Algebra EOC Study Guide Notes

Function Notation

$f(x)$ is a way of representing "y".
$f(3)$ asks for the value of "y" when $x = 3$ . Substitute 3 for x in the function and simplify.
To find $f(x)$ from a graph, locate the y-value corresponding to the given x-value.

Examples of Function Notation

Find $f(2)$ in the provided graph (not included in text, but conceptually, find the y-value when $x=2$ ).
- If when $x=2$ , $y=-2$ , then $f(2) = -2$ .
Find x when $f(x) = -1$ .
- Identify the x-value where $y = -1$ . According to the graph, $x \approx 0.5$ .

Domain and Range

Use braces {} to list the elements of the domain and range.
Domain: List of all x-values.
Range: List of all y-values.
Roots: List of x-intercepts (where $y = 0$ ).

Independent vs. Dependent Variables

Determine which quantity fits best in the sentence:
- (Independent variable) is what.
- (Dependent variable) depends on what.
Time: If time is mentioned, it is almost always the x-value (independent variable).
Independent Variables: x (horizontal axis).
Dependent Variables: y (vertical axis).

Stem and Leaf Plots

An organized chart used to arrange data.
Steps:
- Find the least and greatest values.
- Write the stems (leading digits) in a column.
- Arrange the leaves (trailing digits) from smallest to largest.
- Include an explanation or key.
Example: Data set: 70, 52, 58, 45, 59, 52, 75, 47, 46
- Stem and Leaf Plot:
  - 4 | 5 6 7
  - 5 | 2 2 8 9
  - 6 |
  - 7 | 0 5
- Key: 5|2 = 52

Correlations

Positive Correlation: Both sets of data generally increase together.
Negative Correlation: One set of data generally decreases as the other increases.
No Correlation: Data sets are not related.
A trend line on a scatter plot visually represents the correlation.

Line of Best Fit vs. Trend Line

The line of best fit is the most accurate trend line.
To find a trend line: Identify the y-intercept, find the slope, and write the equation in slope-intercept form: $y = mx + b$ , where b is the y-intercept and m is the slope.

Central Tendencies

Given the data set: 5, 6, 6, 7, 10, 11, 12, 14, 15, 20
Mode: The most common number (6). There may be more than one mode, or no mode if all numbers are different.
Median: The middle number when the numbers are listed in order (10.5). If there are two middle numbers, find their average.
Mean: The average. Add all the numbers and divide by the number of items (10.6).
Range: The difference between the largest and smallest numbers (15).

Percentage vs. Flat Changes

Percentage Increases:
- Measures of Center (Mean, Median, Mode): Increase.
- Measure of Spread (Range): Increases.
Flat or Constant Increase:
- Measures of Center (Mean, Median, Mode): Increase.
- Measure of Spread (Range): No change, because the lowest and highest numbers are raised by the same amount, keeping the difference constant.

Box and Whisker Plots

Shows how data is spread out by dividing it into four groups (quartiles).
Steps (using the data set: 20, 36, 58, 45, 59, 55, 75, 35, 35):
- Draw a line graph with equal intervals.
- Place a dot for the smallest and largest values.
- Place a dot for the median.
- Place dots for the medians of the first and second halves of the data (quartiles).
- Draw a box around the two middle quartiles.
- Place an asterisk (*) for any extreme data items (outliers).

Ordering Real Numbers

Convert numbers in different formats to decimal form.
Compare the decimal values, remembering that 2.01 is less than 2.1.

Isolating a Single Variable

Express one variable in terms of the others; "solve for" means to isolate a variable.
Use the same rules as solving an equation: whatever you do to one side, do to the other.
Example: Solve for t: $abt + 2 = c$ $\frac{3w}{w}$
- $abt = c - 2$
- $abt = 3w(c - 2)$
- $abt = 3wc - 6w$
- $t = \frac{3wc - 6w}{ab}$

Parent Functions & Transformations

Linear: y=mx+b
Absolute Value: $y=a|x-h|+k$
Exponential: $y=a^x$
Quadratic: $y=a(x-h)^2+k$
Rational: $y+\frac{k}{x}$
Square Root (Radical): $y=a\sqrt{x}$

Transformations

h: Horizontal Shift; k: Vertical Shift; a: Stretch; m: Slope
A negative before 'a' indicates upside-down direction (reflection over x-axis).

