Download Unit 2 - Weebly

10th Grade Mathematics Review for Final exam
Don’t be alarmed with the following review guide! Some of these summaries I had already given you
for the specific unit. Use this as a GUIDE to go back and study from along with your text book. Also
useful would be to look at past tests, quizzes, and your notes.
We decided on testing you over the entire year. We realize this is a lot to review. With this in mind,
we will ask easy to medium difficulty questions. (As opposed to if you were tested over JUST semester 2,
the problems would be more difficult) I’m sorry I couldn’t be here today. Please make sure you sent me
your “mathematical investigation” to [email protected]. If you haven’t already done so it will be
considered late! Everyone except for Nora should have completed it. The outline is on my website
(www.howesmath.weebly.com)
Semester 1
Unit 1
Ch.10(G,H), 11,14(f)
10G Speed, Distance and Time
D
s
t
11 Mensuration
 A. Surface area of 2 and 3 dimensional figures
o Unit conversions i.e. (100cm)2 = 10,000cm2 = (1m)2 =1m2
 B. Volume of 3 dimensional figures
o Unit conversions i.e. (100cm)3 = 1,000,000cm3 = (1m)3 =1m3
 C. Capacity
 D. Mass
o

density 
mass
volume
E. Compound solids
14F Lines of symmetry
Unit 2
Ch.5,13,17
5 Graphs, Charts, and Tables



A. Statistical Graphs
o Bar charts,
o Pie charts
o Scatter diagrams (2 variable stats)
o Stem-and-leaf plots
B. Graphs which compare data
o Side by side bar charts
o Back to back bar charts
o Back to back stem and leaf plots
C. Using technology to Graph data
o You should know how to enter data into your lists, find the 1 and 2 variable statistics,
and find the linear regression line that best models the data
13 Analysis of Discrete Data
 A. Variable used in statistics
o Categorical data
 i.e. types of cars (ford, porche, VW, Mercedes, etc)
o Quantitative variable
 Discrete (whole number increments like how many people in a family)..think
counting
 Continuous (could take on any number within a range of values….i.e. weight of
new born baby’s or height, or time)….think measuring
 B. Organizing and Describing data
o Normal distribution (symmetrical)
o Positively skewed
o Negatively skewed
 Grouping Discrete data into class intervals, and analyzing frequency tables is crucial
 C. The centre of a discrete Data set
o
The mean…………. x 
x
where  x is the sum of the data and n is the total
n
number of data values.
o


The median is the middle most number. For an ordered data set use
count the number of places to the median.
o The mode is the most frequently occurring data value
D. Measuring the spread of discrete data:
o Range= max value – min value
o IQR = inter quartile range = Q3 - Q1
E. Data in frequency tables
o
x 
( f  x)
f
n 1
to find the
2
o
Use
n 1
to find the place value of the median. Analyzing the cumulative frequency
2
can help you here.

G. Grouped Discrete Data
o To find the mean, you must first use the midpoints of each class interval and use the
following equation to get an estimate for the mean.

o
Use
x 
( f  x)
f
n 1
to find the place value of the median. Analyzing the cumulative frequency
2
can help you here.
o
Don’t worry about the median  L 
N
 I formula on page 291!...but you might see it
F
next year.
 G. Stats from technology.
o You should know how to enter a data set into a list and obtain the 1 variable statistics.
 See section F on page 21 of you book or come and see me if you forgot how to
do this.
Ch.17 Continuous Data
 A. The Mean of Continuous data
o

( f  x)
to estimate the mean
f
B. Histograms
o …this is like a bar chart except the bars must be touching. See the bottom of pg 355.
o

Use the midpoints of each class interval and x 
Frequency density =
frequency
class interval width
C. Cumulative frequency graph(very important)
o You need to know how to draw one and how to read it!(see page359 in your book for an
example)
 For your x-values, use the upper end point of each class interval
 For you y-values use the corresponding cumulative frequency value.
 Remember to label your axis
 Be able to find the Median (50th percentile), Q3 (75th percentile), and Q1 (25th
percentile)
Unit 3
Ch.16,23
16 Algebraic Fractions

A. Simplifying algebraic fractions (watch out for illegal cancelation)
o



i.e.
a  3 a 1
!!!!!! (see pg 340)

