Question #1 Literal Equations for x Do B.O.D. Start with Begin with C then do the Opposite of Ax y= y=C y = C Ax and last Divide by B y=
for x2 Whenever you have fractions, remove the denominator by multiplying the other terms by it. Start with Begin with x2 then do BOD Question # 4 (Average rate of change from table) 1) The table below shows the average diameter of a persons pupil with age. years mm 30 20 50
4.3 3.5 -0.8 -0.8/20 = -0.04 Find the average rate of change from age 30 to age 50. Include units with your answer. Find the y-values for 30 and 50 Subtract backwards (50 30) (3.5 4.8) and divide the y by x (-0.8/20) The answer is -0.04 mm per year
Question # 2 and Question # 3 (Functions) If you have a table check the x-values. They CANNOT be repeated with a different y-value. Example: If the set of numbers has (1, -2) and (1, 3), one of them must be removed to have a function. If you have a graph, follow or draw vertical lines. If you can cross the graph in more than ONE place, then it is NOT a function. Question #5 Inequalities from Graphs Check the line
< > < > >
< Check where it is shaded Solid: add line under < or > Dashed: keep < or > Check the slope and y-intercept (mx + b) Positive m = rising line Negative m = falling line + b = y-intercept above zero b = y-intercept below zero Questions #6, #27 Transformations G(x) = f(x + 3)
inside parenthesis moves left when positive subtract from x-value, y-value stays the same G(x) = f(x 3) inside parenthesis moves right when negative add to x-value, y-value stays the same G(x) = f(x) + 3 outside parenthesis moves up when positive add to y-value, x-value stays the same G(x) = f(x) 3
outside parenthesis moves down when negative subtract from y-value, x-value stays the same Question #8 Simplifying Radicals Example 1: 3 + = Example 2: + + Example 3: +4 + = 9 Question #7 (Slope and y-intercept) y = mx + b
f(0) = b Slope = m Positive m = rising line Negative m = falling line + b = y-intercept above zero b = y-intercept below zero Question #9 Solving Compound Inequality Solve the following inequalities 1) Split in 2 problems -3 < 2x 3 and 2x 3 < 10 Solve each using B.O.D. Write answer with variable in the middle and numbers listed from least to greatest 0 < x < 13/2
If the x is negative write the answers on the opposite side example -3 < -2x 3 and -2x 3 < 10 -13/2 < x < 0 Question # 10 Convert to vertex form Rewrite into an equivalent equation that reveals the vertex of the parabola. 1) Start by writing the first part like this: Where you a =2 1. To find x: = (Use the Opposite sign when plugging into Vertex Form) 2. To find y: Substitute and evaluate:
3. Complete equation: Question # 11 Solving simple inequalities Solve just like a regular equation 1. 9 > 6n + 7 10 combine like terms 9 > 6n 3 BOD 2 > n Then, write n as the leading term. n<2 Question # 12 (exponential vs linear) Decide whether each of the following scenarios can be represented by an exponential or linear function. Look for Key Words like: Increase or decrease by a % Increase or decrease by a factor
half life, double, triple, quadruple Question # 13 and Question #14 (parabolic motion word problems) f(x) = ax2 + bx + c What does the positive zero of this function represent? Where the function ends, lands, hits ground. You find it by using factoring or the quadratic formula. What does x coordinate of the vertex of this function represent? The time it takes to get to the maximum point. When the maximum occurs. This is the Axis of Symmetry b/2a What does the y-value of the vertex represent?
The maximum value. You find it my substituting the axis of symmetry into equation. What does the value of c represent? The initial amount. The y-intercept. Where the function began. Question # 15 Solutions by factoring (ac method) Find the solutions of the following by factoring You can use formula if you are good at it. Also, practice changing the zeros into factors. If you chose to factor: 1. Find the GCF. 2. If there isnt one, you can use AC (multiply A and C) method (aka Slip and Slide).
