Algebra I Study Guide for EOCE



Algebra I Study Guide for EOCE

What do you need to know?

G. Bailey

Disclaimer: Just because it is not on this sheet, doesn’t mean you shouldn’t study it.

Linear Functions:

x → input → independent variable → domain

y → output → dependent variable → range

How can you tell if an equation is linear?

• no x or y in the denominator of fractions

• x and y are to the first power only

• no product of x and y

• can you put it in slope-intercept form ( y = mx + b) or Standard Form (Ax + By = C)?

Function Notation:

f(x) is read as “ f of x” or “the function of x”

Ex: If f(x) = x2 + 2x – 3, evaluate the following:

f(5) = 52 + 2(5) – 3 = 25 + 10 – 3 = 32

f(-2) = (-2)2 + 2(-2) – 3 = 4 + -4 – 3 = -3

f(3x) = (3x)2 + 2(3x) – 3 = 9x2 + 6x – 3

f(x + 2) = (x + 2)2 + 2(x + 2) – 3 = (x2 + 4x + 4) + (2x + 4) – 3 = x2 + 6x + 5

Expressions with Exponents:

• am ∙ an = am+n Ex: 53 ∙ 58 = 511

• am / an = am-n Ex: 105 / 103 = 102

• (am)n = amn Ex: (74)5 = 720

• (am bx)n = amnbxn Ex: (2x3)2 = 22x6 = 4x6

(Remember, I showed you how to write it out the long way if you forget the rules.)

a0 = 1 for all a, a ≠ 0

a-n = 1/an Ex: 3-4 = 1/34 = 1/81

The exponent only affects only what it is directly next to.

Ex: -32 ≠ (-3)2

-9 ≠ 9

Collecting Like Terms:

You add the coefficients (the big number in front) of like terms (terms with the same variable raised to the same exponent).

Ex: 5x3 + 2y4 + 6x2 + 3x3 + -6y4 = (5x3 + 3x3) + (2y4 + -6y4) + 6x2 = 8x3 + -4y4 + 6x2

(Notice that the exponent did not change – you want to know “How many x3 do I have?”)

Solving Equations: General Steps

1- Add the opposite.

2- Use the Distributive Property to remove the grouping symbols.

3- Clear fractions or decimals by multiplying by a common denominator or some power of 10.

4- Simplify the expressions on each side of the equals sign.

5- Use the Addition and/or Subtraction Property of Equality to get the variables on one side of the equation and the constant(s) on the other.

6- Simplify each side of the equation.

7- Use the Multiplication and/or Division Property of Equality to solve.

8- Special Circumstances:

a. If the solution results in a false statement, there is no solution (Ø). Ex: 5 = 8

b. If the solution results in an identity, the solution is “all real numbers”. Ex: 7 = 7 or x = x

9- ALWAYS check your solution!!

(Use the above rules for inequalities, except remember that when you multiply or divide on both sides of an inequality by a negative number, you must switch the sign around.)

Graphing:

3 methods

1 – table (use at least 3 points, 5 is better)

2 – intercepts (be careful - only 2 points, one where x = 0 and one where y = 0)

3 – y = mx + b (best method)

m = slope = rise

run

b = y-intercept (the starting point)

[pic] [pic]

Positive slope Negative Slope

[pic] [pic]

Zero slope Undefined Slope (or no slope)

A line with a slope of “0” has a slope. An undefined line does not have a slope.

Horizontal Line → zero slope → y = constant (Ex: y = 7)

Vertical Line → undefined (or no) slope → x = constant (Ex: x = -4)

Parallel lines have the same slope.

Perpendicular lines have slopes that are opposite reciprocals (2/3 and -3/2).

Ex: Find the slope of the line containing the points (2, 3) and (-4, 1).

2 ways – graph and count OR use formula

∆y = (y1 – y2) = (3 – 1) = 2 = 1

∆x (x1 – x2) (2 – -4) 6 3

(Remember that it doesn’t matter what order you choose the y’s, as long as you choose the x’s in the same order.)

Solving Systems: (3 methods)

1 - graphing (find the point/area of intersection – the only method you can use for solving systems of inequalities)

2 - substitution (if you already have one equation in the form of “x =” or “y =” or if you can get them in that form easily)

3 - elimination/linear combinations (if you notice that the coefficients of either x or y are opposites; this is the method that can be used for all systems of equations)

Exs:

Substitution

2x + 3y = 12 2(y + 1) + 3y = 12 x = 2 +1

x = y + 1 2y + 2 + 3y = 12 x = 3

5y + 2 =12

-2 -2

5y = 10 (3, 2)

( 5 ( 5

y = 2

Elimination

x + 2y = 9 3(x + 2y = 9) 3x + 6y = 27 1 + 2y = 9

2x – 6y = -22 2x – 6y = -22 2x – 6y = -22 -1 -1

5x = 5 2y = 8 (1, 4)

( 5 ( 5 ( 2 ( 2

x = 1 y = 4

~OR~

-2(x + 2y = 9) -2x + -4y = -18 x + 2(4) = 9 (1, 4)

2x – 6y = -22 2x – 6y = -22 -8 -8

-10y = -40 x = 1

( -10 ( -10

y = 4

Quadratic Functions:

Standard Form Ax2 + Bx + C = 0 (Quadratic functions contain “x2”.)

When asked to solve (find the “roots, solutions, x-intercepts or zeros of”) an equation like x2 – 2x – 15 = 0, there are 3 ways.

