MPM2D0 Final Exam
Grade 9 Review
Like Terms: Terms that can be grouped by adding or subtracting (x+y isn't xy.) (x + 3x=4x)
Exponent Rules
a^b x a^c = a^(b+c)
a^b / a^c = a^(b-c)
Y=MX+B (equation of a line
m=slope of line
b=y intercept
(x,y) can be anything on the x,y plane
BEDMAS (Order of Operations: Brackets, Exponents, Division/Multiplication, Addition/Subtraction)
Exam Tips
Dont stress out
Line up equal signs
Use "Therefore Statements" if it's a word problem
Don't forget Pencil, Eraser, Calculator, Ruler
Eat before
Sleep well
Read questions carefully
Show work
Think of questions fundamentally. If it doesn't make sense, it probably isn't right
Dont rush, take your time
Questions you don't understand, skip and come back to them
Analytic Geometry
Classifying Figures
Tips
To find the ALTITUDE:
1)Find the slope of the side opposite from the vertex
2)Find the slope of the altitude which is perpendicular to the side opposite from the vertex
3)Use the altitude’s slope and the point from the vertex to calculate the y-intercept of the altitude
4)Write the equation of the altitude
To Find PERPENDICULAR LINE (RIGHT BISECTOR):
1: Flip the slope (eg. 2->1/2)
2: Change the sign (eg. 1/2->-1/2)
therefore: (2->-1/2)
To Find The MEDIAN:
1)Find the midpoint of the opposite side
2)Find the slope of the line connecting the vertex to the midpoint of the opposite side
3)Calculatethe y-intercept of the line
4)Write the equation of the line.
Formulas
If Centre is at origin (0,0), then the equation is:
x^2+y^2=r^2
EXAMPLE: Centre at Origin, (x,y)=(2,2)
r^2=2^2+2^2
r^2=8
r=2.83
√((x₂ – x₁)² + (y₂ – y₁)²)
Length of a Line Segment/Distance between Two Points
EXAMPLE: Points (4,1) & -3,-5)
x₂ – x₁=4-(-3)=7
y₂ – y₁=1-(-5)=6
Length of Line=√(7² +6²)
=√83
=9.11
Midpoint of a line segment
EXAMPLE; Points (3,5) & (-7,-7)
midpoint=(-7+3)/2, (-7+5)/2
midpoint=(-2, -1)
Equation of a line: y=mx+b
Slope of a line
EXAMPLE: Points (3,5) & (-2,6)
slope=(6-5)/(-2-5)
slope=-1/7
Altitude: a line segment in a triangle from one vertex to the opposite side, creating a perpendicular line
Parallel: Two lines that never touch
Perpendicular: Two lines that form a 90 degree angle
The line segment from a point in a triangle to the opposite side
Midpoint: The point in the middle of a line segment
Line Segment: A part of a line between two points
Trigonometry
Also:
b^2=c^2-a^2
a^2=c^2-b^2
Similarity/Congruence
Solving Similar Triangles
If the traingles have same side lengths and angles, they are congruent:
AB=PQ, AC=PR, BC=QR
∠A=∠P, ∠B=∠Q, ∠C=∠R
If the triangles have same angles but different side lengths, they are similar:
AB/AC=EG/EF, AC/BC=EF/GF, AB/BC=EG/GF
∠A=∠E, ∠B=∠G, ∠C=∠F
SOH-CAH-TOA
Inverse Signs:
θ=sin-1(o/h)
θ=cos-1(a/h)
θ=tan-1(o/a)
Cosine Law
When solving for cosine(x), use algebra
Use when you have:
SSS
SAS
EXAMPLE: Solving for angle A:
4.8² =8.5²+6.3² -2(8.5)(6.3)(cosA)
cosA=(-8.5²-6.3²+4.8²)/(-2(8.5)(6.3))
cosA=127/153
A =33.9 degrees
*same method for other angles
Sine Law
Use when you have:
AAS
SSA
ASA
EXAMPLE: solving for side b:
a/sin35=42/sin45
a = 42sin35/sin45
a = 34.07
*same method with side c
*SATT for angle C
Quadratic Functions/Polynomials
Transforming Quadratics
Step Pattern
If a=2, the jumps are 2, 6, 10
A parabola with a scale factor of 1 has a "step pattern" of 1, 3, 5
"Step pattern" refering to the jump in y value for each one x value
Reflections and Stretches
Vertical Stretch/Compression:
if a>0, the parabola is stretched
if 0
Reflections:
if a in ax^2 is negative, the parabola opens up and reflects over x-axis (vice versa)
Translations
Horizontal Translations:
y=(x+h)^2: moves the vertex left (h>0)
y=(x-h)^2: moves the vertex right (h<0)
Vertical Translations:
y=x^2 + k: moves the vertex up (k>0)
y=x^2 - k: moves the vertex down (k<0)
Distributive Property (Binomial Multiplication)
(a+b)(c+d)=ac+ad+bc+bd
Quadratic Formula
THE DISCRIMINANT:
if √(b²-4ac)=0, 1 real solution
if √(b²-4ac)>0, 2 real solutions
if √(b²-4ac)<0, no real solutions
e
x=(-b±√(b²-4ac))/(2a)
Shows us the Roots of the Function
Forms of Quadratics, How to Factor
Special Cases (Completing the Square)
The expanded expression is called a DIFFERENCE OF SQUARES. Both terms are perfect squares and the terms are separated by a minus sign
To factor a difference of squares:
1)Two brackets that are the same but one with a plus (+) and one with a minus (-)
2)Square roots of perfect squares go in the brackets
In a perfect square trinomial, the first and last terms are perfect squares.
