Transcript Standard #1: Write an algebraic expression from a word

Standard #1:
Write an Algebraic Expression
from a word problem.
Text Section: 1.1
Reminders
KEY WORDS
• Sum, Increased by, More Than, Plus
• Difference, Less Than, Decreased by
• Product, Per, Groups of, Times
• Quotient, Divided by, Ratio
Examples
Addition:
4 plus a number
5 more than a number
A number increased by 3
The sum of a number and 2
Multiplication:
The product of a and b
5 times a number
Twice a number
Subtraction:
The difference of a and b
*** 3 less than a number ***
A number decreased by 8
A number less 6
Division:
The quotient of a and b
A number divided by 8
The ratio of x and y
Answers
Addition:
4+x
X+5
X+3
X+2
A-b
X-3
X-8
X-6
Multiplication:
ab
5x
2x
Subtraction:
Division:
a/b
x/8
x/y
Standard #2:
Combine like terms in an
expression.
Text Section: 1.7
Reminders
• Distribute First (if necessary)
• Combine ONLY if they have the SAME
variable AND SAME exponent!
Examples
1. 12x + 30x
2. 6.8y2 – y2
3. 4n + 11n2
4. 1/2x3 + 3/4x3
5. 2(x + 6) + 3x
6.
7. -3(-2 – x) + 8
9 + (x – 4)6
Answers
1. 42x
2. 5.8y2
3. 4n + 11n2
4. 1 1/4x3
5. 5x + 12
6.
7. 3x + 14
6x - 15
Standard #3:
Evaluate an expression.
Text Section: 1.6
Reminders
• Use Parentheses when you
substitute in for a Variable.
• PEMDAS!!!
Examples
1. 5(1-2) – (3-2)
2. – 9 – (-18) + 6
3. 16 [5- (3 + 2²)]
4.
7x (3 + 2x) for x = -1
Answers
1. -6
2. 15
3. -32
4.
-7
Standard #4:
Solve a 1 step equation.
Text Section: 2.1-2.2
Reminders
5 Steps!
X + 3 = 10
-3 -3
X = 10 -3
X= 7
7 + 3 = 10
Examples
1. n – 3.2 = 5.6
2.
x+7=9
3.
m = 1.5
3
4. 16 = 4c
Answers
1. n = 8.8
2.
x=2
3.
m = 4.5
4. 4 = c
Standard #5:
Solve a 2 step equation.
Text Section: 2.3
Reminders
D C (no M) S then 8 STEPS!
2x – 3 = 13
+3 +3
2x = 13 + 3
2x = 16
2
2
x = 16/2
x=8
2(8)- 3 = 13
Examples
1. 6x + 3 – 8x = 13
2. 9 = 6 – (x + 2)
3. 2a + 3 – 8a = 8
4. 4(x – 2) + 2x = 40
Answers
1. x= -5
2. x = -5
3. a= -5/6
4. x= 8
Standard #6:
Solve a Multi-step equation.
Text Section: 2.4
Reminders
DCMS
(YES, in that order!)
Examples
1. 7k = 4k + 15
2. 4b + 2 = 3b
3. 2(y + 6) = 3y
4. 3 – 5b + 2b = -2 – 2(1 – b)
Answers
1. K = 5
2. B = -2
3. Y = 12
4. B = 7/5
Standard #7:
Write and solve an equation
from a word problem.
Text Section: 2.1-2.4
Reminders
• Use Key words, write the equation and solve.
• You may need to use DCMS, 5 steps or 8 steps
Examples
A person’s maximum heart rate is the highest
rate, in beats per minute that the person’s heart
should reach. One method to estimate
maximum heart rate states that your age added
to your maximum heart rate is 220. Using this
method, write and solve an equation to find the
maximum heart rate of a 15-year-old.
Answers
15 + x = 220
X = 205
Standard #8:
Solve an Absolute Value
Equation.
Text Section: Ch 2 Extension
Reminders
IS IT ALONE????
IS IT NEGATIVE?
• If an absolute value equation equals a positive
number there are two solutions.
• If an absolute value equation equals 0 there is
one solution.
• If an absolute vale equation equals a negative
number there are no solutions.
Examples
1. 4|x + 2| = 20
2. |x| - 3 = 4
3. |x + 3| + 4 = 4
4. 5 = |x + 2| + 8
Answers
1. x = 3, x = -7
2. x = 7, x = -7
2. x = -3
3. no solution
Standard #9:
Isolate a Variable.
