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Algebra I

Function Families. Linear – constant rate of change Absolute Value – distance from zero Rational Square Root Quadratic Exponential Geometric 7. Power 8. Trigonometric. Algebra I.

gretchen-herrmann
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Algebra I

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  1. Function Families • Linear – constant rate of change • Absolute Value – distance from zero • Rational • Square Root • Quadratic • Exponential • Geometric • 7. Power • 8. Trigonometric Algebra I

  2. Input:x-value independent variable independent quantity domain causeOutput:y-value dependent variable dependent quantity range effect

  3. y = mx+ b A linear relationship m =slope b =y-intercept, where x =0

  4. Linear

  5. Graph 4 Steps 1. Title Equation or Words to Describe Graph 2. Table Use 5 values for x. Two negative, zero, and two positive 3. System Create a Cartesian Coordinate System with a scale to fit the data, and equal intervals 4. Line Should be neatly drawn to go through all points

  6. y-intercept where x = 0 x-intercept where y = 0

  7. y-intercept x-intercept

  8. y-intercept where x = 0 x-intercept where y = 0

  9. Slope m = Subscripts name the point Point 1 and Point 2 (x1, y1) (x2 , y2) Two points define a line.

  10. y x

  11. Positive Slope A lot of work

  12. Wee!!! Negative Slope

  13. Zero Slope No where fast… Ski Bird on a horizontal hill.

  14. Undefined Slope Divide by 0 Undefined!!! Sheer doom awaits or does it? Ski Bird on a vertical hill. Where will this line end?

  15. Slope Bear

  16. Slope

  17. Slope

  18. Undefined Slope No change in x

  19. Washington’s Mountaineering Club This year, 4 climbers competed for the Steepmeister Cup. This award is given to the person who climbs the steepest slope. Each person went up Mount Kilimanjaro by a different route. Determine who the winner is based on two points from each member's climbing log. Climber 1 Desiree (30, 225), (55, 650) Climber 2 Mai (82, -35), (107, 565) Climber 3 Luis (125, 0), (125, 800) Climber 4 Kou (0, 10) (0, 1000)

  20. Mount Kilimanjaro, East Africa • Compelling for its beauty • Highest peak on the African continent • Tallest free-standing mountain in the world • Found in isolation from the coastal area

  21. Function • Matching Rule: for each x there is only one y • Read f(x): “f of x” or “function of x” • Vertical Line Test – is a function if touches graphed line only once http://www.shodor.org/interactivate/activities/VerticalLineTest/

  22. Linear Function y=mx+b • The steady acceleration of a car. (velocity vs. time) V(t)

  23. Linear Function y=mx+b • An airplane flies at an average speed. (distance vs. time)

  24. The regular progress of the tortoise. (distance vs. time) Linear Function y=mx+b d(t)

  25. Monthly deposits of equal paychecks. (amount vs. time) Jan Feb Mar Apr May June July Aug Sept Oct Nov Dec

  26. Linear Function y=mx+b • A bathtub fills at a constant rate. (volume vs. time) V(t)

  27. Problem • Mrs. Kapler’s daughter has $100 in the bank. • She just got a job at the grocery store. It pays $8 per hour. This daughter never spends her money; she prefers to save it all in the bank. • Write an equation that represents the money she has in the bank as a function of hours worked. • What is the slope? The y-intercept? • Create a table. • Graph the relationship. • After how many hours will she have $260 in the bank?

  28. Sara has 4 hours after dinner to study and do her homework. She has homework in math and English. She spends x hours on math and y hours on English. Write an equation that describes this relationship. Graph the equation. If she spends 1 hour doing math, how much time does she have for English?

  29. Vertical Line Test

  30. Functions Matching Rule For each x there is only one y Each element of the domain is paired with exactly one element of the range. For each input (x) there is one and only one output (y) f(x) y x

  31. Function Notation Input Name of Function Output

  32. 2 3 5 4 3 0 2 3 f(x) f(x) f(x) f(x) Is the relation a function? 1. {(2, 3), (3, 0), (5, 2), (4, 3)} YES. For each x there is only one y.

  33. 5 4 5 1 6 3 2 6 9 1 f(x) f(x) f(x) f(x) f(x) Is the relation a function? No for 5 is paired with 2 and 3 2. {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)} For each x there is only one y.

  34. Is this relation a function?{(1,3), (2,3), (3,3)} • Yes • No

  35. Vertical Line Test If touches the graphed line only once is held true. If held true, graphed line is a function. Are these functions? FUNCTION! FUNCTION! NOPE!

  36. Vertical Line Test Yes No No Yes

  37. Is this a graph of a function? • Yes • No

  38. Given f(x) = 3x - 2, find: = 7 1) f(3) 2) f(-2) 3(3)-2 3 7 = -8 3(-2)-2 -2 -8

  39. Given h(z) = z2 - 4z + 9, find h(-3) (-3)2-4(-3)+9 -3 30 9 + 12 + 9 h(-3) = 30

  40. f(3) = 2(3) + 1 = 6 +1 = 7 Given f(x) = 2x + 1, find f(3) • 7 • 1 • -1 • -7

  41. Given g(x) = x2 – 2, find g(4) • 2 • 6 • 14 • 18

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