines

II.

Quadratic Functions a. f(x) = ax2 + bx + c, a ≠ 0 graph  parabolas with vertical axis (of symmetry) b. To find x-intercepts: i. ax2 + bx + c = 0

−b ± √ b 2−4 ac 2a c. How many solutions? i. Does not touch x-axis  no real solution d. Alternate way to find x-intercept i. Complete square 1. Set each “root” to 0 2. Solve for x 3. Get answer (x,0) ii.

x=

III.

Piecewise Defined Functions a ,if x ( ? ) zero a. f (x) b ,if x ( ? ) zero

IV.

Trigonometric Functions a. Basic functions i. Cosine  cos(x) ii. Sine  sin(x) iii. (a, b)  (cos, sin) b. Unit circle i. Whole length of circle = 2π 1. Therefore, first quarter = 2π/4 = π/2 ii. Domain of cos(x)  -1 ≤ cos(x) ≤ 1 iii. Domain of sin(x)  -1 ≤ sin(x) ≤ 1 iv. For every point on unit circle:

{

a2 + b2 = c2  a2 + b2 = 1, where a = cos(x) and b = sin(x) x 0 π/2 π 3π/2

cos(x) 1 0 -1 0

2π 1 v. Example 1: Given that cos(x) = 2/3, 0 < x < π/2, find sin x (2/3)2 + b2 = 1 b2 = 1 – 4/9 = 5/9 b = b =

√

5 9 √5 3

vi. Example 2:

Given sin(x) = -7/10 and 3π/2 < x < 2π a2 + (-7/10)2 = 1 a2 = 1 – 49/100 = 51/100 a = a = cos(x) =

√

51 100 √51 10 √ 51 10...