Absolute Value Transformations

Move up: Add a positive number outside the absolute value bars.
Move down: Add a negative number outside the absolute value bars.
Move left: Add a positive number inside the absolute value bars.
Move right: Add a negative number inside the absolute value bars.
Example:
- $y=|x-2|+3$ : Right 2, Up 3.
- $y=|x+5|-6$ : Left 5, Down 6

Exponential Functions

$y = ab^x$
- a = Beginning amount (if = 1, then invisible)
- b = Rate or multiplier
- x = Periods that the multiplier is used.
Remember that a percentage increase or decrease should be added to or subtracted from 1.
- A 5% increase = Rate of 1.05
- A 2% decrease = Rate of 0.98

Word Problems

"DERT" Formula: Distance = Rate × Time
Consecutive means no skipping between numbers.
Look for a starting or beginning number and use that as your constant or y-intercept.
Look for words like ratio or per to know that you have a rate and use it as a coefficient for x.
Remember that if you are using something up, it will be a negative slope.

Solving "4:5:9" Ratio Type Problems

Example: Three angles in a triangle have the ratio 4:5:9. What is the measure of each angle?
Method #1: Add up the numbers and use as a denominator.
- In the example, the 3 numbers have a sum of 18.
- Use the 18 as a denominator and each number as the numerator.
- Multiply by the total to get each number.

Notes on Sequences

Arithmetic Sequences

(Adding a common difference)
d = common difference (# added)
To find d: Subtract 1st term from 2nd.

Recursive Form for nth term

Uses the prior term ( $a_{n-1}$ )
$an = a{n-1} + d$

Explicit Form for nth term

Does not use prior term.
$an = a1 + d(n-1)$
- $a_1$ = the 1st term
- d = common difference
d = the slope of a line
Note: To find d: Subtract 1st term from 2nd

Geometric Sequences

(Multiplying by a common rate)
r = common ratio (# multiplied)
To find r: Divide 2nd term by 1st.

Recursive Form for nth term

Uses the prior term ( $a_{n-1}$ ).
$an = a{n-1} * r$

Explicit Form for nth term

Does not use prior term.
- $a_1$ = the 1st term
- r = common ratio

Sequences Example

Write a recursive formula for the arithmetic sequence 5, 9, 13, 17,…
Answer:
+4
Slope & d (common difference) are the same.
What is the slope of the line that contains the points associated with these values and their position in the sequence?
How is the slope of the line related to the sequence?

Sequence Given u(0) = 3 and u(n + 1) = u(n) + 7

when n is a positive integer, find:
- a. u(5);
- b. find n so that u(n) = 367;
- c. find a formula for u(n).
  Solution
Slope & d (common difference) are the same.
If u(0) = 3, then 3 is a.
Error Note: Changed from 361 on state website which does not generate an integer answer.
u(n+1)= u(n) +7, then d=7
Therefore a = 10
- a. u(5) = 10+(7*4) = 38
- b. 367 = 10+7(n-1)
  - 367 = 10+7n-7
  - 367 = 3+7n
  - 364 = 7n
  - n=52
- c. a = 10+7(n-1)

Recursive formula for the geometric sequence 5, 10, 20, 40,…

Determine the 100th term.
r=3, a=6 (from 2 x 6)
- a. U(4) = 6*3^3
- u(4)=162
- b. a=6*r^(n-1)

Sequence Given that u(0) = 2 and u(n + 1) = 3u(n)

a. find u(4), and
b. find a formula for u(n).

Consistent/Inconsistent Systems

Inconsistent System: No solution, Lines are parallel.
- Example:
  - 2x-y=-1
  - 4x-2y=4
  - 0-6. Both lines have the same slope
Consistent System:
- Independent: lines intersect, One solution
- Dependent: Infinitely many solutions, Same line!
  - Example
    - x-y=2
    - 2x-2y=4 Second equation is 2x the first equation, 0=0

Graphing Inequality Systems

Type of line.
Where to shade.
y>x+3
- >> above.
y≤x-1
- << below.

