MPM2D0 Final Exam
Trigonometry
Congruent and Similar Triangles
Congruent means all sides and angles are equal / the same
Similar means all angles are the same but sides are different or all sides
One way you can confirm if a triangle is similar or not is by dividing the larger triangles sides by the smaller triangles sides. If all equal the same number then the triangle is similar
Something to keep in mind is that you don't need all lengths or all angles, if you have two lengths you can still do the dividing trick and if you have two angles you can fill it out by doing: 180-x-y = θ
Sine, Cosine and Tangent Ratios
SohCahToa
Sine = Opposite
Hypotenuse
Cosine = Adjacent
Hypotenuse
Tangent = Opposite
Adjacent
Example: Cos25 = 6/x
Only used for right triangles with an angle other than the right angle.
Pythagorean Theorem
Only used in right triangles that show 2 side lengths
Hypotenuse: A² + B² = C²
Side A: C² - B² = A²
Side B: C² - A² = B²
Sine and Cosine Law
Used for any triangle except right triangles
Sine Law is used whenever you have one angle and two sides with one of them being opposite to the angle
This triangle would be solved by:
Sin94/10.5 = Sin40/b (the 'x' can be solved after you find the opposite side to Sin40)
Cosine Law is used whenever you have 2 sides and no opposite angles or whenever you have all 3 sides and no angles.
This triangle would be solved by:
a² = 86² + 65² - 2(86)(65) cos19
This triangle would be solved by:
cosA =46² + 117² - 82² / 2(46)(117)
Quadratic Functions / Polynomials
Polynomials
Degree of a polynomial
The degrees of a polynomial are determined by the monomial with the biggest exponent for example: x³ + 4² would be 3rd degree another example is x²y³ - 4² would be 5th degree
Terms in polynomials
There are 3 simple polynomials with names, these are: Monomial, Binomial and Trinomial. All other polynomials don't have names but can be described by how many terms they have. You can decipher how many terms a polynomial has by the amount of expressions it has, 3x-4³+7y would be a trinomial because it has three expressions (3x, 4³ and 7y)
Quadratics
Quadratic relation standard form
y = ax² + bx + c
First and second differences
First differences
When having to determine first and second differences you will get an equation like y = x² and then you will be given some values that are usually y = -3, -2, -1, 0 1, 2, 3
To determine the first difference you must find the values of x for every y value. Once you have found these go down the line and find the difference of each x value, in this case it would be, x = 9, 4, 1, 0, 1, 4, 9
With these numbers you find the difference of each number left to right, in this case it would be, -5, -3, -1, +1, +3, +5. As you can see there is a pattern in this sequence of numbers, this helps confirm that your values are correct
Second differences
Second differences are deciphered the same way that the first differences are except you take out the first step
To determine the second differences you must go left to right but instead of using the first values use the first differences, in this case our first differences are: -5, -3, -1, +1, +3, +5.
With these numbers you will find another pattern except this time all the differences are the same, in this case all of the differences are +2. The second difference will always be one number if you have done your chart correctly.
Parabolas
Terminology
A parabola is a type of graph that is symmetrical and almost always shaped like a 'U' , all of the parts of a parabola are: the Zeros, where the parabola passes over the x- axis (this is also known as a root or an x-intercept), the Line of Symmetry, divides the parabola into to equal halves (straight vertical line), the Minimum or Maximum is the highest (maximum) or lowest (minimum) point on a parabola, the Vertex where the parabola and the line of symmetry meet (this will always be at the minimum or maximum), the optimal value is the y value of the minimum or maximum, and finally the y-intercept is the point where a part of the parabola crosses over the y-axis
How they are graphed
Parabolas are graphed using the format that first and second differences use, the first and second differences can be used to determine if your graph is correct or not.
