Analytic Geometry Condensed Study Guide

Unit 1A dilation is a transformation that makes a figure larger or smaller than the original figurebased on a ratio given by a scale factor. The dilation is an enlargement when the scale factor is greater than 1, a reduction when it is between 0 and 1. __________________________

Here, the scale factor is 2. If the center of dilation is the origin, you can just multiply all the coordinates by the scale factor. ABC is the original picture or preimage, A’B’C’ is the new figure or image.

__________________________

If a segment of a figure dilated does not pass through the center of dilation, the segment will shift to a parallel segment(see AB and A ' B' above). If the line passes through the center of dilation, it will not move, as seen in the diagram below(note AC is on A ' C ' ¿¿:

Note that if the center of dilation is not (0,0), you do not simply multiply the coordinates by the scale factor.

To determine the center of dilation, connect each of the image points to the preimage points and note the intersection point.

When a figure is dilated, the image and preimage are similar(same shape). Similar figures have congruent angles and proportional sides.

Two triangles can be proven similar if you can find two pair of corresponding angles congruent. This is called AA(angle-angle).

Note the triangles are the same shape but a different size.

To prove that two triangles are similar(or congruent), a two-column proof is often used.

If parallel lines are present within triangles, look for alternate interior or corresponding angles. Look also for vertical angles(opposite in an x shape) and shared sides.

In the figure above, <1 and <5 are corresponding, <3 and <5 are alternate interior, and <1 and <3 are vertical. All of these relationships yield congruent angles. The bold arrows in the lines indicate the lines are parallel.

Notice the diagram and the proof that follows:

Given:

Prove:

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Analytic Geometry Condensed Study Guide

The missing reason would be “Corresponding Angles are Congruent.”

Figures are congruent if they have the same size and shape. Congruent figures have congruent sides and angles.

A rigid motion is a transformation of points in space consisting of a sequence of one ormore translations, reflections, and/or rotations (in any order). This transformation leavesthe size and shape of the original figure unchanged.

Translation,Reflection,Rotation → Congruent and Similar

Dilation→Similar

The following patterns are sufficient for showing two triangles congruent:

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

ASA (Angle-Side-Angle)

AAS (Angle-Angle-Side)

If figures are congruent, they are also similar. Any of the six corresponding parts(three angles and three sides) of two congruent triangles are congruent. This is often called CPCTC (corresponding parts of congruent triangles are congruent.)

A problem on a grid can be shown congruent by counting

blocks.

The two triangles above can be shown congruent by SAS by simply counting off the horizontal and vertical lengths.

The three angles of a triangle add to 180 degrees.

In an isosceles triangle(two sides congruent), the angles opposite the congruent sides are also congruent.

Triangle Midsegment Theorem: If a segment joins the midpoints of two sides ofa triangle, then the segment is parallel to the third side and half its length.

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Analytic Geometry Condensed Study Guide

Above, DE is parallel to BC and is half its length.

The three medians of a triangle meet at a point called the centroid.

Any point on a perpendicular bisector of a segment is equidistant from the endpoints of that segment.

C is exactly equidistant from A and B.

When conducting a proof involving congruent triangles, mark the diagram as you go:

If properly marked, you will see than reason 6 should be SAS.

There are several constructions that you will need to know. The markings are listed below. Go to www.mathopenref.com/ constructions .html for animations.

1. Copy a segment.

2. Copy an angle.

3. Bisect an angle.

4. Perpendicular bisector of a segment.

5. Construct a perpendicular line through a point not on the line.

6. Construct a parallel line through a point not on the line.

In this diagram, you originally had line l and point P.

7. Equilateral triangle in a circle.

8. Square in a circle.

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Analytic Geometry Condensed Study Guide

9. Regular hexagon in a circle.

Unit 2

These are called trigonometric functions.

In a right triangle, the two acute triangles are complementary, or adds to 90. The sine of an angle equals the cosine of its complement an d vice versa.

sin 50=cos40

cos20=sin 70, etc.