Coin Problems/Age Problems

Coin Problems

Usually involved with number of coins and monetary value.
A coin bank has 250 dimes and quarters worth $39.25 How many dimes and quarters are there?
- $d+q=250$
- $.10d+.25q = 39.25$

Age Problems

Represent a person's age in the past, present, or future.
A father is 32 years older than his son. In four years, the father will be 5 times older. How old are they now?
- $s+32= f(age now)$
- $f+4=3+4+(age in 4 years)$
- $f+4=5(s+4)$

Laws of Exponents

Wind/Current and Digit Problems

Wind/Current

Need speed when traveling with the wind and against the wind
Use rate * time = distance
- $(r+w)2.5=750$
- $(r-w)2=750$
  - r=rate; w = wind

Digit

Write value of number in expanded from (t: tens; u: ones)
- Two digit number: 10t+u
- Reverse digits: 10u+t
- Sum of digit: t+u
Product of Powers: $x^m*x^n=x^{m+n}$
Power of Power Property: $(x^m)^n = x^{m*n}$
*(x^5)^4 = x^20
Power of a Product: $(xy)^n =x^n*y^n$
*(xy)^3=x^3*y^3
*Hint: even powers → Positive answer
odd powers → Negative answer
$(-4)^2=16$
$(-4)^3=-64$
m>n \frac{x^m}{x^n} = x^{m-n}
m<n \frac{x^m}{x^n} = \frac{1}{x^{n-m}}
Exception:
Power of Fractions: $(\frac{a}{b})^n = \frac{a^n}{b^n}$
Anything to the 0 power =1, $x^0=1$
$5^0=1$

Scientific Notation

A number written as a product with two factors:
- A number between 1 and 10 and
- A power of 10
Examples
- 463,000,000 = 4.63 x 10^8
- .000597 = 5.97 x 10^-4 Move right for a negative exponent; left for positive
- Add exponents = $(3 x 10^8)(4 x 10^4)$
  - $12 x 10^{12}$
  - $1.2 x 10^{13}$
Count the number of times you need to move the decimal to make a number less than 10
$\frac{2.5 x 10^6}{5 x 10^2}$
$.5 x 10^4$
$5 x 10^3$ subtract exponents.

Slope

$m = \frac{rise}{run} = \frac{y2 - y1}{x2 - x1}$
/ → Positive Slope
\ → Negative Slope
$m=0$ → (y=8) Horizontal Line
Undefined Slope → (x = 8) Vertical Line
Parallel Lines: Same slope
Perpendicular Lines: Reciprocal opposite of the given line

Slope Example

(5,3); (4, -1)
- $m=\frac{3-(-1)}{5-4}$
- $m=4$
- (4,8); (4,2)
  - $m=\frac{8-2}{4-4}$
  - m is undefined

Slope Intercept Form

$y=mx+b$
- b=y-intercept, where the line crosses the y-axis, m=slope
x and y intercepts:
x: Substitute 0 for y and solve for x.
y: b is the y-intercept

Finding a Line Examples

m=3; (2,6)
- $y=mx+b$
- $6=(3)(2)+b$
- $6=6+b$
- b=0
- $y=3x+0$ y=3x
(1,8); (4,2)
- $y=mx+b$
- $m=\frac{2-8}{4-1} = -2$
- $2=(-2)(4)+b$
- 2=-8+b
- 10=b
- $y= -2x+10$

Standard Form

Standard Form of a Linear Equation: Ax + By = C
Fractional Slope m = -A/B
The x will always be positive
No fractions: multiply by common denominator
Examples
- 7y=-5x-35
  - 5x + 7y=-35
- 3x-2y=-18 3x-2y+18 3x-2y=18

Solving Inequalities

Graphing Symbols

Open: < and > ○
Closed: ≤ and ≥ ●
When dividing and multiplying by a negative number, REVERSE the inequality.
15 > -5
Error Note: x>4, not 15
Examples
- -22 < x ≤ 4 Error Note: 1 < -20 x > -1 NOT -2

Absolute Value Equations/Inequalities

Equations: Solve twice using + AND -
Inequalities: Solve twice using + AND - and reverse the inequality

Absolute Value Equation Example

|3x-2|=10
3x-2=10 OR 3x-2=-10
3x=12 OR 3x=-8
$x=4 OR x=\frac{-8}{3}$

Absolute Value Inequality Example

|x-6|>2
x-6>2 OR X-6<-2 reverse inequality Negative added to the 2
x>8 OR x <4

Substitution/Elimination

Substitution Method

Use when you can easily find the value of either the x or y variable.

