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Grade 9 Academic Exam Review Algebra: Algebraic terms have a constant and a variable term: 3x 2 Adding and subtracting polynomials o you have to have like terms (both variable and exponent are the same) o Cannot add/subtract unlike terms i.e. 3 y 2 y 2 Multiplying polynomials o 3(4 x 3) 12 x 9 Dividing polynomials o Can always multiply/divide polynomials using the distributive property Cancel out common terms 24 x 4 16 x 3 3x 2 2 x 2 8x Exponent laws ALWAYS NEED THE SAME BASE o Multiplying – add the exponents x 4 x 2 x 6 o Dividing – subtract the exponents y 8 y 3 y 5 o Power of a Power – multiply the exponents ( x 3 ) 3 x 9 Solving polynomial equations o Isolate term – sings change when being moved over the equal sign!!! o Do BEDMAS backwards 2(x+2) – 6 = 4 2(x+2) = 4 + 6 x+2 =10/2 x=5–2 x=3 Investigating Relationships: Scatterplots – a graph that shows the relationship between 2 sets of data Dependent variable: variable that depends on the other Independent variable: variable that doesn’t change based on the other Lines of best fit: a line the passes through as many of the points possibly with the remaining points grouped equally above and below the line. Mean fit line: A more accurate line of best. This line has to pass through the mean point (the average point) Correlations: o Positive – slopes up to the right o Negative – slopes down to the right o No correlation – points all over the scatter plot o Strong correlation – points nearly create a straight line o Weak correlation – points are widely dispersed but still show a general trend Bias: The data is not an accurate representation of the truth. Can be caused by sample size, sampling procedure Models of Movement – shows the distance travelled over a period of time. You can analyze direction, distance, time, and speed. x y 1st differences: determines if the data is linear. 1st diff’s o To calculate the first differences from a table of values: 6 8 10-8 = 2 7 10 12-10 =2 8 12 14-10 = 2 9 14 Linear Relationships: 2nd 1st (x,y) x-axis (independent) 3rd 4th y-axis (dependent) rise y y 2 y1 run x x2 x1 Slope: m Tables of values, graphing from table of values 1. If necessary complete the table to values for unknown values. 2. Each set of x and y values are coordinates, plot these points 3. Connect the points Creating an equation in slope, y-int form (y = mx + b) 1. Find the slope 2. Sub the slope and a coordinate (x,y) into y=mx+b to solve for b 3. Sub slope(m) and y-int(b) into equation Graphing a line with slope and y-int, 1. Plot the y intercept 2. Move from the y-intercept the rise/run (slope) and make more points 3. Drawing a line connecting all points Parallel lines have equal slopes Perpendicular lines have negative reciprocal slopes. To find negative reciprocal flip and change sign. Horizontal lines are parallel to the x-axis, have the form y = # and a slope of 0. Vertical lines are parallel to the y-axis, have the form x = # and an undefined slope Writing equations with two variables o Combine two qualities represented by the equation ax by c o To rearrange this equation to slope, y intercept form you need to isolate y 2 x 5 y 500 i.e. 5 y 2 x 500 y 2 100 5 Standard form of a line Ax + By + C = 0 *no fractions and x must be positive* Finding x and y intercepts (and graphing using that method) To find x- int (set y=0) To find y-int (set x=0) 2 x 5 y 500 2 x 5 y 500 2 x 500 5 y 500 x 250 y 100 (250,0) (0,100) Finding the POI of two lines o Graph two set of lines (using table of values, slope y-int, or x and y intercepts) o Find the POI by looking at the graph o Sub the POI into BOTH equations to ensure that LS=RS o A line can intersect: once – different slopes never – same slope, different y-intercept infinite # of times – same slope and y-intercept (same line) Geometry: Angle theorems o OAT – opposite angles are equal o SAT – angles form 180° o CAT – angles form 90° o Patterns (need parallel lines and a transversal) Z, F, C Pythagorean theorem c a a2 b2 c2 *** c must always be the hypotenuse (the longest side) make sure you sub in and solve the equation correctly b Sum of interior angles of a polygon o 180n 360 or 180(n 2) where n= # of sides Exterior angles always add to 360, no matter the number of sides of the polygon Polygon properties o Bisectors – splits in half o Diagonals – cut diagonally through a shape o Midsegments – line joining the mid-points of adjacent sides o Midpoints – the middle point of a line How to calculate areas and perimeters of 2 dimensional shapes (You have the EQAO formula sheet) Area of regular polygons = How to calculate the volume and surface area of 3 dimensional shapes - prisms, cylinders, pyramids, cones, spheres (You have the EQAO formula sheet) Slant height = a 2 h 2 REMEMBER HOW TO USE PYTHAGOREAN THEOREM PROPERLY If dealing with 4 sides, the max area and min perimeter occurs with a square Optimizing a Cylinder Optimizing a Rectangular Prism Occurs when h=2r occurs when all sides are equal (a cube) Max. Volume Max. Volume r SA then SA 6l 2 6 Min SA r V then SA 6r 2 2 Pa , where P=perimeter, a = apothem 2 l 3 V then SA 6l 2 Min SA l SA then V l 3 6