2022 ACP Pre Calculus Summer Packet
Introduction
Welcome to the 2022 ACP Pre Calculus Summer Packet! This packet is designed to help you review important algebra and geometry concepts that are essential for success in Pre Calculus. Completing this packet will strengthen your mathematical skills and prepare you for the upcoming school year.
Please work through the problems systematically, showing all work where necessary. This packet should be completed before the first day of class and will be collected during the first week of school.
Materials Needed
- Pencils and erasers
- Lined or graph paper
- Scientific or graphing calculator (TI-84 or similar recommended)
- Ruler and protractor
Topic 1: Functions and Graphs
Key Concepts
A function is a relation where each input has exactly one output. Functions can be represented in various ways: algebraically (as an equation), graphically, numerically (as a table), or verbally (in words).
Types of Functions
- Linear functions: f(x) = mx + b
- Quadratic functions: f(x) = ax + bx + c
- Exponential functions: f(x) = ab
- Logarithmic functions: f(x) = logx
- Rational functions: f(x) = P(x)/Q(x), where P and Q are polynomials
Example 1
Find the domain and range of f(x) = x + 2.
Solution:
Domain: All real numbers, or (-, ). Any real number can be squared.
Range: Since x 0 for all real x, x + 2 2. Therefore, the range is [2, ).
Example 2
Given f(x) = 2x + 3 and g(x) = x - 1, find (f g)(x).
Solution:
(f g)(x) = f(g(x)) = f(x - 1) = 2(x - 1) + 3 = 2x - 2 + 3 = 2x + 1
Topic 2: Polynomial and Rational Functions
Key Concepts
Polynomials are expressions consisting of variables and coefficients, involving only addition, subtraction, multiplication, and non-negative integer exponents.
Properties of Polynomials
- The graph of a polynomial function is smooth and continuous.
- The number of turning points is at most (n - 1), where n is the degree.
- The end behavior depends on the degree and the sign of the leading coefficient.
Solving Polynomial Equations
- Factoring: Completely factor the polynomial, then set each factor to zero.
- Rational Root Theorem: If P(x) = 0 has integer coefficients, then any rational solution p/q has p as a factor of the constant term and q as a factor of the leading coefficient.
- Quadratic Formula: For ax + bx + c = 0, x = (-b (b - 4ac))/(2a)
Example 3
Solve x - 2x - 5x + 6 = 0.
Solution:
By the Rational Root Theorem, possible rational zeros are 1, 2, 3, 6.
Testing x = 1: 1 - 2 - 5 + 6 = 0, so (x - 1) is a factor.
Dividing by (x - 1), we get x - x - 6, which factors to (x - 3)(x + 2).
Therefore, the zeros are x = 1, x = 3, and x = -2.
Rational Functions
Rational functions are ratios of polynomials. Key features to identify:
- Vertical asymptotes: Values of x that make the denominator zero (but not the numerator).
- Horizontal asymptotes: Determine the behavior as x approaches infinity.
- Holes: Values where both numerator and denominator are zero (common factors).
Example 4
Find the asymptotes of f(x) = (x - 4)/(x - 9).
Solution:
Vertical asymptotes: x - 9 = 0, so x = 3 and x = -3.
Horizontal asymptote: The degrees of numerator and denominator are equal, so y = (leading coefficient of numerator)/(leading coefficient of denominator) = 1/1 = 1.
Holes: None, since x - 4 and x - 9 have no common factors.
Topic 3: Exponential and Logarithmic Functions
Exponential Functions
The general form of an exponential function is f(x) = ab, where a 0, b > 0, and b 1.
Where A = final amount, P = principal, r = interest rate, n = times compounded per year, t = time in years
Logarithmic Functions
A logarithmic function is the inverse of an exponential function. If y = logx, then x = a.
log(xy) = logx + logy
log(x/y) = logx - logy
log(x) = n logx
log(1) = 0
log(a) = 1
Solving Exponential and Logarithmic Equations
Example 5
Solve 2 = 16.
Solution:
2 = 16
Write 16 as a power of 2: 2 = 2
Since the bases are equal, the exponents are equal: x + 2 = 4
x = 2
Example 6
Solve log(x + 3) + log(x - 1) = 3.
Solution:
log(x + 3) + log(x - 1) = 3
log[(x + 3)(x - 1)] = 3
(x + 3)(x - 1) = 2 = 8
x + 2x - 3 = 8
x + 2x - 11 = 0
Using the quadratic formula: x = (-2 (4 - 4(1)(-11)))/(2)
x = (-2 (4 + 44))/(2)
x = (-2 48)/(2) = (-2 43)/(2) = -1 23
Since the domain requires x + 3 > 0 and x - 1 > 0, x > 1.
Only x = -1 + 23 2.46 meets this condition.