3
1
B. Multiplying and dividing algebraic fractions
o To multiply, multiply across the numerator, and multiply across the denominator.
o Dividing is the same as multiplying by the inverse
C. Adding and subtracting algebraic fractions.
o First find a common denominator.
o Keep the denominator and add or subtract across the numerator.
D. More complicated fractions ( I won’t ask you to do a really complicated one on the exam,
but definitely you should know how to add, subtract, multiply and divide two algebraic
fractions)
o You should also be familiar with certain x values that may be a problem in you
expression. For example in the expression
x3
, when x =-2 or 1, I will get a
( x  2)( x  1)
zero in the denominator…….we call this “undefined”
o Also note when x = -3 the expression will be equal to zero!
Ch. 23 Further Functions
 A. Cubic Functions
o
General form: f ( x)  ax3  bx 2  cx  d where a  0 and a,b,c, and d are constants.


(see blue box at bottom of page 470 and top of page 471) You should be
familiar with the general shape of the cubic function
B. Inverse Functions
o
o

The inverse function of f ( x ) is f 1 ( x)
The inverse function can be found algebraically by interchanging x and y and then
making y the subject of the resulting formula.
C. Using Technology
o Using your GDC, you should be able to do the following:
 Obtain:
 A table of values for a function
 A graph of the function
 The “zeros” or “x-intercepts” for the function
 The y-intercept of the function
 Any asymptotes of the function
 The “turning points” or in other words the max (peak) or min (trough)
values
 The points of intersections of two functions( This is just the points
where two graphs overlap)
 Note: You can find instructions to do any of the above calculator
functions starting on pg 22 of your book….or just ask me!!!


Also, to solve “unfamiliar functions” you can always use your calculator. For
example to solve 3x  5  x  2 for x, plug 3x  5 into your Y1 and x  2
into your Y2 and use the 2nd trace (intersect) buttons on your GDC to find where
these two functions meet! The x-coordinate of the points of intersection will be
your solution. (you should get x 1.763 if you try this)
D. Tangents to curve
o Just know what a tangent line is, and how to find the slope of a line (rise over run)
Unit 4
Ch.28&31
28 Exponential Functions and equations
 A. Rational Exponents
o (Know and love the exponent laws on page 566 and 567)
 B. Exponential Functions
o An exponential function is a function in which the variable occurs as part of the
exponent i.e. f ( x)  2 x




All graphs of the form f ( x)  a x

Have a horizontal asymptote at y=0 (the x-axis)

Pass through (0,1) since f (0)  a 0  1
C. Exponential equations
D. Problem solving with exponential equations
o Growth and Decay problems
E. Exponential Modeling
o Using your GDC, enter your x data into L1, your y data into L2, and use the “ExpReg”
(press Stat, calc, ExpReg) to get a “model” that fits your data.
o Be familiar with the “r-value”
 1  r  1 and it tells you how closely your “model” fits the data. (see page
576)
Ch. 31 Logarithms
A. Logarithms in base a.
-writing equivalent logarithmic and exponential statements
B. The logarithmic function
-The logarithmic function is the INVERSE of the exponential function.
- The graphs will be symmetrical over the line y=x.
-You can find the inverse of an exponential function (it will be logarithmic)
-You can find the inverse of a logarithmic function (it will be exponential)
-use your GDC to solve log(x-2)=3-x
C. Logarithm Rules
log( xy )  log( x)  log( y)
x
log    log( x)  log( y)
 y
log( x n )  n log( x)
Remember that putting an inverse function into the original function will always give you the same value
back. i.e.
f  f 1 ( x)   x . Therefore, for logarithmic and exponential functions we have;
log a (a x )  x
a loga ( x )  x
Your calculator only thinks in base ten. The CHANGE OF BASE FORMULA is;
log a ( x) 
log( x)
log(a)
D. Logs in base 10
Logarithmic equations
E. Exponential AND logarithmic equations
F. The NATURAL LOG …….ln(x)
Semester 2
Unit 5
Ch.22&25
Chapter 22 (2 var stats)

A. Correlation between two variables.
o Analyzing scatter plots
o Some words to remember(Strong, moderate, weak, non-linear, linear, positive,
negative)

