Precalculus Review Package PDF

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Download Precalculus Review Package PDF

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1. x2 + 3x − 10

4. 2x2 + 7x + 6

2. x2 − 12x + 36

5. 90x4 + 180x3 − 270x2

3. 25 − 9y 2

6. 4(x − 3)3 (2x + 1)5 + 10(x − 3)4 (2x + 1)4

Helpful resources on factoring: • Paul’s Online Math Notes: Factoring • Khan Academy: Factoring Quadratics

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Solving Equations

Solve the following equations √

1. 3(x + 2) − 2(5x + 1) = 7x + 9

4.

2. x2 = x + 6

5. 2x = |x| + 5

3. x3 + 4x2 = 12x

6. |x − 2| = x2 − 4

Helpful resources on solving equations: • Paul’s Online Math Notes: Solving Equations 1

x= x−2

• Khan Academy: – Solving basic equations – Solving quadratic equations by factoring – Solving square root equations – Solving absolute value equations

3

Simplifying Rational Expressions

Simplify each of the following expressions, and state the domain 1.

x2 + x − 6 x2 − 9

x2 + 10x + 25 2. 2 x + 7x + 10

3.

3 1 + x−2 x−1

4.

x2 − 36 x2 + x − 12 x2 + 7x + 6 16 − x2

Helpful resources on rational expressions: • Paul’s Online Notes: Rational expressions • Khan Academy: Rational expressions, equations and functions

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Working with Logarithmic Expressions 1. Evaluate: (a) log2 8 (b) log9

1 3

2. Expand: log10



x−2 x+3



Helpful resources on logarithmic expressions: • Paul’s Online Notes: Logarithm functions • Khan Academy: – Introduction to logarithms – Properties of logarithms

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5

Lines

For each item below, write an equation for the line with the given properties in both pointslope form and slope-intercept form. 1. Passing through the points (1,2) and (3,-1) 2. x-intercept of 5, and y -intercept of -2 3. Passing through the point (3,-1) and parallel to the line with equation 3x + 2y = 5 Helpful resources on lines: • Paul’s Online Math Notes: Lines • Khan Academy: Linear equations, functions and graphs’

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Solving Logarithmic and Exponential Equations

Solve the following equations 1. 3x = 9x−1

3. log3 (2x + 5) = 3   2 = −1 4. log4 x+3

4x+3 = 8x−5 16x−1 Helpful resources on logarithmic and exponential equations:

2.

• Paul’s Online Math Notes: – Solving exponential equations – Solving logarithmic equations • Khan Academy: – Logarithmic equations – Solving exponential equations with logarithms

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Solving Trigonometric Equations (The Unit Circle)

Find all solutions to each of the following equations θ 

1. sin(θ) = 0

3. cos

2. tan(2x) = 0

4. 2 sin(x) = 1

2

Helpful resources on solving trigonometric equations • Paul’s Online Math Notes: Trigonometric equations • Khan Academy: Solving basic sinusoidal equations 3

=0

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Trigonometric Functions

Suppose that sin θ = 53 and

π 2

< θ < π. What are the values of each of the following:

1. cos θ

3. sec θ

2. tan θ

4. csc θ

5. cot θ

Helpful resources on evaluating trigonometric functions • Paul’s Online Math Notes: Trigonometric functions • Khan Academy: The unit circle definition of sine, cosine and tangent

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Word Problems

Rewrite each of the following as an algebraic expression/equation. Make sure that you describe any variables that you use. 1. The square of a number is equal to three less than twice the number. 2. The difference of two numbers is equal to the cube of the sum of the numbers 3. The difference of two numbers is equal to the sum of the cubes of the numbers 4. The absolute value of two more than a number is greater than the square root of the number.

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Algebraic Properties

For which values of a and b is it true that a2 + b2 = (a + b)2 ?

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Graphing Functions

Sketch the graphs of the following functions (find and label x- and y-intercepts and any maximums/minimums, if they exist) 1. f (x) = sin(x)

3. h(x) = 3x − 1

2. g(x) = cos(x) + 1

4. F (x) = log2 (x)

Helpful resources on graphing functions: • Paul’s Online Math Notes: – Common graphs Examples 1, 2, 11, 12 and 13 – Exponential functions 4

– Logarithm functions • Khan Academy: – The Graphs of sine, cosine and tangent – Graphs of exponential functions – Graphs of logarithmic functions

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Composition of Functions

1. Let f (x) = 2x and let g(x) = x + 1. Are the composite functions (f ◦ g)(x) = f (g(x)) and (g ◦ f )(x) = g(f (x)) equal? 2. Consider the following functions f (x) = x + 1, g(x) = x2 + x, P (x) = x3 . Evaluate (find a formula for) the following expressions, simplify completely. You can assume that h 6= 0 and x 6= −1, 2 g(x + h) − g (x) h P (x) − P (2) (e) x−2 P (x + h) − P (x) (f) h

(a) f (x + 2) (b)

(d)

f (x) − f (−1) x+1

(c) g(x + h) Helpful resources on composition of functions:

• Paul’s Online Math Notes: Functions Examples 2, 6 and 7 • Khan Academy: Composition of functions

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