Text Section: 2.5
Reminders
• Use Opposite Operations
to get the Letter all by
itself.
Examples
1. Given d = rt, solve for t
2. Given A = ½ bh, solve for b
3. Solve m – n = 5 for m
4. Solve m = x for k
k
Answers
1. t = d/r
2. B = 2a/h
3. M = 5 + n
4. K = m
x
Standard #10:
Write an inequality from a
word problem.
Text Section: 3.1
Reminders
<
>
≤
≥
≠
A < B
A>B
A≤B
A≥B
A ≠ B
A is less than
B
A is greater
than B
A is less than
or equal to B
A is greater
than or equal
to B
A is not equal
to B
Examples
Write in Words
1. b < - 1.5
2. r ≥ 2
3. 5 ≥ w
4. -1/2 < a
Answers
1. All real numbers less than - 1.5
2. All real numbers greater than or equal to 2
3. All real numbers less than or equal to 5
4. All real numbers less than - 1/2
Standard #11:
Solve an inequality by
adding and subtracting.
Text Section: 3.2-3.3
Reminders
Same 5 Steps as solving an Equation.
X + 3 < 10
- 3 -3
X < 10 -3
X< 7
Examples
1. x + 9 < 15
2. d – 3 > - 6
3. 0.7 ≥ n – 0.4
4. 2 ½ ≥ - 3 + t
Answers
1. x < 6
2. d > - 3
3. n< 1.1
4. T < 5 ½
Standard #12:
Solve an inequality by
multiplying and dividing.
Text Section: 3.4-3.5
Reminders
SAME 5 or 8 Steps with
1 TRICK
If you Multiply or Divide BY
(not into) a Negative- you
MUST flip the inequality
SIGN!
Examples
1. -50 ≥ 5q
2. -42 ≤ 7x
3. 10 ≥ -x
Answers
1. q≥ -10
2. x ≥ -6
3. x ≥ -10
Standard #13:
Solve an Absolute Value
Inequality.
Text Section: Ch 3 Extension
Reminders
IS IT ALONE????
Set up TWO inequalities: Flip the sign AND Negative!
**Tip: Remember “less thAND”**
**Tip: Remember “greatOR”**
Examples
1. |x|-3<12
2. |x-4|+7≤-2
3. |x|-20>-13
4.
|x-8|+5≥11
Answers
1. X < 15 AND x > -15
2. No solution
3. X > 7 OR x < -7
4.
X > 14 OR x < 2
Standard #14:
Graph an inequality on a
number line.
Text Section: Chapter 3
Reminders
Graph on a Number Line
Open Circle when it is < or >
Closed Circle when it is < or >
Shade Left or Right???
Make sure your solution has the Variable
on the left side BEFORE you Graph.
Examples
Graph
1. b < - 1.5
2. r ≥ 2
3. 5 ≥ w
4. -1/2 < a
Answers
1.
2.
3.
4.
Standard #15:
Interpret/Describe
solutions of an inequality.
Text Section: Chapter 3
Reminders
AT LEAST >
AT MOST <
MORE THAN >
LESS THAN <
Examples
1. Give three possible solutions. 5s > 10
2. Which inequality has the solution shown? d < -3
a. 4 > d + 7
c. d – 8 < - 5
b. 9 + d > 6
d. 2 < - 1 + d
3. Which inequality has -2 as a solution?
a. 2x > 4
b. -2x < 4
c. -2x > 4
4. Which statement justifies the given inequality?
a. You spent more than $300
b. You spent at least $300
c. You spent less than $300
d. You spent at most $300
x ≥ $300
d. -2x > -4
Answers
1. 3,4,5 etc..
2. A
3. D
4. B
Standard #16:
Recognize a function in a
variety of ways.
Text Section: 4.2
Reminders
The x values, MAY NOT REPEAT!!!!
Determine if it is a function from:
Graph
VERTICAL
Table
LINE TEST!
Chart
Ordered Pairs
Mapping Diagram
Equation
Examples
Is it a FUNCTION?
1. {(-4,2),(2,3),(0,7),(-4,-1)}
2.
3.
4.
x
−2
−1
0
1
y
5
5
5
5
Answers
1. No
2. No
3. Yes
4. yes
Standard #17:
Identify inputs and outputs.