B. Line of best fit by eye
o Must include the point (xbar, ybar)!!!

C. Linear regression, (regression line, line of best fit etc. different names same thing)
o Know how to enter 2 data sets into your calculator
o Know how to create a scatter plot
o Use your stat button and select linreg(y=mx+b) to get the regression line
o Make sure the diagnostic is turned on so you can get Pearson’s correlation coefficient(rvalue)
o r-value close to 1 means strong POSITIVE linear correlation
o r-value close to -1 means strong NEGATIVE correlation
o r-value close to 0 means no correlation
 Note: 1  r  1
o Interpolating and extrapolating data using your regression line
Chapter 25 (probability)

A. See power point (www.howesmath.weebly.com) for a review of some probability
terminology

B. Relative frequency=




o Relative frequency is based on experiments and is and estimate for the probability
C. Probabilities from two-way tables
D. Expectation=np where n is the number of trials and p is the probability. We can use
expectation to predict the future!!!
E. Combined Events:
o Sample space is the set of all possible outcomes of an experiment
 Represented using
 Lists
 2D grids
 tree diagrams
 Venn Diagrams
F. Theoretical probability
o
o

frequency
number of trials
P(event happens)=
number of ways it can happen
total number of possible outcomes
use lists, tree diagrams, 2Dgrids or Venn Diagrams to assist you in figuring out “the
number of ways it can happen” and “the total number of possible outcomes.”
G. Compound events
o Independent events (A doesn’t affect the outcome of B and vice versa)
 P( A  B ) = P(A)  P(B)
o Complementary events (you either hit the door or you miss it……..1 MUST occur)
 P(E) + P( E' ) = 1
o Dependent events
 P(A THEN B) = P(A)  P(B given that A has occurred)





H. Using tree diagram
o Place probabilities on the branches and multiply across branches to find probabilities
I. Sampling with and without replacement.
J.
o Mutually Exclusive event
 (disjoint, or no overlapping)
 P(A  B) = P(A) + P(B)
o Non-Mutually Exclusive events
 (at least one overlapping event)
 P(A  B) = P(A) + P(B) - P(A  B)
K. Miscellaneous probability questions
o Putting EVERYTHING together to solve problems involving probabilities
Sets, set notation,
o A,B,C,E,F (use capitals to represent a set)
 i.e. A= the set of odd numbers
B= the set of even numbers
means AND
means or
A’ with the apostrophe means the compliment of A or everything outside of A. If A is the set of
odd numbers then A’ is any number that is not odd.
Unit 6
Ch.15(D), 27, 29
15D The first quadrant of the unit circle.
See exercises 15D.1 and 15D.2 for examples
 sine, cosine, and tangent (Soh Cah Toa)
 Know your 30,60,90 and 45, 45, 90 triangles and how to find EXACT values for all of these
angles.
 Coordinates for any point on the unit circle are (cosθ, sinθ)
 Find values for unknown sides of a triangle using trig functions (see #3 pg 329)
 Simplifying trigonometric expressions and equations (see #1,2 pg 329)
27 Circle Geometry
Review HW from sections 27A.1, 27A.2, 27B.1 and 27B.2 and ch. 27 quiz
 Circle Theorems
o Terminology:
 Circle, circumference, arc, chord, semi-circle, diameter, radius, tangent
(definitions on pg 547,548)
o Know and be able to apply the 4 circle theorems (pg 549,550)

Arc Theorems
o Terminology:
 Minor arc, major arc, major segment, minor segment, SUBTENDS
(definitions on pg. 552)
o Know and be able to apply the 2 arc theorems (pg 553)

Cyclic Quadrilaterals
o Know and be able to apply the 2 cyclic quadrilateral theorems (pg 556)
o Know and be able to apply the 3 tests for cyclic quadrilaterals (pg559)
29 Further Trigonometry
Review HW from sections 29A-H and extra packet problems on radian measure
 Section A: The unit Circle. This section extends upon 15D. You already know how to find exact
values for sin, cos, and tan for 0,30,45,60, and 90 degree angles using your two special triangles.
Find reference angles (i.e. reference angle for 210 is 30) and decide on the value of sine, cosine,
and tangent depending on which quadrant you are in. (see examples from 29A.1, 29A.2)

Area of a triangle: A 
1
ab sin C where a and b are the two legs and C is the INCLUDED
2
angle.

