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1. 11th and 10th grade Math TAKS Review

2. General Calculator Stuff • The memory will be cleared on the calculator before the TAKS test. You will want to go to 2ndTablset and change the independent variable to ask. Do NOT change the dependent variable, leave it on Auto. • Anytime you are graphing and you need a standard size graph, go to zoomstandard for a grid which will be -10 to 10 for both the x- and y-axis. • If you use SOH-CAH-TOA then, then you will also need to change the mode to degrees

3. More General Stuff Adding units which are not like What is the sum of 7ft 3in and 26in 7 feet 3 inches 26 inches 7 feet 29 inches there are two feet in 29 inches so add 2 feet to the 7 feet and lose 24 inches from 29 inches 9 feet 5 inches

4. Objective 1 Start of Objective 1 • Objective 2 Start of Objective 2 • Objective 3 Start of Objective 3 • Objective 4 Start of Objective 4 • Objective 5 Start of Objective 5 • Objective 6 Start of Objective 6 • Objective 7 Start of Objective 7 • Objective 8 Start of Objective 8 • Objective 9 Start of Objective 9 • Objective 10 Start of Objective 10

5. Start of Objective 1

6. Objective 1 • X values also known as: • Domain • Independent Variable • Y values also known as: • Range • Dependent Variable Which is the independent and the dependent variable between number of shirts bought and the total cost? Which makes more sense: The # of shirts I buy depends on the total cost The total cost depends on the # of shirts I buy Ans: the total cost depends on the # of shirts I buy, so The dependent variable is the total cost The independent variable is the # of shirts

7. Objective 1 cont X-intercept: Find X when Y = 0, the answer will appear as (X,0) Y-intercept: Find Y when X = 0, the answer will appear as (0,Y) Graphing Lines: Slope intercept formula: y = mx + b m is the slope and b is the y-intercept Slope: Rise over run Positive slope is up and to the right Negative slope is down and to the right Another word for slope is rate of change Positive slope Negative slope

8. Objective 1 cont Given the equation 4x – 3y = 12, find the x-intercept, y-intercept, slope, and graph the line. X-intercept Find X when Y = 0 4X – 3(0) = 12 4X = 12 X = 3 X-Intercept is (3,0) Find the slope Solve the equation for y Y = MX + B M is the slope & B is the Y-int 4X – 3Y = 12 – 3Y = –4X + 12 –3 –3 Y = 4/3X – 4 Slope = 4/3 and Y-int = (0,–4) Y-intercept Find Y when X = 0 4(0) – 3Y = 12 –3Y = 12 Y = –4 Y-Intercept is (0, –4)

9. Graph each of the following equations 4x – 3y > 12 4x – 3y < 12 Objective 1 cont. 0 – 0 < 12 true 0 – 0 > 12 false Should the line be solid or dotted? After graphing the line, shade the TRUE side true 4x – 3y > 12 4x – 3y < 12 0 – 0 > 12 false 0 – 0 <12 true true

10. Objective 1 cont For a relation to be a function: the x-values must all be different Is the following relation a function? {(2,3), (4,5), (5,3)} Is the following relation a function? {(2,3),(4,5), (2,8)} All the x-values are different so it is a function The x-values are NOT different so it is NOT a function

11. Objective 1 cont Quadratics of the form Y = AX2 + C If A is positive the parabola opens __________________ If A is negative the parabola opens __________________ Ignoring the +/- sign for the value of A, The larger the value of A, the _________________ your graph The smaller the value of A, the _________________ your graph The value of C at the end does what to the graph? upward downward narrower wider Shifts it up/down

12. Objective 1 cont • What is the difference between the graphs of y = ½x2 and y = 5/3x2 • Describe the vertical shift from y = 2x2 – 7 to the graph of y = 2x2 + 5 The graph ½x2 is wider than the graph of 5/3x2 The graph 5/3x2 is narrower than the graph of ½x2 The 2nd graph is shifted 12 units above the 1st graph

13. Start of Objective 2

14. Objective 2Parent Functions • Linear functions • Y = MX + B Graph is a line • Quadratic functions • Y = AX2 + BX + C Graph is a parabola • Cubic functions • Y = AX3 + BX2 + CX + D Graph • Absolute functions • Y = A│X - H│+ K Graph is V-shaped • Square root functions • Y = A √X – H + K Graph • Exponential functions • Y = A(B)X Graph

15. Objective 2 cont Expand (2X – 3Y)2 Many student will choose the following incorrect answer 4X2 + 9Y2 You need to write the binomial twice and then FOIL (2X – 3Y)(2X – 3Y) = 4X2 – 6XY – 6XY + 9Y2 Ans: 4X2 – 12XY + 9Y2