Text Section: 4.2
Reminders
INPUT- X Values
OUTPUT- Y Values
Examples
1. List the Inputs
x
5
4
3
2
y
-12
-10
-8
-6
2. List the Outputs {(4,0),(2,-3),(0,-6),(-2,-9)}
Answers
1. (5,4,3,2)
2. (0,-3,-6,-9)
Standard #18:
Identify domain and range.
Text Section: 4.2
Reminders
DOMAIN- X Values
RANGE- Y VALUES
Example
What is the Domain? Range?
Answers
Domain: (-2,3,4,10)
Range: (-1,0,2,4,6)
Standard #19:
Write a rule for a given
function.
Text Section: 4.3
Reminders
ALL rules Start with:
Y=
Ask yourself:
“what do I have to do to x to get y???”
Examples
1. Write an equation (rule) for the following function.
x
y
2
−3
4
−1
6
1
8
3
2. A caricature artist charges his clients a $10
setup fee plus $15 for every person in a picture.
a. Write a rule for the artist’s fee.
b. Write ordered pairs for the artist’s fee when there
are 1, 2, 3, and 4 people in the picture.
Answers
1. Y = x – 5
2. A. y = 15x + 10
B. (1,25)(2,40)(3,55)(4, 70)
Standard #20:
Evaluate an equation in
function notation.
Text Section: 4.3
Reminders
Substitute in for the given Variable
Follow PEMDAS
DCMS
Examples
Answers
1. 14
2. 7
3. 15
4. -1
Standard #21:
Graph a linear function.
Text Section: 4.4
Reminders
LINEAR means LINE
Make sure your graph is a LINE!
Examples
1. Graph y = -4x + 2
2. Graph y = 2x – 5
3. Graph y = -x + 3
4. Graph y = 7
Answers
1.
3.
2.
4.
Standard #22:
Determine if a Relation is a
Linear Function.
Text Section: 5.1
Reminders
LINEAR Function (linear is the KEY word)
DO NOT just look at x values, you have to
see if there is a common difference in the x
AND the y values!
No Absolute Values
Exponents
Sq Roots
Variable in the Denominator
Examples
Is it LINEAR (not just a function)!
1. {(-4,3), (-1, 1), (2, -1), (5, -3)}
2. x
y
-3 -1 1
3
7
12
3
18
5
25
3. Tell which equation is linear.
a. y = x+ 2
c. -2y + 5x = 8
b. y = 2x3
d. y = |x|+ 2
Answers
1. yes
2. no
3. A
Standard #23:
Write a function in
Standard Form
Text Section: 5.1
Reminders
Ax + By = C
“A” can NOT be a fraction OR negative.
If A is negative- change ALL signs.
If there is a fraction, multiply all by the
DENOMINATOR!
Examples
Write in Standard Form
1. 5x + 3y = -2
2. x-y = 1
3. -9x = 2y -7
4. 2y = ½ x – 5
Answers
1. 5x + 3y = -2 a = 5, b = 3, c = -2
2. x – y = 1
a = 1, b = -1, c = 1
3. 9x + 2y = 7 a = 9, b = 2, c = 7
4. x – 4y = 10
a= 1, b = -4, c = 10
Standard #24:
Identify Values of A,B, and C
Text Section: 5.1
Reminders
It has to be in Ax + By = C
A, B, and C are Real numbersNOT variables!!
Examples
Give values of A,B, and C
1. 5x + 3y = -2
2. x – y = 1
3. 9x + 2y = 7
4. x – 4y = 10
Answers
1. a = 5, b = 3, c = -2
2. a = 1, b = -1, c = 1
3. a = 9, b = 2, c = 7
4. a= 1, b = -4, c = 10
Standard #25:
Find the x and y-intercept in a given
situation. (equation, graph, word prob)
Text Section: 5.2-5.3
Reminders
Answer MUST BE an ordered pair!
Cover the y, solve for x
Cover the x, solve for y
Examples
Find the x and y intercepts of the following.
1. 2x + 5y = 10
2. –x + 6y = 18
3. You can earn $12 an hour babysitting
and $15 an hour raking leaves. You want to
make $360 in one week
Answers
1. (5,0),(0,2)
2. (-18,0)(0,3)
3. (30,0)(0,22)
Standard #26:
Interpret Rate of Change
(slope in a word problem).