1 – Factor x2 – 2x – 15 = 0

(x + 3)(x – 5) = 0

x + 3 = 0 x – 5 = 0

-3 -3 +5 +5

x = -3 x = 5 (Remember, this is NOT an ordered pair.)

( 2 roots)

2 – Graph on your calculator (if using a graphing calculator)

First press [Y=], then type in the equation using [X,T,(,n] for the “x” and either [x2] or [(] for the exponent. Then [GRAPH].

Notice the parabola crosses the x-axis at “-3” and “5”. These points are where y = 0.

3- Quadratic Formula

a = 1

b = -2

c = -15

x = 2 ± √ ((-2)2 + -4(1)(-15)) = 2 ± √ (4 + 60) = 2 ± √ (64) = 2 + 8 = 10 = 5

2(1) 2 2 2 2

= 2 – 8 = -6 = -3

2 2

The Quadratic Formula shows the point(s) (solutions, roots, zeroes) where the parabola crosses the x-axis.

You can use the discriminant to determine the number of solutions or roots (the number of times that the parabola crosses the x-axis) that a quadratic equation has.

The Discriminant is b2 – 4ac (no √ sign).

|If the Discriminant is: |There are _____________ roots. |

|0 |1 |

|Positive |2 |

|Negative |no |

Graphs:

If the Discriminant is 0, there is one root and your graph will look like:

The graph hits the x-axis once. (It may be turned the other way

or moved over.)

If the Discriminant is a positive number, there are two roots and your graph will look like:

The graph goes through the x-axis twice. (It may be turned the other way

or moved over.)

If the Discriminant is a negative number, there are no roots and your graph will look like:

The graph does not touch the x-axis. (It may also be turned the other way

or moved over.)

Think about the reasons for the above results in terms of the quadratic formula and what happens when you take the square root of 0 (one answer), a positive number (two answers), or a negative number (no answers).

Factoring: General Steps

1- Always look for a common factor.

2- If the problem is of the form:

a. x2 + bx + c → what will multiply to give you “c” and add to give you “b”.

b. ax2 + bx + c → Use Master Product: multiply “a” and “c”, and then find factors of this number that add up to “b”. Rewrite as a polynomial with 4 terms and factor by grouping.

c. Special Products

• a2 – b2 = (a + b)(a – b)

• a2 + 2ab + b2 = (a + b)2 or (a + b)(a + b)

• a2 – 2ab + b2 = (a – b)2 or (a – b)(a – b)

d. If your problem is a polynomial with 4 terms, factor by grouping.

3- ALWAYS factor completely!!!

4- Check using FOIL or the distributive property.

Examples:

3x3 + 6x2 – 15x = 3x(x2 + 2x – 5)

4x2 – 25 = (2x + 5)(2x – 5)

x2 + 6x + 9 = (x + 3)2 or (x + 3)(x + 3)

4x2 – 4x + 1 = (2x – 1)2 or (2x – 1)(2x – 1)

x2 – 9x + 20 = (x – 5)(x – 4)

6x2 – x – 2 = (2x + 1)(3x – 2)

3xy – 6y + 5x – 10 = (3xy – 6y) + (5x – 10)

3y(x – 2) + 5(x – 2)

(x – 2)(3y + 5)

How do you know if you should just factor the problem or get number answers?

If the problem has an equal sign in it, you should solve it and get number answers. Otherwise, just factor.

Linear Equations – writing equations of lines

Example 1: Write the equation of the line with a slope of -3 and y-intercept of 7.

Use y = mx + b and substitute in for m and b.

Ans: y = -3x + 7

Example 2: Write the equation of the line with a slope of 2 and passing through the point (1, -4).

Use either y = mx + b or y- y1 = m (x – x1).

Using y = mx + b, substitute 1 in for x, -4 in for y, and 2 in for m. Then solve for b. Finally, substitute m and b into y = mx + b and you are finished. x and y should appear in the final answer.

-4 = 2(1) + b

-4 = 2 + b

-2 -2

-6 = b

Therefore, your equation is y = 2x + -6. This is the general equation that fits all points on the line.

Using y- y1 = m (x – x1), substitute 1 in for x1, -4 in for y1, and 2 in for m. Then distribute and get y by itself. You will get the same answer as above.

Example 3: Write the equation of the line through (3, 8) and (5, 14).

Find the slope first and then use the steps from above, using the slope and one of the points. Or, you can use your graphing calculator to go to [STAT], “Edit” and enter the ordered pairs into the lists, and then [STAT], “Calc”, #4, Enter, Enter. The “a” is your slope and the “b” is the y-intercept. You can also use this when finding just the slope.

Example 4: Write the equation of the line in point-slope form, slope-intercept form, and Standard form for the line through (4, 9) and (12, 25).

Use the flow chart given to you previously.

Strategies for Taking Multiple Choice Tests

• If the question is a factoring question, work out all the answers and match to the problem.

Ex: What is the correct factorization of 2x2 – 5x – 3?

A. (2x – 1)(x +3) B.(2x + 3)(x – 1) C. (2x + 1)(x – 3) D. (2x – 3)(x +1)

• If the question is a “match the graph to the equation” question, use your calculator.

• If the question is a “which ordered pair satisfies the equation” question, use the [TABLE] function of your calculator. Solve for y first.

Ex: Which ordered pair satisfies the following equation: -2x + 3y = 7

A. (0, 2) B. (5, -2) C. (4,5) D. (-3, 3)

You can also use the [TABLE] function to do the opposite type of question – if you are given a table or set of ordered pairs and asked to match with the equation.

• Use your calculator to solve 2(x + 3) = -8x – 4

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