16. Objective 2 cont Find the equation of the line given the x-intercept of 5 and y-intercept of 3 X-intercept means find x when y = 0 so the point is (5,0) Y-intercept means find y when x = 0 so the point is (0,3) Using the slope formula on the TAKS card, the slope is m = y2 – y1 = 3 – 0 = 3 = -3/5 x2 – x1 0 – 5 -5 Using y = mx + b the answer is y = -3/5x + 3 Check it with the graph on the calculator Show the students: stat edit… stat calc linreg on the calculator X Y 5 0 0 3

17. Start of Objective 3

18. Objective 3 • X-intercept: Find x when y = 0 so the answer should appear as (x,0) • Y-intercept: Find y when x = 0 so the answer should appear as (0,y) • Graphing lines in the form of y = mx + b m is the slope of the line and b is the y-intercept

19. Objective 3 cont • Slope • Rise over Run • Up/down then to the right (never to the left) • If the slope is positive, then up and right • If the slope is negative, then down and right • The words “rate of change” is another way of saying slope. They may use words like: “cost per minute,”, or “miles per gallon,” or “cost per ounce” for the slope. N Negative slope Positive slope

20. Objective 3 cont • Parallel lines • Parallel lines will never intersect • Parallel lines have the SAME slope • Perpendicular lines • Perpendicular lines intersect at a right angle • The calculator won’t show a right angle because the window isn’t squared • Perpendicular lines have OPPOSITE and RECIPROCAL slopes

21. Objective 3 cont • A direct variation is a line which passes through the origin (0,0) and has the following equation y = kx • Ex: Y is directly proportional to X and passes through the point (10,-5), Find the equation for the line. y = kx -5 = k(10) -1/2 = k Answer: y = -1/2 x

22. Objective 3 cont • Ex: Y is directly proportional to X and passes through the point (-2,10). Find the value of X when Y = -30. y = kx 10 = k(-2) -5 = k Equation: y = -5x Find x when y = -30 y = -5x -30 = -5x 6 = x Answer: X = 6

23. Start of Objective 4

24. Objective 4 • Graphing Inequalities: Solid/dotted & Shading Y < MX + B & Y > MX + B Dotted lines and shade the true side Y < MX + B & Y > MX + B Solid lines and shade the true side Graph 3x – 2y < 12 Test (0,0) 0 – 0 < 12 true – 2y < –3x + 12 –2 –2 y > 3/2x – 6 slope = 3/2 and y-int (0,-6)

25. Objective 4 cont • Solving a system of equations: 2x + 3y = 1 4x – 5y = 13 Sub 2 for x 5 2(2) + 3y = 1 4 + 3y = 1 3y = -3 y = -1 5 3 3 10x + 15y = 5 12x – 15y = 39 22x = 44 x = 2 Answer: (2,-1)

26. Objective 4 cont • Solving a system of equations: 2x + 3y = 1 3y = -2x + 1 3 3 y = -2/3x + 1/3 4x + 6y = 2 6y = -4x + 2 6 6 y = -2/3x + 1/3 Notice these two lines coincide with one another 2x + 3y = 1 4x + 6y = 2 -2 -2 1 1 -4x – 6y = -2 4x + 6y = 2 0 = 0 Always true so Infinite solutions

27. Objective 4 cont • Solving a system of equations: 2x + 3y = 1 3y = -2x + 1 3 3 y = -2/3x + 1/3 4x + 6y = -12 6y = -4x – 12 6 6 y = -2/3x – 2 Notice these two lines are parallel and will never intersect 2x + 3y = 1 4x + 6y = -12 -2 -2 1 1 -4x – 6y = -2 4x + 6y = -12 0 = -14 Never true so No solutions

28. Objective 4 cont • Beware of the following words: • Fewer than • Less than • Subtracted from For example: “five subtract some number” = 5 – n but, “five less than some number” = n – 5

29. Start of Objective 5

30. Objective 5 Parabola’s: y = Ax2 + Bx + C A is positive opens upward A is negative opens downward The larger │A│ the narrower the graph The smaller │A│ the wider the graph Congruent parabolas will have the same size and shape Congruent parabolas will have equal or opposite A values The constant at the end of the equation, shifts the parabola up and down A parabola that opens upward has a minimum A parabola that opens downward has a maximum Roots are also known as the x-intercepts The Ax2 term is known as the quadratic term The Bx term is known as the linear term The C is known as the constant term

31. Objective 5 cont Compare and Contrast: Y = 2X2 + 3 and Y = 2X2 – 5 Both equations have a positive 2 for the coefficient for X2 so both graphs open upwards, both have the same steepness, therefore the two graphs are congruent The constants at the end of the equations are different which means the 2nd graph is shifted down 8 units from the 1st Compare and Contrast: Y = 3X2 and Y = – 3X2 Both equations have opposite signs for the coefficient for X2 so the graphs have the same steepness which makes them congruent, and the 1st parabola opens upward while the 2nd opens downward

