Math 12 Pre-Calculus Midterm Review (Chapters 1 6)

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1 REVIEW SCHEDULE: Date: Topics Covered: Suggested Practice: Feb. 10/11 Chapters 1 3 Unit 1 Test : Pg All Feb. 12/13 Chapter 4, 5 Unit 2 Test : Pg # 1 6, 9, 10, 12 17, 20 Feb. 16/17 Chapter 6 Pg # 7, 8, 11, 19 Chapter 1: Transformations Given any function y f ( x) we can analyze its transformed function as follows: y af b( x c) d Where: a = vertical expansion/compression by a (if a < 0 vertical reflection) b = horizontal expansion/compression by 1/b (if b < 0 horizontal reflection) c = horizontal translation d = vertical translation Whether reading transformations or doing a transformation, follow the order Stretch, Reflect, Translate (SRT) the order of horizontal/vertical does not matter. Inverse Functions are those reflected over the line y=x. To find them, swap x and y. Ex.1: Given the description, write the following transformations for f(x) in function notation: VE of 2, HC of 1/3, VT down 3, HT right 1 Ex.2: Describe the transformations (in order) applied to y 2 f (3x 2) 1 Write the equation for both the base function and the transformed function. 1

2 Chapter 2: Radical Functions & Equations Radical functions can be transformed as in Chapter 1. We can take the square root of a function y=f(x) by taking the square root of all the y values. The new function y f ( x) will have invariant points wherever f(x)=0 and f(x)=1. We can use the corresponding graph of a radical function to solve a radical equation by finding its zeros. To solve a radical equation algebraically, isolate the radical and square both sides. Check for extraneous roots. Ex.1: If the point (-2,7) is on the graph of y=f(x), name a point on the graph of y f ( x). Ex.2: Given the graph of y=f(x) graph the function y f ( x). State both domains and ranges, as well as any invariant points. Given 2 x 4 3 0, solve for x both algebraically and using graphing technology. 2

3 Chapter 3: Polynomial Functions To divide any two polynomials use either long or synthetic division. We can write a division statement: P( x) Q( x) D( x) R Remainder Theorem: given a polynomial P(x), we can find the remainder when P(x) is divided by (x-a) by P(a). If P(a) = 0, the Factor Theorem states that (x-a) must be a factor of P(x). Graphing polynomials of degree > 2 ODD Functions EVEN Functions Ex.1: Given the polynomial y x x x 2 2 2( 1) ( 2)( 3), determine the following (without graphing): a) Describe the end behaviour of the function. b) Determine the x-intercepts of the function and their multiplicities. c) Determine the y-intercept of the function. 3 2 Ex.2: Factor f(x) = 2x 5x 4x 3 completely. 3

4 Sketch the graph of the following polynomial function without technology. f x x x x 3 ( ) ( 2) ( 4) Ex.4: Determine the equation of the following polynomial. Chapter 4: Trigonometric Equations Converting between radians and degrees: or 180 Drawing angles in standard position; finding reference angles and co-terminal angles Arc length a r where must be in radians. Using the unit circle (radius = 1) to find either coordinate points on the circle in a given quadrant, or a standard position given a point. Knowing the special triangles: Find an exact or decimal value for any trigonometric ratio. Solve a trigonometric equation (1 st or 2 nd degree) in a specified domain, or for all real numbers. 4

5 Ex.1: Draw an angle of 235 in standard position. Convert to radian measure. State the quadrant the angle lies in, the reference angle, and another co-terminal angle. Ex.2: Determine the exact coordinate on the unit circle for 5 P 6. Determine the exact value of the other five trigonometric ratios if 1 cos x and tan x < 0. 2 Ex.4: Solve the following trigonometric equations over the domain specified. a) cot x 1 0, 0x 2 b) 2 sin x sin x 2 0, general solution 5

6 Chapter 5: Trigonometric Functions Base functions for y sin x and y cos x Transformations to sinusoidal functions acts the same as in Chapter 1, different terminology: y asin b( x c d Base function y tan x has asymptotes every n 2 Can solve a trigonometric equation using a graph to find the corresponding zeros. Applications of sinusoidal functions (ferris wheel, pendulums, ocean tides, ) **DRAW A GRAPH** Ex.1: Sketch the graph of the function y 2cos( )1 over two cycles. Ex.2: Determine a sinusoidal function for the following graph: 6

7 The number of hours of daylight, L, in Lethbridge, Alberta, may be modelled by a sinusoidal function of time, t. The longest day of the year is June 21, with 15.7 h of daylight, and the shortest day is December 21, with 8.3 h of daylight. a) Determine a sinusoidal function to model this situation. b) How many hours of daylight are there 103 days after June 21? Chapter 6: Trigonometric Identities Determine non-permissible value(s) of a trigonometric expression. Use sum or difference identities to determine the exact value of a trigonometric expression with an angle other than the special angles. Simplify trigonometric expressions by any of the following methods: - Substituting a known identity - Re-writing in terms of sine and cosine - Finding a common denominator - Factoring - Multiplying by a conjugate Prove a trigonometric identity algebraically. Solve a trigonometric equation algebraically by substituting known identities. Ex.1: Determine the non-permissible values, in radians, for the expression tan x 1 cos x Ex.2: Use sum or difference identities to find the exact value of the expression 5 sin 12 7

8 Simplify each of the expressions: a) sec x tan x b) csc2x cot 2x 2 2 Ex.4: Solve the following trigonometric equations over the domain 0x 2 a) sin 2xsin x 0 b) sin x1 cos 2 2 x 8