The Pythagorean Theorem helps find the third side of a right triangle if you know two sides. a2+b2=c2 The hypotenuse is always c.

tanA= sinAcosA

If you know one trig function, you can find the other two by building a right triangle. For

example, if sin A=¿ 35

¿, draw

a right triangle, label one acute angle A, label the opposite as 3 and the hypotenuse as 5, find the missing side with the Pythagorean Theorem, then find cos or tan.

To find a missing side in a right triangle with one side and one angle, use the appropriate trig function and solve.

To find x, use the ration

sin 75= x12

x=12 ∙sin 75

x=11.59

Typically, if the variable is on top, you multiply. If it’s on the bottom, you divide.

The angle of depression is 3 degrees in the diagram:

The angle of elevation is 32 degrees in the diagram:

Unit 3A radius of a circle is perpendicular to a tangent at the point of tangency.

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Analytic Geometry Condensed Study Guide

Two tangents from a common external point are congruent.

The measure of an arc of a circle equals its central angle.

The measure of an arc is one half of an inscribed angle that intercepts it.

It follows than an inscribed angle that intercepts a semicircle is a right angle.

An angle formed by a tangent and chord resembles an inscribed angle and is also half the intercepted arc:

When two chords intersect inside a circle, the vertical angles formed are the average of the two intercepted arcs they face:

When any combination of tangents and secants intersect a circle, the outside angle is half the difference of the intercepted arcs:

When two chords cross in a circle, the product of each portion of each chord is equal to the product from the other chord:

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Analytic Geometry Condensed Study Guide

When you have tangents or secants, the portions of segments obey the rule “outside * whole = outside * whole.

The opposite angles of an inscribed quadrilateral are supplementary:

__________________________

The area of an entire circle can be found by the formula

A=π r2 . The area of a sector of a circle can be found with the

formula AS=m

360∙ π r2

.

Notice how you are just finding a fraction of the area.

The entire circumference of a circle can be found by the formula C=πd or C=2 πr .To find the arclength of an arc, use the formula L=¿ m

360∙2 πr . Notice how you are

just finding a fraction of the circumference.

Angles and arcs can also be measured in radians. πradians= 180 degrees. To convert from radians to

degrees, multiply by 180π

.

To convert from degrees to

radians, multiply by π

180 . An

entire circle is 2 π radians.

Volume measures the amount of space enclosed by an object.

Cylinder: V=π r2 h (V=Bh ¿

Cone: V=13

π r2 h (V=13

Bh

)

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Analytic Geometry Condensed Study Guide

The variable B refers to the area of the base, which is π r2 for a circle.

Pyramid: V=13

Bh

Prism: V=Bh

Pyramids and Prisms can have any polygonal base.

Sphere: V= 43

π r3

By Cavalieri’s Principle, two figures have the same volume if each of their cross sections are equal. The cylinders below have the same volume:

Unit 4Some basic root examples:

Rational exponents are exponents that are fractions. Below, b is called the index of the root.

Some basic exponent rules:

Rational numbers can be written as the ratio of two

integers pq

. The decimal

equivalent of any rational number is a terminating or repeating decimal. Examples include:

If you add, subtract, multiply, or divide rational numbers(except by 0), the result is a rational number. This is called closure.

Irrational numbers cannot be written as the ratio of two integers. Examples include:

If you add, subtract, or multiply a rational number and an irrational number, the result is always irrational.

A polynomial consists of constants, variables, and exponents. Examples include:

If you add or subtract polynomials, just combine like terms. When subtracting, be careful to change all of the signs in the second polynomial.

To multiply polynomials, distribute:

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Analytic Geometry Condensed Study Guide

Like rational numbers, polynomials are closed under addition, multiplication, and subtraction.

Polynomial operations are involved with perimeter and area problems:

Perimeter

Area

The imaginary unit is i=√−1. Any number containing i is called an imaginary number.

The powers of i fall in a repeating pattern:

i0=1

i1=i

i2=−1

i3=−i

i4=1

i5=i

Etc.

Complex numbers are written in the form a+bi . When adding or subtracting complex numbers, treat them as if they are polynomials and as if i were a variable(though it’s not).

When multiplying complex numbers, it is especially helpful to remember that i2=−1.

