Summer Review. For Students Entering. Algebra 2 & Analysis
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1 Lawrence High School Math Department Summer Review For Students Entering Algebra 2 & Analysis
2 Fraction Rules: Operation Explanation Example Multiply Fractions Multiply both numerators and denominators straight across. You don t need the same denominator Divide Fractions Invert the second fraction (turn upside-down), then multiply straight across 4 5 x 2 3 = 4 x 2 5 x 3 = 8 15 step = 4 5 x 3 2 Step 2. 4 x 3 5 x 2 = = 6 5 Adding Fractions Subtract Fractions Step 1: Denominators must be the same, if not, find the common denominator. Step 2: add the numerator, put the answer over the denominator Step 3: Simplify Same as addition, however, you are subtracting the two numerators. step 1: = 4(3) 5(3) + 2(5) 3(5) Step 2: = step 1: = 4(3) 5(3) 2(5) 3(5) Step 2: = 2 15 Direction: Simplify each fraction. Multiply Fractions x x x c 6 Divide Fractions c 6 Add Fractions c 6
3 Subtract Fractions c 10 Order of Operations: The order of operations refers to what operation should be performed in which order. The notation tells you which operation to do first as it appears in the equation. To remember the conventional order of operations, you can think of PEMDAS (You might remember this as "Please excuse my dear Aunt Sally.") 1. ( ) Parentheses or Grouping Exponents 3. X and Multiplication and Division (perform the operation from left to right in the equation) 4. + and - Addition and Subtraction (perform the operation from left to right in the equation) Examples: x 7 =? 2. (1 + 3) x (8-3) =? x 7 = = 44 (1 + 3) x (8-3) = 4 x 5 = 20 Evaluate each expression: Show all work to receive credit 1. 8 x x (8 +13) x ( ) x (9-4) 2 7. (9-7) - (16 8) ((13-6) )
4 Combining Like Terms What Does Combining Like Terms Do? Algebraic expressions can be simplified by Combining like terms, consider the algebraic expression below: 15x x = 18x + 4 As you can see, 15x and 3x are like terms (they both have x as a variable). Therefore the coefficients, 15 and 3, can be added. We cannot add 4 to 18x because 4 does not contain the variable x. 4 is the constant term. Example: 16c + 2(8 7c) = 16c c 2c + 16 Distribution Property combine like terms Noticed: 2c + 16 DOES NOT become 18c Simplify each Expression 1. 12k + 11(3k - 5) 2. -5(3b + 4) - 6(3 + 6b) 3. 4(-3a + 13) + 3(a + 4) x + 2(9 + 7x) 5. 2x x 2 + 6x 6. x 3 + 3x + x (5a 3 4a 2 ) - (2a 3-7a 2 ) 8. (3x 2 4x) + (-x 2 + 6x) 9. (2b 3 b 2 + 3) (b 3 4b 2 ) 10. (2c 4 + 3c 2 5) + (c 3 + 5c 2 + 6c)
5 We use distance formula to learn the distance between any two point in space (x 1, y 1) and (x 2, y 2). Distance = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 We use midpoint (M) formula to find the middle of two points in space (x 1, y 1) and (x 2, y 2). Midpoint = ( x 1+x 2, y 1+y 2 ) 2 2 We use slope formula to find the slope (m) of the line through the points (x 1, y 1) and (x 2, y 2). Slope= change in y (rise) change in x (run) = y 2 y 1 x 2 x 1 Example: Find the distance and the midpoint between the two points (3, 9) and (8,-2). (It doesn t matter which point is which, but be consistent) Distance = (x 2 x 1 ) 2 + (y 2 y 1 ) 2 Distance = (8 3) 2 + ( 2 9) 2 Distance = (5) 2 + ( 11) 2 Distance = (25) + (121) Distance = (146) Distance = 12.1 Midpoint = ( x 1+x 2, y 1+y 2 ) 2 2 M = ( , ) 2 2 M = ( 11 2, 7 2 ) slope = y 2 y 1 x 2 x 1 m = m = 11 5 Find the distance (nearest tenth), midpoint and slope of two points. Two points Distance Formula Midpoint Formula Slope Formula 1. (2, 6), (8, 18) 2. (2, 3), (4, -11) 3.(5, -8), (5, -1) 4. (5, 2), (-4, -1)
6 Point Slope form and Slope- intercept form When given the slope (m) and a point (x 1, y 1), we can use point slope formula to determine the equation of the line. Point slope form: y y 1 = m(x x 1 ) Once we plug in to the point slope form, we need to simplify into Slope intercept form: y = mx+ b Example: Write an equation of the line through (1, 4) with the slope of 3. Step 1 y y 1 = m(x x 1 ) Point Slope Form Step 2 y 4 = 3(x 1) Substitute 3 for slope and (1, 4) for (x 1, y 1) Step 3 y 4 = 3x 3 Distribution the slope (3) Step Add for to both sides, to get y by itself Step 5 y = 3x + 1 Slope intercept form When given two points, you need to find the slope (using slope formula). Then use either of the two given points for the point slope form. Practice: Write an equation of the line in slope intercept form that contains the given points and slope. 1. m = 2, (5,3) 2. m = 3, (4,-2) 3. m= 1, ( 6,4 ) 4. 2 m=3, (-5, 6) 2 5. (-4, 7), (-6,-4) 6. (19,-16), (-7,-15) 7. (12, 2), (-7,5) 8. (8, 10), (-4, 2)
7 Graphing Linear Equations: If the equation is not in slope intercept form, then you need to solve for y. Graph the following A. y = 7x + 3 B. y = 2 C). y = 1 2 x D. y = 3 x 1 E. y = 5 x + 2 F. 3y = 2x G. x + y = 4 H. x = 2 I. 4x + 5y = 10
8 Exponent: Simplify using positive exponents. 1. (-4) 7 (-4) (9 3 ) y y x (x -4 y 2 ) p 2 (3p 5 ) (8mn 3 ) 0
9 Multiply Polynomial Example1: (x 2 2x + 1)(3x 3 + 4x 2 + 5x) Practice: Find the product. 3. 2n(3n 2 5n 8) 4. (x 1)(x + 3) 5. (p + 5)(p 5) (x 3) 2 8. (d + 2) (3d 2 + 2d + 4) Hint: NOT x (x + 3)(x 2 + x 9) 10. (d 2 + 4d + 1) (3d 2 +2d +5) Simplify Radical:
10 You need to remember, the list of perfect square, perfect squares are the number that time itself. It s called a perfect square because for the object to be a square all sides are congruent. You also need to remember that square root ( ), get rid of the square. Perfect Square 1 2 = = = = = = = 49 And so on Practice Multiply/Divide Radicals Outside #, times/divide Outside #, Inside the radical times/divide Inside the radical. Then simplify the radical To Add or Subtract Radicals, we need to have the same number under the radical. If not, we might have to simplify. a = =
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