# The Battle of the Atlantic Essay

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Math Review Sheet - Grade 10 - Ontario

Chapter 1 – INTRODUCTION TO FUNCTIONS
-The independent variable goes on the x-axis, and the dependent variable on the y-axis eg. Time – x-axis and Distance – y-axis
-A function is a rule that gives a single output number for every valid input number
-The graph of y=mx+b is a straight line with slope m and y-intercept b. -slope is Rise/Run
-Constant Slope Property -The slope of all segments of a line are equal.
-Properties of a Linear Function -The equation y=mx+b represents a linear function. -m is the slope and b is the y-intercept. -The points on the graph of a linear function lie on a straight line.
-For consecutive values of x, the differences in the values of y are constant. -To find the y-intercept, substitute 0 for x then solve for y. -To find the x-intercept, substitute 0 for y then solve for x.

Chapter 2 – COORDINATE GEOMETRY -Distance Formula and Length of a Line Segment
-The distance between any two points A(x_1,y_1 ) and B(x_2,y_2 ) is given by this formula. The distance is also the length of line segment AB. AB=√(〖(x_1-x_2)〗^2+〖(y_1-y_2)〗^2 )
-Midpoint Formula
-When M is the midpoint of a line segment with endpoints A(x_1,y_1 ) and B(x_2,y_2 ), the coordinates of M are ( 〖 [x〗_1+x_2]/2 , [y_1+y_2]/2 ) -Equation of a Line with Slope m passing through (p,q) y=m(x-p)+q -Properties of a Parallelogram -The opposite sides of a parallelogram are equal. AB=DC AD=BC -The diagonals of a parallelogram bisect each other. -E is the midpoint of AC and BD -Equation of a Circle -The equation of a circle with a center at (0,0)/the origin and the radius is x^2+y^2=r^2 -Lines -Parallel Lines -Slopes are equal -Perpendicular Lines -Slopes are negative reciprocals -Constant Slope Property -The slope of all segments of a line are equal

Chapter 3 – LINEAR SYSTEMS
-To graph an equation, express it in the form y=mx+b. Then we can identify the slope m and the y-intercept b. -Properties of Linear Systems
1.Multiply both sides of either equation of a linear system by a constant does not change the solution.
2.Adding or subtracting the equations of a linear system does not change the solution
-Solving Linear Systems by Algebraic Modeling eg. A principle of \$700 is investedat 7% per annum and the rest at 10% per annum. In one year, the total interest earned was \$55.80. How much was invested at each rate?

Let a represent the money invested at 7%, in dollars Let p represent the money invested at 10%, in dollars

(1) a+p=700--→ p=-a+700 (2) 0.07a+0.1=56.80 0.07a+0.1(-a+700)=56.8 p=-a+700 0.07a-0.1a+70=56.8 p=(-1)(440)+700 -0.03a=56.8-70 p=-440+700 -0.03a= -13.2 p=260 -------- ------ -0.03 -0.03 a= 440
So therefore \$440 was invested at 7% per annum and \$260 was invented at 10% per annum.
-Solving Linear Systems by Elimination eg. Solve for y and x values. (1) 2x+5y=16 (2) x-y=1 Multiply (2) so that at least one value x or y are equal, and then we can subtract the equations to eliminate x values.

So, (1) 2x+5y=16 Then, x-y=1 (2) - 2x-2y=02 x-(2)=1 7y=14 x=1+2 y=2 x=3 -Solving Problems Using Graphing
-To graph an equation, express it in the form of y=mx+b. Then we can identify the slope m and the y-intercept b.
-This means that that we would have to convert any other equation so that it would be y=mx+b.

Chapter 4 - INTRODUCTION TO QUADRATIC FUNCTIONS -The Step Property -it is: a, 3a, 5a, 7a
-The graph of y=x2 -The graph is a parabola opens upward. -The origin is the lowest point of the parabola. -The graph is symmetrical about the y-axis.