(8+6 i ) (2−3 i )

16−24 i+12i−18 i2

16−12i−18 (−1 )

34−12 i

Unit 5Quadratic equations are equations of the form

a x2+bx+c where a≠ 0.

There are several ways to solve a quadratic equation:

1. graphing(not efficient unless graph is given)

2. square roots (only if b=0)

3. factoring(only if factorable)

4. quadratic formula

Methods 2 and 3 are easier than 4, but each is limited as to when they can be used.

Sometimes, the solutions, or roots, are complex.

Method 1- Graphing

If a graph is given, the x-intercepts are the solutions. You may have 2 real solutions, 1 repeated solution, or 2 complex(0 real) solutions.

Here, the number 0 is a repeated solution.

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Analytic Geometry Condensed Study Guide

Here, 1 and -1 are the two solutions

There are no x-intercepts. The solutions are complex and can not be determined from the graph.

Method 2- Square Roots

The following illustrates the method. Notice that b=0 from

a x2+bx+c. (No x term)

Method 3- Factoring

There are several factoring techniques.

Always look for a GCF before you look for a pattern:

2 x2+4 x=2 x( x+2)

Difference of Two Perfect Squares

x2− y2=( x− y ) ( x+ y )

x2−9=(x−3)(x+3)

4 a2−25 b2=(2 a+5 b)(2a−5 b)

x4−36 y4=(x2−6 y2)(x2+6 y2)

Trinomials with Leading Coefficient 1:

x2+9 x+20=(x+4)(x+5)

x2+3 x−18=(x+6)(x−3)

Trinomials with Leading Coefficient not 1:

2 x2+15 x+7

(x+ 142 )(x+ 1

2 )( x+7 ) (2 x+1 )

When you have an equation, set each factor equal to zero and solve:

x2−15 x+56=0

( x−7 ) ( x−8 )=0

x−7=0∨x−8=0

x=7∨x=8

Method 4-Quadratic Formula

To find the zeros of the above functions, make h (t )=0 and solve. You must use the quadratic formula as you can’t use the other methods.

An example of a quadratic expression is 7 x2+8x−9.

7,8, and -9 are called coefficients.

7 x2,8 x, and -9 are called terms.

7 is called the leading coefficient.

-9 is called the constant term.

The degree, or highest power, or any quadratic equation is 2.

Completing the Square changes a quadratic equation from standard form (no parenthesis) to vertex form (one parenthesis.

y=x2+6 x+8

y−8=x2+6 x

y−8+9=x2+6 x+9

y+1= (x+3 )2

y=(x+3)2−1

The nine comes from taking half of 6 and squaring it.

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Analytic Geometry Condensed Study Guide

The significance of vertex form is that you can easily determine the vertex of the parabola. In this case it’s (-3,-1). Change the sign inside the parenthesis, but not outside the parenthesis.

The vertex can be determined without completing the square.

To find the x-coordinate of the

vertex, use the expression −b2 a

.

Put this value back into the equation to find y:

x2+6 x+8 Here b=6∧a=1

The x-coordinate is

−b2 a

=−−62 (1 )

=−3. Plug the -3

back in to get -1.

One common application of quadratic equations are area problems. For example, a rectangle has a width 5 more than its length. Find the dimensions if the area of the rectangle is 50 ¿2.

x = length, x−5 = width

A=lw

x (x−5 )=50

x2−5 x=50

x2−5 x−50=0

( x−10 ) ( x+5 )=0

x=−10 or x=5

Granted that length can’t be negative, it follows that the length is

5 and the width is 10.

The formula

is also a common application problem. Typically the variables vo

(initial velocity) and ho (initial height) are provided. -16 is a constant due to gravity. With metric units, -4.9 may be used in place of -16.

How long would it take for a ball launched with an initial velocity of 48 ft/s from an initial height of 160 feet take to hit the ground? Use the formula

h ( t )=−16 t2+48 t+160

h (t )=0 when the ball is on the ground.

0=−16 t 2+48 t+160

0=−16 (t 2−3 t−10 )

0=−16 (t−5 ) ( t+2 )

t=5∨t=−2

The answer is 5 as time can’t be negative.

