2016 2017 Summer Math Preparation for Students Who Will Take College Algebra
2016–2017 Summer Math Preparation
for Students Who Will Take College Algebra
What is the aim of this assignment? Doing this assignment will help you review math that you will need to know for your math course next year. It will help you realize what topics you need to fix up before starting that course so that we can spend less time reviewing and more time learning new material.
Printing: Please print this file out. Do your work right on the sheet. If you need additional sheets, you can work on the back or staple additional pages.
Showing your work:
- You must work in pencil, not pen. Edit your mistakes and be neat.
- Show all work, even for any multiple-choice questions, so that your teacher can see what you were thinking, give you feedback, and help you.
- Write your work clearly, so that someone else can follow your thinking. Use correct mathematics notation as well as English words.
Due date: This assignment is due the first day of school. You have all summer to do this packet, but you really should start in July and work gradually so that you have time to do a great job.
Grading: This assignment will be graded as part of your Quarter 1 grade. You may also be assessed on these topics in September.
Getting Help: You are expected to work independently on this. You can ask classmates, friends, or others for suggestions, but you may NOT look at their written work. If someone asks you for help, don’t show them your work. Instead, guide them hints and ask questions that will help them.I have read this description and agree to do this assignment independently.
NameDate
Algebra Skills Needed to Be Successful in College Algebra
- Simplifying Polynomial Expressions —You need to be able to:
●Simplify an algebraic expression using the operations and properties
●Simplify polynomial expressions using addition and subtraction.
●Multiply a monomial and polynomial.
- Solving Equations —You need to be able to:
●Solve multi-step equations.
●Solve a literal equation for a specific variable, and use formulas to solve problems.
- Rules of Exponents —You need to be able to:
●Simplify expressions using the laws of exponents.
●Evaluate powers that have zero or negative exponents.
- Binomial Multiplication —You need to be able to:
●Multiply two binomials.
- Factoring —you need to be able to:
●Identify the greatest common factor of the terms of a polynomial expression.
●Express a polynomial as a product of a monomial and a polynomial.
●Find all factors of the quadratic expression ax2 + bx + c by factoring and graphing.
- Radicals —you need to be able to:
●Simplify radical expressions.
- Graphing Lines —you need to be able to:
●Identify and calculate the slope of a line.
●Graph linear equations using a variety of methods.
●Determine the equation of a line.
- Functions —you need to be able to:
●Work with functions in different forms — graphs, tables of values, words, and equations —including reading points, interpreting the function in various ways, finding domain and range, tc.
●Understand function notation and use it in different ways.
●Determine whether a point satisfies an equation and explain what that means for a graph.
A. Simplifying Polynomial Expressions
Combining Like Terms: You can add or subtract terms that are considered “like”, or terms that have the same variable(s) with the same exponent(s).
Example 1: 5x –7y + 10x + 3y = 15x – 4y
Example 2: –8h2 + 10h3– 12h2– 15h3 = –20h2 –5h3
Applying the Distributive Property: Every term inside the parenthesis is multiplied by the factor outside of the parentheses.
Ex. 1: 3(9x – 4)= 3•9x – 3•4Ex. 2: 4x2(5x3 + 6x) = 4x2• 5x3 +4x2 • 6x
=27x –12 = 20x5 + 24x3
Combining Like Terms AND the Distributive Property (Problems with a Mix!): Sometimes problems will require you to distribute AND combine like terms.
Example: 3(4x – 2) + 13x = 3• 4x –3• 2 + 13x
= 12x –6 + 13x
= 25x – 6
Practice Set A: Simplify each expression. Show your steps clearly using the method shown above.
- 8x – 9y + 16x + 12y2. 14y + 22 – 15y2 + 23y
3. 5n – (3 – 4n)4. –2(11b – 3)
5. 10q(16x + 11)6. –(5x – 6)
B. Solving Linear Equations
Solving Two-Step Equations:To solve an equation, you can UNDO the order of operations and work in the reverse order.
Addition is “undone” by subtraction, and vice versa. Multiplication is “undone” by division, and vice versa.
Ex. 1: 8x + 4 = 4x + 28
–4 –4
8x = 4x + 24
–4x –4x h
4x = 24
x = 6 / SimplifyingBefore Solving:
If variables are on both sides of the equal sign, get all terms with variables on one side and all terms without variables on the other side. You also may need to use the distributive property.
5(4x – 7) = 8x + 45 + 2x
20x – 35 = 10x + 45
–10x –10x
10x – 35 = 45
+ 35 +35
10x = 80
x=8
Practice Set B1:Solve each equation. Show all steps like the examples above.