The sine rule:
o
o


sin A sin B sin C


a
b
c
Use the sine rule when you are given:
 Two sides and an angle not included between these sides, or
 Two angles and a side.
Note. You should be aware the ambiguous case.
The Cosine Rule: a  b  c  2bc cos A where b and c are the two sides and A is the
included angle, and a is the side across from A. (Know how to label a triangle.)
Problem solving using areas of triangles, sine and cosine rules, Pythagorean theorem. This may
involve
BEARING PROBLEMS, and problem solving with compound shapes.
(see HW from sections 29E and 29F)
2
2
2
Trigonometric graphs
 Terminology:
o Periodic function, period, principal axis, maximum point, minimum point, crest, trough,
amplitude. (see definitions on pg 596)
 See “properties of trig graphs.” Green boxes on page 597 and 598.

For y  a sin bx  c and
o
o
y  a cos bx  c
a is the amplitude (stretches vertically)
b affects the period and is related by

o
period =
360
b
c shifts the entire graph up or down.
Unit 7
Ch.24, 26, 30
Chapter 24 Review
A. Vector notation and the directed line segment.
B. Vector equality- two vectors are equal if and only if they have the same magnitude and
direction.
C. Vector addition (tip to tail method)
D. Vector subtraction (subtracting is the same as adding the negative)
a. Note> For two Vectors, a and -a will have the same magnitude but opposite direction.
E. Vectors in Component form
F. Scalar multiplication
G. Parallel vectors
H. Vectors in Geometry
See Review sets 24A, 24B for practice problems, and the quiz over chapter 24, and past HW assignments
for review.
Chapter 26 Review
A. Number sequences…..finding the rule from the given pattern.
B. Algebraic rules for sequences.
a. Be familiar with the un notation.
C. Geometric sequences.
a. The next term is generated multiplying the previous term by some constant.
D. VERY IMPORTANT ……..The Difference Method for Sequences.
a. This method will generate the rule for you.
i. 1st difference constant = linear sequence
ii. 2nd difference constant = quadratic sequence
iii. 3rd difference constant = cubic sequence
Also see my notes on www.howesmath.weebly.com for my notes on the difference of squares method.
Review for Ch. 30
A. Direct variation
a. y  x means y  k  x where k is some constant
B. Inverse variation
C.
y
1
1
means y  k
where k is some constant.
x
x
D. Variation Modeling
a. Finding a correct model to fit given data. Data will either fit a direct variation model or
an in-direct variation model
E. Power Modeling
a. The same as section c but fitting the data to a power model. The equation will be of the
form y=axb where a and b are constants.
You should be able to find a and b analytically and by the use of your GDC. (see directions at the
front of your book or come and see me if you have questions about how to do this)
Unit 8
Ch.21&32
Review your ch. 21 and ch. 32 quizzes
21 Quadratic equations
 A. Quadratic equations
o
f ( x)  ax 2  bx  c (general form)
f ( x)  a( x   )( x   ) (factored form)
o
f ( x)  a( x  h)2  k ( vertex form where (h,k) is the coordinate for the vertex)
o

B. The Null factor law
o If a  b  0 then either a = 0 or b = 0.
o You should know and be able to use the null factor law to solve a quadratic equation

Ex.
x2  7 x  6  0
( x  6)( x  1)  0
 either (x+6) = 0 or (x+1) = 0
 x  6 or x  1

C. The quadratic Formula
o
For ax 2  bx  c  0 , then x 
b  b2  4ac
2a




D. Quadratic Functions
E. Graphs of quadratic functions (see green box with summary of the shape of a quadratic on
page 434)
F. Axes Intercepts
o Know how to find the x and y- intercepts for a quadratic.
o You can use this information to make a “sketch” of the graph
G. Line of symmetry
o Remember the general shape for a quadratic is a parabolas
o The parabola is symmetrical about a line of symmetry
o A parabola always has a turning point we call the vertex
o
The equation for the line of symmetry of f ( x)  ax 2  bx  c is x 
 H. Finding a quadratic function
 I. Using technology.
NOTE: I expect you to be able to solve a quadratic equation by
1. Factoring
2. Using the quadratic formula
3. Using your graphics calculator
 J. Problem solving
o Read the problem carefully
o Draw a diagram!! A picture is worth 1000 words!
32 Inequalities (we just covered this so I will keep it brief)




A. Solving on variable inequalities with technology
B. Linear inequality regions in the Cartesian plane
o Shade out the UNWANTED regions
C. Integer points in Regions
D. Problem solving
b
.
2a