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X-intercept Roots Zeros Hits ground/water ~= 0 Crosses x-axis (~,0)

X-intercept Roots Zeros Hits ground/water ~= 0 Crosses x-axis (~,0). Table Method:. y1= equation    y2=0  2nd  Trace 5 MOVE Enter Enter Enter . The x values when y=0 (or y goes from + to -). Graphing Method:. y-intercept Initial Value Start at.

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X-intercept Roots Zeros Hits ground/water ~= 0 Crosses x-axis (~,0)

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  1. X-intercept Roots Zeros Hits ground/water ~=0 Crosses x-axis (~,0) Table Method: y1= equation    y2=0  2nd  Trace 5 MOVE Enter Enter Enter  The x values when y=0 (or y goes from + to -) Graphing Method:

  2. y-intercept Initial Value Start at Equation: Table: Calculator: The number without a variable y= 3x2 +5x + 3or y = 6(2)x The y value when x = 0 2nd Trace 1(value) 0 Enter

  3. Exponential Growth (Equation): y=abx or y=a(1 + r)x y=final amount   a=initial amount  b=base   r=rate as a percentage  (4.5%= .045) If b > 1 then it’s increase;  If 0<b<1 then it’s decay

  4. Decimals of Depreciate (Decrease) 5% &Appreciate (Increase) 13.5% (1- .05) = .95 (1 + .135) = 1.135

  5. % & % Increase/Decrease 50$ shirt with 7% increase (tax) 50$ shirt with 20% decrease (off) Final = Beginning ( 1 +  r) 50(1.07) $53.50 50(0.8)$40

  6. Linear Growth: Slope of y =3x + 2 Y-intercept of y = 3x + 2 y= mx + b  m=slope (rate of change, change per x)  b=y-intercept (crosses y-axis & initial value) 3 (also rate of change ) 2 (also starting value)

  7. The equation p(x) = .5x + 2 where x is the year since 2000 and p(x) represents the population (in thousands) of Pineboro. What is the slope and what does it represent? What is the y-intercept and what does it represent? .5 thousand; increase in population per year 2 population in 2000

  8. Find rate per mile: $2 per .4 mile Find rate per mile: $1.20 per .2 mile Find rate per minute: $3 per 30 seconds Find rate per foot: $12 per 10 feet $2/ .4 = $5 per mile $1.20/.2 = $6 per mile 3/.5 = $6 per minute $12/10 = 1.2 per foot

  9. Write equations of the following: Balloon company charges flat-fee of $10 per delivery and $2 per mile Taxi charges a flat-fee of $3 and $2 per 0.5 mile Car rental charges $20 a day plus 65 cents a mile Car rental charges $50 per day C = 2m + 10 F= 2/.5 m + 3 Or F= 4m + 3 C=0.65m + 20 C = 50

  10. Regressions & Coefficient of Correlation Coefficient of correlation: STAT   EDIT   Enter x’s into L1 Enter y’s into L2  STAT   CALC  Choose Linreg or Expregr ENTER Write equation on paper and then type in y= ~1 Positive      ~ -1 Negative Find:   2nd 0 Scroll to DiagnosticON Enter  then do regression 

  11. The number of schools in Turnstown can be represented by the following table: According to the line of best fit, how many schools are predicted for 2014? 0 10 30 44 50 • Change 1960 to 0 and the other years (only do on year problems) • STAT EDIT Put the numbers in STAT CALC Linregression • 3) Write the equation. Y = 2x + 20 • 4) Type Equation into Y= y = 2x + 20 • 5) Look at table for x=54 (2014-1960 = 50) 54  128

  12. Exponent Examples 2a2*3a5 (3a5)2   = 2aa 3aaaaa = 6a7 Multiply: Mult. Coefficients, add exponents = (3aaaaa)(3aaaaa)= 9a10 Power: Write it out then perform rule

  13. Exponent Examples Continued: Write out the variables; reduce the coefficients; cancel letters Negative exponents: Switch the location and make positive; 5xo = 5 Zero power makes exponent disappear

  14. Write it out twice Move negative exponents Write out exponents; remove Zero exponent Combine exponents Subtract exponents

  15. Dividing by a monomial Divide each by the bottom Simplify 6x3 Divide each by 6x3 6x3 6x3 6x3 Simplify

  16. Greatest Common Factor (GCF): 42x2yz4 and 28x5 y3 • Largest integer that goes into each coefficient • Smallest of each exponent • 14x2 y Calculator (for coefficient) Math Num GCD (#1,#2) z

  17. Least Common Multiple (LCM): Example:10x2 yz4 and 15x5 y3 Smallest integer they go into, largest of each exponent LCM: 30x5y3z4