The middle term is twice the product of the square roots of the first and last terms
Vertex Form: y=a(x-h)²+k
a = +, opens up. a = -, opens down
Vertex (h,k)=(-b/2a, substitute h for k)
k=f(h)=y
h=-b/2a, which is the Axis Of Symmetry
Vertex (h,k)
Factored Form: (x-r)(x-s)
To get to Vertex Form: Get to standard form and then (refer to "to get from Standard Form to Vertex Form)
To get to Standard Form: Use Distributive Property, collect like-terms
Use distributive property to get to standard form
Derrives from standard form (ax²+bx+c)
Standard Form: (ax²+bx+c)
To get from Standard Form to Vertex Form:
4: Write in vertex form (in the bracket is a perfect square trinomial.
2(x²+4x+4)-5
=2(x+2)²-5
3: Simplify the Equation by removing the negative in the end of the bracket
2(x²+4x+4)+3-8
=2(x²+4x+4)-5
*dont forget to multiply the negative by a (2)
2: Complete the square by:
Take half of the coefficient of x, square it, and add it inside the parentheses. Subtract the same value to balance the equation
ex: x²+4x, half of 4 is 2, and 2²=4, so
2(x²+4x+4-4)+3
1: if a isn't 1, factor out a so that the other steps work (ex, 2x²+8x+3=2(x²+4x)+3
To get from Standard Form to Factored Form:
1: ex. x²+5x+6
b=5, c=6
numbers that add to 5 and multiply to 6:
3 and 2
Therefore x²+5x+6=(x+3)(x+2)
Find two numbers that add to "b" and multiply to "c"
Factoring Complex Trinomials
Parabolas
Properties of a Parabola
A parabola is a symmetrical graph, a 'U' shape. Its key components include:
Zeros: The points where the parabola intersects the x-axis (also called roots or x-intercepts).
Line of Symmetry: A vertical line that divides the parabola into two equal halves.
Minimum or Maximum: The lowest (minimum) or highest (maximum) point on the parabola.
Vertex: The point where the parabola meets the line of symmetry, which is also the minimum or maximum point.
Optimal Value: The y-coordinate of the vertex (either the minimum or maximum value).
Y-Intercept: The point where the parabola crosses the y-axis.
Roots
if (x,y) = (a,0) & (b,0), then y=(x-a)(x-b)
TO GET THE ROOTS:
1.Get rid of any fractions by multiplying each term by the LCD.
2.Write equation as ax2+ x + c = 0
3.Fully factor.
4.Set each factor equal to 0 and solve using the zero product propert
ax²+bx+c
"Golden Formula": Everything derives from this
Terms
1st/2nd Differences
If 1st relations jump by the same, it is linear
If 2nd relations jump by the same, it is expontial
Degree of Term: The highest exponent in a term
Degree of polynomial: the exponent after the variable in a polynomial (eg. x^2y^3. degree=5 (2+3)
Term: A set of numbers that multiply with each other (eg. 5x^2)
Coefficient: A number in which the variable multiplies with eg. coefficient = 5: (5x)
Variable: A number used to represent an unknown quantity that can vary
Linear Systems
Application
1)Assign variables to each of the unknowns
2)Write 2 equations showing the relationships between the variables. Each equation should include both variables.
3)Solve the system of equations using any method (graphing,substitution,elimination)
4)Check your solution
5)Clearly communicate your final answer
Graphing
Only works when isolated for y for the y=mx+b format
m=rise/run
b=y intercept
Substitution
Elimination
can only eliminate when the coefficient infront of both x's or both y's are equal and opposite signs
A Linear Systemis composed of two or more linear relations.
To solve a Linear System you must determine which values of x and y are common to all lines. This occurs at the point of intersection.