Factored form
y = a (x - r) (x - s)
The sign of a determines rather the parabola opens up or down, if the a value is a negative it opens downwards (maximum value) but if it is a positive it opens upwards (minimum value)
The 's' and the 'r' in this equation tell us what the zeroes of the parabola are, whenever you have a value in one of these spots it is the opposite of that number, for example 3 would become -3
(x - 4)² = (x - 4) (x - 4)
To find y-int substitute x for 0
Binomial Multiplication
Binomial Multiplication is multiplying multiple terms at a time
A common abbreviation used to remember the order of how to multiply these is "FOIL" this stands for: First, Outer, Inner, Last.
(9x - 4) (x-6)
9x² - 54x - 4x + 24
This kind of multiplication can be applied to parabolas
Factoring by Grouping
Factoring by grouping is the opposite of binomial multiplication
There are 2 requirements to see if we can factor by grouping or not
The first factor is that the polynomials have 4 terms
The second factor is that if you multiply the first term by the last and multiply the 2 middle ones together you should get the same product
This is true because the right bracket are the highest common factors, and the right bracket are the result of the right bracket
Factoring Trinomials
Simple Trinomials
x² + bx + c
Find two numbers that have a product of c and a sum of b. b= (_+_) c =( _ x _)
The answer to this would be 3 and 2 because 3x2 = 6 and 3+2 = 5
This would become (x + 3) (x + 2) You can verify your answer is correct by using foil to get back to the original equation
Simple trinomials can also be in the forms: x² y² + bxy + c or
x² + bxy + cy²
Complex Trinomials
ax² + bx + c as long as 'a' doesn't equal 1
Find two numbers with the product of ac and the sum of b
The two numbers here are -12 and -2, this is because -12 -2 = -14 (b) and -12 x -2 = 24 (ac)
Complex trinomials can also be in the forms: ax² y² + bxy + c or
ax² +bxy + cy²
Factoring by Special Cases
Perfect Squares
A perfect square is when you have a variable that you can get the square root of (any variable with an exponent) and another number that has to be a negative that you can also get the square root of.
An example could be x² - 16 or 4x² - 4
These would become (x - 4) (x + 4)
Once you have a perfect square you make it into a positive and a negative bracket then factor to the fullest
Perfect Square Trinomials
Perfect square trinomials are slightly different then normal perfect squares, there will be 3 terms, the first and last terms have to perfect squares but the middle term has to be twice the product of the square roots of the first and last term
An example of this could be 9x² + 42x + 49
This is correct because 2(3x)(7) = 42x
For this perfect square you must put your two square rooted outside numbers/variables and put them into a bracket with a ² sign. Make sure to make the sign on the inside based on the sign of the second number/variable
Solving Quadratic Equations
ax² + x + c = 0
Remove any fractions by finding the LCD, make sure the equation is in the form above, fully factor, then set each factor to =0 and solve using zero factor property.
ax² + bx + c = 0
To solve you substitute the values (very simple once you know the formula)
Real solutions mean zeroes (make sure to not square root it)
Transforming Quadratics
Translations
y = x² + k means there is a positive translation (vertex moves up)
y = x² - k means there is a negative translation (vertex moves down)
y = (x + h)² means there is a horizontal translation right (vertex moves to the right)
y = (x - h)² means there is a horizontal translation left (vertex moves to the left)
Reflections and Stretches
If the 'a' in y = ax² is a negative '-a' that means the parabola is reflected over the x-axis
If the 'a' in y = ax² is a value above 0 that means a parabola has a vertical stretch but if 'a' is less than 0 that means the parabola has a vertical compression
Red is stretched and blue is compressed
Step Pattern
1, 3, 5
Whatever your 'a' value is it what you multiply the step pattern by, your step pattern helps determine how many units you go up for every one unit moved horizontally. Example 2 become 2, 6, 10
Completing the Square
Rewrite the equation from highest to lowest degree. Common factor a to make the coefficient of x² = 1. Then use (b/2)² to find the trinomial value that goes inside the bracket, finally use this to find the vertex
For the final step you would switch the sign on the 2 to make it (2,5)
Linear Systems
Solving Linear Systems by Graphing
Used whenever you have 2 line equations
The slope in this case the first slope would be 2/3 and the first point you can plot is (0,-1). The second slope is -1/0 and the second point is (0,4). You would then graph these lines and find the point of intersection.