  18. Slope of Parallel Lines: What is the slope of the line parallel to y = 3x + 2? What is the slope of the line parallel to 3x + 2y = 8 Same slope (5/6 & 5/6) 3 2y = -3x + 8 y = -3/2x + 8/2 so -3/2

  19. Slope of Perpendicular Lines: What is the slope of the line perpendicular to y = 3x + 2? What is the slope of the line perpendicular to 3x + 2y = 8 Negative Reciprocal slope (5/6 & -6/5) -1/3 2y = -3x + 8 y = -3/2x + 8/2 so 2/3

  20. Direct Variation: (Set up Proportion then solve) Y varies directly as x. y is 8 when x is 3. Find x when y is 40 Y = Y XX Write proportion 8 = 40 3X Substitute in Cross multiply 8x = 3(40) Solve by dividing x = 120/8 = 15

  21. The number of dolls made varies directly as the number of hours worked. 100 dolls are made in 25 hours. Find how many dolls are made in 75 hours dolls = dolls hourshours Write proportion 100 = d 2575 Substitute in Cross multiply 25x = 75(100) Solve by dividing x = 7500/25 = 300

  22. Systems of Equations 2x – y = 10 Find x + y x = y + 4 2x – 1y = 10 1x – 1y = 4 Put in matrix ready mode with x’s and y’s together 2nd x-1   1 ENTER 2 X 3 [ 2 -1 10 ] [ 1 -1 4 ] Put into matrix Solve with matrix 2nd MODE 2nd x-1  Scroll to RREF 2nd x-1 1 ENTER You get [ 1 0 6 ] so x = 6 and y = 2 so x + y = 8 [ 0 1 2 ]

  23. Midpoint of (x1, x2)  and (y1, y2)  Midpoint:  Mx =     My = Or Graph to find Midpoint Midpoint is ½ way Find midpoint of (3,2) and (-1,5) (1, 3.5)

  24. Mary’s house is (3, 1) and is the exactly halfway between Jack’s house (-1,3) and Gary’s house? Find the coordinates of Gary’s house. Jack’s House Down 2 Midpoint Right 4 Down 2 Gary’s House (5, -1) Right 4

  25. What does An+1=2An + 3 mean? Next number = 2 * (now number) + 3

  26. The number of butterflies, of a particular day, is related to the previous day’s number of butterflies, Bn by the function Bn+1 = 1.12Bn. • Describe the change in butterflies • Increases by 12% every day • (112% - 100% = 12%) • B)Find the number of butterflies on the 4th day if there was 50 butterflies on the first day • Day 1: 50 • Day 2: 50(1.12) = 56 • Day 3: 56(1.12) = 62.7 • Day 4: 62.7(1.12) = 70.24 butterflies

  27. Factor trick with Calculator (only for multiple choice) Use if given area or “Which is a factor? • 37 STO X ENTER Store 37 as x • (Problem) / (Answer A) If no decimal then it could be a factor • (Problem)/(Answer B) • (Problem)/(Answer C) • (Problem)/(Answer D) Example: Area is 10x2 – x – 2. Which could be a length? A) 5x – 2 B) 2x + 1 C) 10x – 3 D) 2x - 1 • 37 STO X ENTER Store 37 as x • (10x2 – x – 2 ) / (5x – 2) Decimal so not a factor • (10x2 – x – 2)/(2x + 1) Decimal so not a factor • (10x2 – x – 2)/(10x – 3) Decimal so not a factor • (10x2 – x – 2)/(2x – 1) NOT A DECIMAL so it’s the answer!

  28. Domain & Independent Variables: All x values (numbers from left to right) Domain of {(2,3),(5,6),(1,8)} Domain of y = x2 + 3x {1,2,5} All reals

  29. Range & Dependent Variables: All y values (numbers from down to up) Range of {(2,3),(5,6),(1,8)} Range of y = x2 + 6x {3,6,8} y> -9 (-3,-9)

  30. Error on calculator: If you have an error that you can’t fix immediately then reset your memory by:   2nd    +     7    1 2 If calculator keeps saying 2nd Curve when finding intersection then make sure you have y2 = 0

  31. Slope of (x1, x2)  and (y1, y2)  m=

  32. Distance between (x1,x2) &(y1,y2)  Distance:    don’t forget to close the parenthesis

  33. Pythagorean Theorem  Pythagorean Theorem:   a2 + b2 = c2 c a b

  34. Area of rectangle and triangle: Rectangle A=lw Triangle A = ½ bh

  35. Area &Circumference of Circle: Area: A= (3.14) r2 Circumference=2(3.14)r  or  (3.14)d

  36. Other Reminders: 4 less than a number: 4 is less than a number: Sum of 3 and x: Twice the difference of x and 4: There are twice as many cats as dogs Quadrants: x-4 x<4 (x+3) 2(x-4) c = 2d  II           I III           IV

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