Text Section: 5.4
Reminders
Rate of Change
=
SLOPE
=
y- y
x-x
Examples
The table shows the average
temperature for five months. Find
the rate of change for EACH time
period.
x
2
3
5
7
8
y
56
56
63
71
72
Answers
2-3 = 0
3-5 = 7/2
5- 7= 4
7- 8= 1
Standard #27:
Identify Slope as being:
(positive, negative, zero,
undefined)
Text Section: 5.4
Reminders
Positive
Negative
Zero
Undefined
Examples
Tell whether the slope of each line is
positive, negative, zero, or undefined.
1.
2.
6
6
4
4
2
2
-10
-5
3.
5
10
-10
-5
5
-2
-2
-4
-4
-6
-6
10
4.
6
4
6
2
4
-10
-5
5
10
2
-2
-10
-5
5
-4
-2
-6
-4
10
Answers
1. positive
2. undefined
3. negative
4. zero
Standard #28:
Identify Slope in a given
situation (ordered pairs, table, graph)
Text Section: 5.4
M=
Reminders
Rise
Run
M=
Up and Over
M=
Y-y
X-x
M=
Examples
1. (-2, -2) and (7, -2)
2.
3.
x
y
1
18.5
2
22
3
25.5
4
29
Answers
1. M = 0
2. M = 2/3
3. M = 7/2
Standard #29:
Write an Equation in Slope
Intercept Form from a
given situation
Text Section:5.6
Reminders
y = mx + b
m = slope
b = y intercept
Examples
Write an equation in SLOPE INTERCEPT FORM
1. m = 4; (-3, 5)
2. (3, -2)(12, 1)
3. 8x – 4y = 16
Answers
1. y = 4x + 17
2. y = 1/3x – 3
3. y = 2x - 4
Standard #30:
Graph from Slope
Intercept Form
Text Section: 5.6
Reminders
Change to y = mx + b
Plot your b (your beginning
point)
x =
Vertical Line
Y =
Horizontal Line
Up and Over for your slope
Examples
1. x = 2
2. y = - x - 4
3. y = 3x
8. y = -3
Answers
1.
3.
2.
4.
Standard #31:
Graph from Standard Form
Text Section: 5.2
Reminders
Change to y = mx + b
Plot your b (your beginning
point)
x =
Vertical Line
Y =
Horizontal Line
Up and Over for your slope
Examples
1. 6x + 3y = 9
2. -4x + 12y = -24
Answers
1.
2.
Standard #32:
Write an Equation to a Line
Parallel
Text Section: 5.8
Reminders
Parallel Lines= SAME Slope
1. Slope
2.Pt Slope
3.Slope intercept
Examples
Give all 3 Answers for Each.
PARALLEL
1. y = 3x + 4; (2, -5)
2. 5x -10y = 20; (-4,2)
Answers
1. m = 3
y + 5 = 3(x -2)
y = 3x -11
2. m = ½
y – 2 = ½ (x + 4)
y=½x+4
Standard #33:
Write an Equation to a Line
Perpendicular
Text Section: 5.8
Reminders
Perpendicular Lines= Opposite Inverse Slopes
1. Slope
2. Pt Slope
3. Slope intercept
Examples
Give all 3 Answers for Each.
PERPENDICULAR
1. y = 3x + 4; (9, -5)
2. 5x -y = 12; (-10,2)
Answers
1. m = -1/3
y + 5 = -1/3(x -9)
y = -1/3x -2
2. m = -1/5
y – 2 = -1/5 (x + 10)
y = -1/5 x
Standard #34:
Graph a Linear Inequality
on a Coordinate Plane
Text Section: 6.5
Reminders
Dashed or Solid?
Shade Above or Below?
Positive or Negative?
Examples
1-2 Graph each linear inequality.
1. x ≤ -2
2. y ≥ x + 4
Write a linear inequality for the given graph.
1.
3. Y > -x + 4
Answers
2.
Standard #35:
Gather data from a
Scatter Plot.
Text Section: 4.5
Reminders
Do NOT Connect the dots
Correlations:
Positive, Negative, No Correlation
Examples
1. Describe the correlation
2. Predict typos in 12 chapters
Answers
1. Positive Correlation
2. Approx 14
Standard #36:
Model a Scatter Plot
Text Section: 4.5
Reminders
Dots- don’t connect
Titles- x and y axis
Examples
Graph a scatter plot using the table.
Remember to include all aspects of the
graph.
Hours Studied
Test Grade
3
5
2
6
4
1
2
7
1
7
0
1
3
65
80
70
80
75
50
65
80
45
95
20
40
70
Test Grade
Answers
Hours Studied
Standard #37:
Estimate a Line of Best Fit
Text Section: 4.5
Reminders
y = mx + b
Check your slope
Check your y intercept
*especially on multiple choice!