The slope is the coefficient of 'x' or 'M'
Determine where to put your first point on the line by looking at the 'b' value
Solving Linear Systems by Substitution
Used whenever you have 2 line equations
In this case you would use the bottom equation and substitute it for 'y' in the top equation, making it, 2x + 5 ( 3x - 3 ) = 19
From there you would have the 'x' value, with that you can then find 'y' with the same method
In this case you would isolate one of the variables. If you did this in the first equation it would become x = -3.5y + 17
You would then substitute that into the other equation to find your answer
Solving Linear Systems by Elimination
Used whenever you have 2 line equations but two coefficients with the same variable must be multiples of each other. So 2x and 4x would work but 2y and 4x would not
In this case you would add the top to the bottom because the signs are different on our variables we are trying to cancel out. If the signs were the same we would subtract instead
This would result in us having one equation, this equation being: 8x = 28. From there divide each side by your coefficient to find your final answer
Analytic Geometry
Finding a Line
Slope
This is one of the most important parts of this unit as you need this to find almost everything
Distance
The length of a line
Points: (4 , 7) and (12 , 19)
This would become:
√ (12-4)² + (19 -7)²
√ 8 + 12
√20
4.47cm
Rise and Run can be found by doing:
Run = x2 - x1
Rise = y2 -y1
Midpoint
The middle of a line
Points: (4 , 7) and (12 , 19)
This would become:
4 + 12 / 2 , 7 + 19 /2
16/2 , 26/2
8 , 13
Median
The line segment that connects a vertex to the midpoint of the opposite side
Points: a(16,17) b(12,8) c(18,8)
Midpoint of BC = 15 , 8
z
17 = 1/8(16) + b
y-int = 15
Final equation: y = 1/8x + 15
1. Find the midpoint of the opposite side
2. Find the slope of the vertex to the midpoint
3. Calculate the y-intercept
Put them together into y = mx + b to get the equation of the median
Right Bisector
The line segment that intersects through the mid point of another line segment and makes a 90° angle
Points: z(16,17) x(12,8) y(18,8)
Midpoint of xy = 15 , 8
Slope of xy = 6
Perpendicular Slope = -1/6
y-int of right bisector = 10.5
y = -1/6x + 10.5
1. Find the midpoint of the original line
2. Find the slope of the original line
3. Find the perpendicular slope (opposite of normal slope)
4. Calculate the y-intercept of the right bisector
5. Put them all into y = mx + b to get your right bisector equation
Altitude
The altitude is the line from a vertex that is perpendicular to the opposite side
Points: a(16,17) b(12,8) c(18,8)
Slope of opposite side = 6
Perpendicular slope = -1/6
y-int = 19.6
y = -1/6x + 19.6
1. Find the slope of the opposite side from the vertex
2. Find the perpendicular slope (that is the slope of the altitude)
3. Find the y-intercept using the point from the vertex
4. Put them together into y = mx + b to get the equation of the altitude
Finding a Circle
Radius
The radius of a circle is the line that connects from one part of the circle to the very middle
The radius is important because it is used in both circle equations
If points (x , y) are on the circle that means: x² + y² = r²
If points (x , y) are outside the circle that means: x² + y² > r²
If points (x , y) are inside the circle that means: x² + y² < r²
Is the point (−5,9) inside, outside or on the circle of x² + y² = 100
25 + 81 = 100
106 > 100
Circle
Equation of a circle with center at origin:
x² + y² = r²
If the origin is (0,0) and the radius is 6 the equation would be x² + y² = 36
Equation of any circle: (x - h)² + (y - k)² = r²
The h and k are the center of the radius, h is always the first number and k is the always second one
If the origin is at (7,8) and the radius is 11 the equation would be
(x - 7)² + (y - 8)² = 121 ²
Diameter
The diameter is the distance of one point on a circle to the opposite point (must pass through midpoint
If the diameter is 6 the radius would be 3
If you have the diameter and need the radius you can use diameter/2 to find the radius