Topic 4: Trigonometry Basics
Key Concepts
Trigonometry studies relationships between the sides and angles of triangles.
sin = opposite/hypotenuse
cos = adjacent/hypotenuse
tan = opposite/adjacent = sin /cos
cosecant csc = 1/sin
secant sec = 1/cos
cotangent cot = 1/tan = cos /sin
sin + cos = 1
1 + tan = sec
cot + 1 = csc
Unit Circle
The unit circle is a circle with radius 1 centered at the origin of the coordinate plane. Each point on the unit circle corresponds to an angle's sine and cosine values.
| Angle (radians) | Angle (degrees) | sin | cos | tan |
|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 |
| /6 | 30 | 1/2 | 3/2 | 1/3 |
| /4 | 45 | 2/2 | 2/2 | 1 |
| /3 | 60 | 3/2 | 1/2 | 3 |
| /2 | 90 | 1 | 0 | undefined |
Trigonometric Functions Graphs
- Sine function: Period = 2, Range = [-1, 1]
- Cosine function: Period = 2, Range = [-1, 1]
- Tangent function: Period = , Range = (-, )
Topic 5: Systems of Equations and Inequalities
Solving Systems of Linear Equations
Methods:
- Graphing: Graph both equations and find the intersection point.
- Substitution: Solve one equation for a variable and substitute into the other.
- Elimination: Add or subtract equations to eliminate one variable.
- Matrix methods: Use inverse matrices or Cramer's rule.
Example 7
Solve the system:
3x + 2y = 7
2x - y = 4
Solution using elimination:
Multiply the second equation by 2: 4x - 2y = 8
Add this to the first equation: (3x + 2y) + (4x - 2y) = 7 + 8
7x = 15, so x = 15/7
Substitute x into the second equation: 2(15/7) - y = 4
30/7 - y = 28/7
y = 30/7 - 28/7 = 2/7
The solution is x = 15/7, y = 2/7
Systems of Nonlinear Equations
These involve at least one nonlinear equation. Methods include substitution and graphing to identify intersections.
Example 8
Solve the system:
y = x
y = 2x + 3
Solution:
Set the equations equal to each other: x = 2x + 3
Rearrange: x - 2x - 3 = 0
Factor: (x - 3)(x + 1) = 0
Solutions: x = 3 and x = -1
Find corresponding y-values:
When x = 3: y = 3 = 9
When x = -1: y = (-1) = 1
Solution pairs: (3, 9) and (-1, 1)
Systems of Inequalities
Graph each inequality, indicating the region that satisfies all inequalities.
Example 9
Graph the solution set of the system:
x + y 4
x - y 2
Solution:
Graph the line x + y = 4 (intercepts at (0,4) and (4,0)). Shade below the line since y -x + 4.
Graph the line x - y = 2 (intercepts at (0,-2) and (2,0)). Shade above the line since y x - 2.
The solution is the region where both shadings overlap.
Topic 6: Sequences, Series, and Probability
Sequences and Series
Arithmetic Sequences
Sum of first n terms: S = (n/2)(a + a) = (n/2)(2a + (n-1)d)
Geometric Sequences
Sum of first n terms: S = a(1 - r)/(1 - r) when r 1
Sum to infinity: S = a/(1 - r) when |r| < 1
Example 10
Find the sum of the first 10 terms of the arithmetic sequence where a = 3 and d = 5.
Solution:
First, find a = a + (10-1)d = 3 + 9(5) = 48
Then, S = (10/2)(3 + 48) = 5(51) = 255
Probability Basics
Combinations and Permutations
Example 11
A committee of 3 is to be selected from 7 people. How many different committees can be formed?
Solution:
This is a combination problem because order doesn't matter.
7C3 = 7!/(3!(7-3)!) = 7!/(3!4!) = (765)/(321) = 35
Therefore, 35 different committees can be formed.
Practice Problems
Complete these problems to test your understanding. Show all work and check your answers.
- Find the domain of f(x) = (x - 16).
- Solve for x: 3^(2x+1) = 27.
- If f(x) = 2x + 5 and g(x) = x - 3, find (f g)(2).
- Given that sin = 4/5 and /2 < < , find cos and tan .
- Solve the system: 2x + y = 5 and x - 3y = 6.
- Find the sum of the first 15 terms of the geometric sequence: 2, 6, 18, ...
- Find the inverse function of f(x) = 3x + 7.
- Solve: log(x + 2) = 3.
- Find the slant asymptote of f(x) = (x + 3x + 2)/(x + 2).
- A password consists of 2 letters followed by 3 digits. How many different passwords can be formed if letters and digits cannot repeat?
Final Words
This summer packet covers key concepts you'll need for Pre Calculus. If you find any of these topics challenging, please use online resources, textbooks, or ask for help before the school year begins. Taking the time to review now will put you in a strong position for success in Pre Calculus.
Have a wonderful summer, and we look forward to seeing you in the fall!
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