To solve a system of equations including a linear equation and quadratic equation, isolate y from the linear equation and set the result equal to the quadratic equation and solve. There are often two solutions:

Get y by itself:

Set this expression equal to the quadratic equation:

Now plug each x back into the linear equation to find the y’s that go with them:

You can also graph such a system and observe their intersection points:

The solutions are approximately

(-3,-6) and (2,-1).

To find the average rate of change for a quadratic equation over the interval [a,b], apply the

formula r=f (b )−f (a)

b−a. This is

basically the slope formula.

Find the average rate of change from −1 ≤ f (x)≤ 2 using the table.

x f ( x )-1 50 41 5

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Analytic Geometry Condensed Study Guide

2 8

r= 8−52−(−1 )

=33=1

When the leading coefficient of a quadratic function is positive, the graph is a u-shape, has a vertex that is a minimum of the graph, and first decreases then increases:

When the leading coefficient of a quadratic equation is negative, the graph is and upside-down u, has a vertex that is a maximum of the graph, and first increases then decreases:

A recursive process can show that a quadratic function has second differences that are equal to each other.

Notice the second differences for the quadratic equation

A recursive function is one where each value is based on the previous one. When working with multiple choice problems involving formulas, it is often best to work backwards:

Find the equation that matches the sequence: 8,17,32,53,80,…

a) 3 x2 b) 3 x2+5

c) 5 x2 d) 5 x2+3

Plug 1 into each formula. 1 should give you 8(first term)

Plug 2 into each formula. 2 should give you 17(second term).

Etc.

B is the correct answer here….

A graph can be transformed in several ways.

Consider f ( x )=x2

Adding outside a function makes it go up. f ( x )=x2+1

Subtracting inside a function makes it go down. f (x)=x2−1

Adding inside a function makes it go left. f ( x )=(x+1)2

Subtracting inside a function makes it go right. f ( x )=¿

(x−1)2

Multiplying by a number whose absolute value is larger than one makes a function vertically stretch (get more narrow) f ( x )=3 x2

Multiplying by a number whose absolute value is less than one makes a function vertically shrink

(get wider) f ( x )=13

x2

Multiplying by a number that is negative will make a graph reflect over the y-axis (turn upside-down)

f ( x )=−x2

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Analytic Geometry Condensed Study Guide

Even functions have variable exponents that are all even, are symmetric to the y-axis, and have the property f ( x )=f (−x ) .

f ( x )=7 x4+9 x2−3 is an even function. Notice that its variable exponents are all even(4,2,and 0). If you put any number in, say 2, its opposite,-2, would yield the same value if plugged in.

f (2 )=145 , f (−2 )=145

Odd functions have variable exponents that are all odd, are symmetric to the origin, and have the property f ( x )=−f (−x ) .

f ( x )=2 x3−11 x is an odd function. Notice that its variable exponents are all odd(3 and 1). Notice that f (2 ) and f (−2 ) are

now opposites. f (2 )=−6

f (−2 )=¿ 6.

Remember from coordinate algebra that exponential functions may start out small, but their values will eventually exceed those of linear or quadratic functions.

A quadratic regression is a curve that best fits a set of data. If a set of data takes on the approximate shape of a parabola, you can draw a parabola through the data to make observations.

Unit 6

The equation of a circle on a coordinate plane is

where (h,k) is the center and r is the radius. For the equation

(x−3)2+( y+2)2=9, the center is (3,-2) and the radius is 3.

The area formula for a circle is derived from the Pythagorean Theorem.

Sometimes the equation is not given with parenthesis. In this case, you have to complete the square in order to find the center:

The center is (1,2), the radius

√2.

A parabola is not only the graph of a quadratic function, but can be defined as all of the point that are equidistant from a fixed point called the focus and a fixed line called the directrix.

When the parabola opens vertically, its formula is

Where (h,k) is the vertex and p is the distance that the focus and directrix is from the vertex.

When the parabola opens horizontally, the formula is

The picture in this situation:

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Analytic Geometry Condensed Study Guide

Write the equation of the parabola whose focus is (6,4) and whose directrix is y=2.