Substitution Example

Solve:
- 15x-5y=30
- y=2x+3
  Solution
- 15x-5(2x+3)=30
- 15x-10x-15=30
- y=(2)(9)+3
- y=21
- 5x – 15 = 30
- 5x=45
- x=9
Answer: (9,21)

Elimination Method

Use when there is no easy way to substitute. Multiply one or both equations by a corresponding factor. (Use opposites so you can add equations together.)

Elimination Example

Solve:
- 2x+3y=1
- 5x+7y=3Multiply equation by -2
  - 10x+15y=5 AND -10x-14y=-6
    Solution
- y=-1
- 2x+(3)(-1)=1
- 2x-3=1
- 2x=4
- x=2
Answer: (2,-1)

Equations

$3x=27$ Divide for answer → x=9
$\frac{x}{3} =18$ Multiply for answer → x = 54

Two-Step Equations

Solve in reverse order of operations
- +/- first
- then x/÷
Watch for negatives

Two-Step Equations Examples

-7-13y=32
- -13y=39
- y=-3
$\frac{x}{4}+5=12$
- $\frac{x}{4} =7$
- $x=28$

More Equations

Combine Like Variables

4n-2+7n=20
11n-2=20
11n=22
n=2

Distribute to Eliminate Parentheses

3(r-4)=9
3r-12=9
3r=21
r=7

Multi-Step Equations

Distribute
Combine
Move variables to the same side of the equation
Solve remaining equation
Example

(+3y) 5y-10=14-3y (+3y)
8y-10=14
8y=24
y=3

Variables on Both Sides of Equation

Example

3x-2(x+6)=4x-(x-10)
3x-2x-12=4x-x+10
x-12=3x+10
-12=2x+10
-22=2x
-11=x

BIG TIP

Look to get rid of fractions on variables. Multiply all terms by the common denominator.

Proportions

Proportion: An equation showing that two ratios are equal.
Cross Products are equal $\frac{a}{b} = \frac{c}{d}$
Example
$\frac{4}{9} = \frac{3}{12}$
49=312
36=36
## Proportion Example 2
11/m = 7.5/100
10m=135
m=13.5
## Proportion Example 3
22/9 = x/27
22=x/27
4 9x = 2227
x = 66

Literal Equations

Treat variables like numbers…
Examples
- T+M=R, solve for T
  - T=R-M
- A=lw, solve for w
  - $\frac{A}{l} = w$
- ax+r=7, solve for x
  - ax=7-r
  - $\frac{ax}{a} = \frac{7-r}{a}$
    *Proportion Review
    *1 5/10 = x/15
    *2 5 * 15 = 10 * X
    *3 5/10 = x\15
$\frac {75}{10}= x$
### Proportions With Percent Review
$\frac{25}{30} = \frac{9x}{27}$
*1 5 * 27 = 9x * 30
*2 \frac{675}{9} = \frac{9x\675}{9}
Test correction #14, x=75
Percent Problems
Converting Percents
PD: Move decimal two to the left (2←)
DP: Move decimal two to the right (2→)
PF: Write number over 100 and reduce
Hint: Write percents with one decimal over 1000, two decimals of 10,000, etc.
FD: Divide numerator by denominator
Percent Problems using Equations
Write the equation as you read the problem.
is → =
of → x
what → x
Percent Change
Percent change=subtract amounts and divide by original number
Test Taking Tips
Skip problems that you don't know how to do and come back.
Write down all your steps. In other words, don't do the work in your head.
When you panic, stop, and do something that relaxes you, like close your eyes and take a deep breath.
If you finish early, go back and do the problems again. Don't look at your previous work.