Examples
1. Estimate the line of best fit
2. Which equation represents
the line of best fit for the
given scatter plot?
Answers
1. y = -x + 5
2. y = -2x - 1
Standard #38:
th
n
Find the
term of a
Sequence
Text Section: 4.6
Reminders
an= a1+ (n-1)d
Term you need
to find
Common Difference
in the sequence
Examples
Find the given term of each arithmetic sequence.
1. 5,2,-1,-4,…; 23rd term
2. -1.1,0,1.1,2.2…; 51st term
3. 407,402,397,392…; 17th term
4. 11,21,31,41,…; 33rd term
Answers
1. A23= -61
2. A51= 53.9
3. A17= 327
4. A33= 331
Standard #39:
Find the Common
Difference in a Sequence
Text Section: 4.6
Reminders
Common Difference is the d in
the formula
Look at the sequence, do the
numbers go up (+) or down (-), and
by what value?
Examples
Find the common difference (d) in
each arithmetic sequence.
1.
2.
3.
4.
107,105,103,101,…
4.85, 5, 5.15, 5.3, …
3 ½ , 2 ¼ , 1, -3/4 , …
2, 15, 28, 41, …
Answers
1. d= -2
2. d = .15
3. d = - 1 ¼
4. d = 13
Standard #40:
Simplify Exponential
Expressions
Text Section: 1.4
Reminders
Words
Multiplication
Power
Value
3 to the first
power
3 to the second
power or 3 squared
3
3
3
3*3
32
3 to the third
power, or 3 cubed
3*3*3
33
27
3 to the fourth
power
3 to the fifth
power
3*3*3*3
34
81
3*3*3*3*3
35
243
9
Examples
1-3 Simplify each expression
1. (-2)3
2. -52
3. (2/3)2
Answers
1. -25
2. -8
3. 4/9
Standard #41:
Write Numbers as a Power
of the given base
Text Section: 1.4
Reminders
Use the given base
Then find the exponent for
that base to get the given
answer.
Examples
4-6 Write each number as a power of the
given base.
4. 8; base 2
5. -125, base -5
6. 64, base 8
Answers
4. 23
5. -53 or (-5)3
6. 82
Standard #42:
Zero and Negative Exponents
Text Section: 7.1
Reminders
NO NEGATIVE
Exponents EVER!!!
ANYTHING Raised to
the Zero Power is = to 1
Examples
1. m-3n
2. -3f-3
3. x-7y2
r3v-4
4. 4x-5
y-6
5. (g0h0)7
Answers
1. n
m3
2. -3
f3
3. v4y2
r3x 7
4. 4y6
x5
5. 1
Standard #43:
Multiplication of Exponents
Text Section: 7.3
Reminders
You can only multiply powers that have
the same base- if they do you ADD
exponents
When you raise a power to a power you
MULTIPLY the exponents and leave the
base the same
Check for Negative and Zero Exponents
Examples
1. (f4)6 • g
2. m • (h3)4 • (m-2)3
3. (6y8)2
4. (k4)2 • (m-1)-4
Answers
1. f24g
2. h12
m5
3. 36y16
4. k8m4
Standard #44:
Division of Exponents
Text Section: 7.4
Reminders
When we divide with exponents we
subtract the exponents.
You can only divide powers
with the same base
Examples
38
32
a5b9
(ab)4
y
y4
m5n4
(m5)2n
Answers
1. 36 = 729
2. ab5
3. 1
y3
4.
n3
m5
Standard #45:
Standard Form to
Scientific Notation
Text Section: 7.2
Reminders
The Number has to be > 1 but < 10
Be careful where you put your
decimal.
Is your exponent – or + ?
Examples
1. .000000802
2. 8127
3. .678
4. 60228
Answers
1. 8.02 x 10-7
2. 8.127 x 103
3. 6.78 x 10-1
4. 6.0228 x 104
Standard #46:
Scientific Notation to
Standard Form
Text Section: 7.2
Reminders
Just move the Decimal
Negative Exponent- Move Left
Positive Exponent- Move Right
Examples
1. 6.09 X 104
2. 53.8 X 10-5
3. 0.07 X 108
4. 8.1 X 10-2
Answers
1. 6090
2. .000538
3. 7000000
4. .081