Since the directrix is a horizontal line, the graph opens vertically and the x-coordinate of the focus and vertex will be the same. So h=6. k, the y-coordinate of the vertex, is halfway between the focus and directrix. Halfway between 2 and 4 is 3. So k=3. p is the distance between the directrix(or focus) and the vertex. The distance from 3 and 2 is 1, so p=1. Using the correct form (

x2¿ our answer is

When solving a system of equations involving a line and a circle, the method is similar to that of a line and quadratic.

Find the intersection points of a line with slope of 2 that passes through the origin and a circle with radius √5 that is centered at the origin.

The system of equaitons:

x2+ y2=5

y=2 x

Plug 2 x in for y:

x2+ (2 x )2=5

x2+4 x2=5

5 x2=5

x2=1

x=± 1

Plug both 1 and -1 back into y=2x to get 2 and -2. So the solutions are (1,2) and (-1,-2).

Like before, there could be 2 solutions, 1 solution, or no solutions.

__________________________

To show that a point is on a circle with given radius and center, first find the equation of the circle and then plug the point into the equation to see if it works.

Is (8,2) on the circle whose center is (5,-2) and whose radius is 5? Find the equation of the circle:

( x−5 )2+( y+2 )2=25

Plug (8,2) in:

(8−5 )2+(2+2 )2=25

32+42=25

25=25

It works, so (8,2) is on the circle.

__________________________

If asked to show any geometric property on a coordinate grid, you may need to use the distance or midpoint formulas:

M=¿

Unit 7 Probability=¿of favorable outcomes

total¿of outcomes¿

The symbol for the probability of A is P ( A ) .

A sample space is the set of all possible outcomes. A subset from this group is called an event. For example, the sample space when rolling a die is {1,2,3,4,5,6}. Rolling a 4 would be an event as would rolling an even number.

The intersection”and” of two events is what the events have in common. The symbol for the intersection of events A and B is A ∩ B . If two events are independent, then P ( A ∩B )=P( A)∙ P(B).

The union”or” of two events is all of the outcomes of either event. They symbol for the union of events A and B is A∪B . If there is no overlap, then P ( A∪B )=P ( A )+P (B)

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Analytic Geometry Condensed Study Guide

Conditional Probability, P ( A|B ) , is the probability of an event happening given that another event has occurred. The formula for conditional probability is

Find the probability that a randomly selected student would be a junior granted that the student owns a car.

Answer:

Notice that in many cases it is easier just to note that 18 students have a car, 6 of which are juniors, so you get the same

answer: 6

18=1

3.

The complement of an event is the set of events not included in the favored outcome. In symbols, the complement of A is denoted A'and is sometimes read “not A”. For example, the probability that a randomly

selected day starts with “T” is 27

. The complement of this event is all other days, and the probability of the complement is

1−27=5

7.

Use the Venn diagram:

If A represents those with a bicycle, and B those with a skateboard, then

A ∩ B contains Joe, Mike,Linda, and Rose

A∪Bcontains everyone except Amy, Gabe, and Abi

( A∪B)' contains Amy, Gabe and Abi.

Two separate events are independent if the outcome of one event does not affect the outcome of the other. If one outcome does affect the other, the events are called dependent.

If two events are independent, then

P ( A∧B )=P (A )∙ P(B)

What’s the probability that a randomly selected day would start with an S and a randomly selected month would start with an A?

Since one event doesn’t affect the outcome of the other, then

the answer is 27

∙ 212

= 121

Another fact about independent events is that if A and B are independent, then

P ( A|B )=P ¿A) and

P (B|A )=¿ P (B ) .

Mutually exclusive events have no overlap. If A and B are mutually exclusive, then P ( A orB )=P ( A )+P (B)

If two events are not mutually exclusive, then P ( A∨B )=P ( A )+P ( B )−P( A∧B)

Suppose you have the following classmates:

Boys: George,John,Thomas, James

Girls: Martha, Abigail,Dolly

One student is randomly selected. Find:

P(boy or Martha). There is no overlap, so P(boy or Martha)= 47+ 1

7=5

7

P(girl∨name has5 letters). There is overlap here(Dolly is both) so P(girl∨name has5 letters)

.= 47+ 1

7−1

7=4

7

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