1. 5x –2 = 332. 140 = 4x + 36
3. 8(3x – 4)=1964. 45x –720 + 15x = 60
5. 132 = 4(12x – 9)6. 198 = 154 + 7x –68
Solving Literal Equations: A literal equation is an equation that contains more than one variable.You can solve a literal equation for one of the variables by isolating the specified variable (getting that variable by itself).
Practice Set B2: Solve each equation for the variable specified. Show all steps like the examples above.
7) Solve for a in the Pythagorean Theorem a2 + b2 = c2.
8) Solve for r in A = πr2
C. Rules of Exponents
Practice Set C: Simplify each expression.
D. Binomial Multiplication
When multiplying two binomials (an expression with two terms), you can distribute or use “FOIL” method. (The “FOIL” method is actually the result of using the distributive property twice.)
= F + O + I + L = x2 + 10x + 6x + 60 = x2 + 16x + 60.
Practice Set D
Multiply. Write your answer in simplest form.
Explain why (x + a)2 is always equivalent to x2 + 2ax+ a2.
E. Factoring
Using the Greatest Common Factor (GCF) to Factor
Applying the difference of squares: a2 –b2 = (a –b)(a + b
Practice Set E
F. Radicals
To simplify a radical, we need to find the greatest perfect square factor of the number under the radical sign (the radicand) and then take the square root of that number.
Example:
If you don’t find the greatest square first, you may still be able to simplify the radical, but it will take more steps, such as below.
Same Example:
Practice Set F: Simplify each radical.
G. Graphing Lines
Slope: There are many ways to represent slope. List as many ways as you can think of. You can use words to describe what slope tells us about a line or a linear function. You can also use symbols and formulas.
Slope =
= [use words]
= [use other words]
= [use symbols]
= the of the line [use a word]
= the of of the y-value with respect to its x-value.
Use the graph patch to the right to sketch the points and slope below.Then explain your list about slope using some sentences and this example.
Practice Set G1: Find the slope between the given points.
Practice Set G.2
III. Graphing a Line given in Standard Form —TWO OPTIONS:
Practice Set G3: Graph each line
H. Functions
1)This problem is about the function g(x) = –0.5x + 3.
a)Compute the following:
g(–4)
g(–2)
g(0)
g(0.5)
g(2)
g(4)
b)Make a list of the points (x, y) associated with part (a) above. Your list can be separated by commas.
c)Plot those points (x, y) on the grid to the right.
d)Do you think the pattern continues in between the points? Check and then change your graph accordingly.
e)How would you express g(k)?
f)Challenge: Express g(–x2).
2)The equation x = 3 is NOT a function of x. Give as many reasons as possible.
3)Determine the values below using the graphs of f and g shown. If a value is not defined, state “not defined.” Assume the ends of f and g continue the shape you see beyond the graph.
a)f(0)
b)g(0)
c)2 –5(g(0) –3) (Read carefully.)
d)g(3)
e)g(–2)
f)f(1) + g(1)
g)f(–3) + g(5)
h)For what values of x does g(x) = –2?
i)For what values of x isf decreasing?
j)For what values of x isg increasing?
k)Give three values that g(x) can never be? (Is this asking for an x-value or a y-value?)
l)Describe all the possible output values of f in a sentence.
4)What does the notation f(x) really represent? Is it a function? An x-value? A y-value? An expression? (for what?) A complete equation? (for what)? Explain.
5)Use the graph below of y = f(x) to answer the following questions.
a)What does the closed point at the left end of the graph tell you?
b)What does the arrow tell you?
c)What is the domain of f?
d)What is the range of f?
e)For what value(s) of x does y = 0?
f)An x-intercept is an x-value where the graph crosses the x-axis. Identify the x-intercepts of f.
g)What is the y-intercept?
h)A function can have multiple x-interceptsbutonlyone y-intercept? Explain why using the definitions of the word function, x-intercept, and y-intercept.
i)For what values of x is f(x) < 0?
j)Graph g(x) = –3. (Reason this out — this equation states that no matter what x is, g(x) is…?)
k)For what values of x is g(x) ≥ f(x)?
l)Explain why it is impossible to write up a complete table of values for this graph — and why that’s not really a problem.
6)For any function f, Does the point (x, f(x)) satisfy the equation that represents f(x)?
7)Which of the following equations does the point (x, y) = (5, 10) satisfy?
a)f(x) = 10x – 50
b)g(x) = 2x
c)h(x) = x2 – 15
d)j(x) = 5
e)k(x) = 10
8)Is the point (–9, 3) on the graph of ? Explain.
9)Is the point (4, 3) on the graph of ? Explain.
10)What does it mean for a point to “satisfy” an equation?
Answer Key
Practice Set A
Practice Set B1
Practice Set B2
Practice Set C
Practice Set D
Practice Set E
Practice Set F
1. 112. 3.
4.5. 86.
7. 8. 9.
Practice Set G1
Practice Set G2
Practice Set G3