3. Then, find two numbers that multiply to give you C, but add or subtract to give you B Question # 16 (finding a) In each case below, the graph of f(x) = x2 was transformed into g(x) which has the equation g(x) = ax2. Find the value of a. The value of a is positive if the graph opens up and negative if it opens down. 1 unit a=3 3 2 a is negative
but this point is Not a lattice pt. Find a lattice point. 1 1 unit (2, 6) Question # 18 System of Equations (value; amount) 1) There are a total of 14 coins when dimes and nickels are combined. The total amount is 80 cents. How many dimes and nickels are there, respectively? 12 2 _______
nickels _______ dimes x = nickels (.05) x + y = 14 .05x + .10y = .80 y = dimes (.10) ________________ -.05x 0.5 y = -.7 .05x + .10y .05y = .80 .10 y=2 Then, plug in y into one of the original equations to solve for x.
Question # 19 (interpreting linear equations) In y = mx + b format The m is always the rate of change, the slope, the price of one item, the value at which the y changes. The b is always the initial amount, the y-intercept, the fee, the one-time cost. The x-value is always the independent variable The y-value always depends on the x-value. Thy y-value will always represent a total. Question # 20 (equivalent expression; vertex to standard) Convert to standard form Example: F(x)
-3(x 4)2(x+ 14)2 First square the =parenthesis (x)(x) x2 x(-4)(2) 8x (-4)(-4) = + 16 Now multiply -3(x2 8x + 16) (if needed) -3x2 + 42x 48 Last, combine with the last term (if needed) -3x2 + 42x 48 + 1
Final answer: -3x2 + 42x 47 Question # 21 and Question #22 (multiplying binomials) Simplify the following: Sum and difference: (ax + b)(ax b) (ax)2 b2 Regular multiplication (ax + b)(cx + d) acx2 + adx + bcx + bd acx2 + (ad + bc)x + bd Squares: (ax + b)2 or (ax + b)(ax + b) (ax)2 + 2abx + b2 Question # 23 (factor; difference of two squares) Rewrite into equivalent equations that reveal the zeroes of the function.
y = x2 36 This is a sum and difference equation. Reveal the zeroes means write with factors. =x =6 y = (x 6)(x + 6) Question # 24 (Arithmetic Functions) Write two explicit rules for the following arithmetic sequences. For each, find the value of the 50th term. 19, 27, 35, 43, 51 First term: 19 Rate of change: 51 43 = 8 First explicit rule: f(x) = 19 + 8(x 1) Simplify the first:
f(x) = 19 + 8x 8 f(x) = 8x + 11 This is equivalent rule Question # 25 For each, determine whether the points are solutions to the given equation. (2, 34) f(x) = 3x2 2 Substitute point (2, 34) into the function: x=2 f(x) = 34 f(x) = 3x2 2 34 = 3(2)2 2 34 10 (2, 34) is not a solution. Question # 26 (systems of equations; standard form)
Matt and Ming are selling fruit for a fundraiser. Customers can buy small boxes of oranges and large boxes of oranges. Matt sold 3 small boxes and 14 large boxes for a total of $203. Ming sold 11 small boxes and 11 large boxes for a total of $220. Find the cost each of one small box of oranges and one large box of oranges. $7 Cost of a Small Box _______ $13 Cost of a Large Box _______ S = cost of small box 3S + 14L = 203 11S + 11L = 220 L = cost of large box 33S + 154L = 2233
________________ -33S 121L 33L ==-660 1573 L = 13 3S + 14(13) = 203 S=7 Then, plug in y into one of the original equations to solve for x. Question # 28 Solving equations (fractions) Solve the following equations. Can you figure out what I did? 2x 6 = 15x + 1 -13x 6 = 1
x = -7/13 Question # 29 substitution For the function f(x) = 3x 2, find the value of the following: Can you figure out what I did? f(2) = 3(2) 2 = 4 f(4) = 3(4) 2 = 10 f(2) - f(4) = (4) (10) = -6
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