32. Objective 5 cont Compare and Contrast: Y = 2/3X2 and Y = 3X2 Both equations have positive coefficients for X2 so both of the parabolas open upward The coefficients of X2 are different which means the 1st graph is wider and the 2nd graph is narrower than the parent function of y =x2

33. Objective 5 cont Simplify: Reduce by 8

34. Start of Objective 6

35. Objective 6 Tessellation – no gaps, no holes – covers the entire plane – all quadrilaterals make pure tessellations Tangent lines are perpendicular to the radius at the point of tangency Collinear – means points that lie on the same line Coplanar – means points that lie on the same plane Pythagorean Theorem A2 + B2 = C2 on TAKS card

36. Objective 6 • Dilation – make the picture larger or smaller • Reflection – mirror (flipped over a line) • Translation – slide the picture • Rotation – turn the picture

37. Objective 6 cont 30-60-90 Rule and 45-45-90 Rule Find the missing lengths 30˚: 60˚: 90˚ X:X√3:2x 45˚: 45˚: 90˚ X:X:X 2 (30-60-90) c 2 4 1 3 30˚ b

38. Objective 6 cont 30-60-90 Rule and 45-45-90 Rule Find the missing lengths 30˚: 60˚: 90˚ X:X√3:2x 45˚: 45˚: 90˚ X:X:X√2 (30-60-90) c a 2 1 3 30˚ 8

39. Objective 6 cont 30-60-90 Rule and 45-45-90 Rule Find the missing lengths 30˚: 60˚: 90˚ X:X√3:2x 45˚: 45˚: 90˚ X:X:X√2 (30-60-90) 10 2 a 1 3 30˚ b

40. Objective 6 cont 30-60-90 Rule and 45-45-90 Rule Find the missing lengths 30˚: 60˚: 90˚ X:X√3:2x 45˚: 45˚: 90˚ X:X:X√2 (45-45-90) c 4 2 1 1 45˚ b

41. Objective 6 cont 30-60-90 Rule and 45-45-90 Rule Find the missing lengths 30˚: 60˚: 90˚ X:X√3:2x 45˚: 45˚: 90˚ X:X:X√2 (45-45-90) 8 1 2 a 1 45˚ b

42. Start of Objective 7

43. Objective 7 Isosceles: at least two sides of equal length Label each of an isosceles triangle sides and angles Label each of an isosceles trapezoid sides base Vertex angle leg leg leg leg Base angles base base Isosceles Trapezoid Isosceles Triangle

44. Objective 7 cont Midpoint and Distance Formula: both are on the TAKS card Find the distance and the midpoint between (–2,3) and (4,–5) (x1,y1) (x2,y2)

45. Objective 7 cont Given point M is the midpoint of segment AB. If point A is located at (-8,11) and point M is located at (-2,3) find the other endpoint. –4 = –8 + x 4 = x 6 = 11 + y –5 = y Answer: (4,–5)

46. Objective 7 cont same Parallel lines have the ___________ slopes and never intersect Perpendicular lines have the ____________ & ____________ slopes and form four right angles Complementary angles have a sum of ____________ Supplementary angles have a sum of _____________ opposite reciprocals 90˚ 180˚

47. Objective 7 cont What is the name of each shape? Rectangular Prism Pyramid Better name: Pentagonal Pyramid The 1st word in the name tells you the shape of the base Pyramids extend to a point like the Egyptian Pyramids Cylinder What are: Vertices, Edges, and Faces Vertices: the points Edges: the line segments Faces: the flat sides Cone

48. Start of Objective 8

49. Objective 8 • To be able to prove that two shapes are similar: • Show that all pairs of corresponding sides are proportional • Show that all pairs of corresponding angles are equal • To say that two shapes are congruent: • All corresponding sides are equal in length • All corresponding angles have equal measures • Scale Factor • Scale factor: a:b • Perimeter ratio: a:b • Area ratio: a2:b2 • Volume ratio: a3:b3

50. Objective 8 cont Given isosceles triangle ABC is similar to triangle DEF Find the area of triangle DEF B E To find the height of ∆DEF use Pyt. Thm. 4.82 + h2 = 62 h2 = 62 – 4.82 h2 = 12.96 take the square root h = 3.6 Area of ∆DEF = ½ bh = ½ (9.6)(3.6) 6 10 4.8 F D 9.6 A C 16 Scale factor: 10:6 10(DF) = 6(16) DF = 9.6 Area of ∆DEF = 17.28 un2