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Functions, Equations, and Graphs Chapter 2

IA. Functions, Equations, and Graphs Chapter 2. In this chapter, you will learn:. What a function is. Review domain and range. Linear equations. Slope. Slope intercept form y = mx+b. Point-slope form y – y1 = m(x – x1). Linear regression. What is a function?. FUNCTION. FUNCTION.

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Functions, Equations, and Graphs Chapter 2

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  1. IA Functions, Equations, and Graphs Chapter 2

  2. In this chapter, you will learn: • What a function is. • Review domain and range. • Linear equations. • Slope. • Slope intercept form y = mx+b. • Point-slope form y – y1 = m(x – x1). • Linear regression.

  3. What is a function? FUNCTION FUNCTION A function is a special type of relation in which each type of domain (x values) is paired of with exactly one range value (y value). NOT A FUNCTION NOT A FUNCTION FUNCTION NOT A FUNCTION FUNCTION

  4. Relations and Functions Suppose we have the relation { (-3,1) , (0,2) , (2,4) } -3 0 2 1 2 4 FUNCTION ONE – TO – ONE DOMAIN x - values RANGE y - values

  5. Relations and Functions Suppose we have the relation { (-1,5) , (1,3) , (4,5) } -1 1 4 5 3 5 FUNCTION NOT ONE – TO – ONE

  6. Relations and Functions Suppose we have the relation { (5,6) , (-3,0) , (1,1) , (-3,6) } 5 -3 1 0 1 6 NOT A FUNCTION

  7. Domain and Range Domain The set of all inputs, or x-values of a function. It is all the x – values that are allowed to be used. Range The set of all outputs, or y-values of a function. It is all the y – values that are represented.

  8. Example 1 • Domain = ________________ • Range = _________________ All x – values or (-∞ , ∞) Just 4 or {4}

  9. Example 2 Just -5 or {-5} All y – values or (-∞ , ∞) Domain = ________________ Range = _________________

  10. Example 3 All x – values or (-∞ , ∞) From -6 on up or [-6 , ∞) Domain = ________________ Range = _________________

  11. Example 4 From -6 on up or [-6 , ∞) All y – values or (-∞ , ∞) Domain = ________________ Range = _________________

  12. Example 5 All x – values or (-∞ , ∞) All y – values or (-∞ , ∞) Domain = ________________ Range = _________________

  13. Function Notation What is function notation? • Function notation, f(x) , is called “f of x” or “a function of x”. • It is not f times x . • Example: if y = x+2 then we say f(x) = x+2. • If y = 5 when x = 3, then we say f(3) = 5

  14. Example 1f(x) = 3x + 1 3 (5) + 1 = 16 f( 5) = ____________________ 3 (13) + 1 = 40 f( 13) = ____________________ 3 (-11) + 1 = -32 f( -11) = ____________________

  15. Example 2f(x) = x² + 3x - 5 5² + 3 (5) – 5 = 35 f( 5) = ____________________ 0² + 3 (0) – 5 = -5 f( 0) = ____________________ 4² + 3 (4) – 5 = 23 f( 4) = ____________________

  16. SLOPE SLOPE RISE RUN

  17. Slope Formula Given points (X1,Y1) and (X2,Y2) Is the same as ?

  18. Example( 4 , 0 ) and ( 7 , 6 ) (X1,Y1) (X2,Y2)

  19. Example( -6 , 5 ) and ( 2 , 4 ) (X1,Y1) (X2,Y2)

  20. Example( 5 , 2 ) and ( -3 , 2 ) (X1,Y1) (X2,Y2)

  21. Example( 2 , 7 ) and ( 2 , -3 ) (X1,Y1) (X2,Y2)

  22. Recap of slope

  23. Linear Forms of Linear EquationsStandard Form (AX + BY = C) FINDING INTERCEPTS • A and B cannot be fractions. • A cannot be negative To find y-intercept, set x = 0 Y-intercept (0,-5) To find x-intercept, set y = 0 X-intercept (5/3,0) y = 3x – 5 y – 3x = – 5 – 3x + y = – 5 3x – y = 5

  24. Linear Forms of Linear EquationsSlope Intercept Form (y = mx + b) FINDING INTERCEPTS • Y is isolated on one side. • Y is positive To find y-intercept, set x = 0 Y-intercept (0,-3) To find x-intercept, set y = 0 X-intercept (6,0) x – 2y = 6 –2y = – x + 6 y = ½ x + 6/-2 y = ½ x – 3

  25. Find a line|| to 2x + 4y = -8and passes thru (8,3) Slope intercept • First find slope • 2x + 4y = -8 • 4y = -2x – 8 • y = - ½ x – 2 • Slope • Use y = mx + b • y = - 1/2x + b • Plug in point • 3 = - ½ (8) + b • 3 = -4 + b • 7 = b FINAL EQUATION y = mx + b y = - ½ x + 7

  26. Find a linePerpendicular to 3x – 2y = 10and passes thru (-6,2) Slope intercept • First find slope • 3x – 2y = 10 • - 2y = -3x + 10 • y = 3/2 x – 5 • Slope • Use y = mx + b • y = -2/3 x + b • Plug in point • 2 = -2/3 (-6) + b • 2 = 4+ b • -2 = b FINAL EQUATION y = mx + b y = -2/3x - 2

  27. Find a lineThat has slope = 2and passes thru (-4,7) Slope intercept • Use y = mx + b • y = 2x + b • Plug in point • 7 = 2(-4) + b • 7 = -8 + b • 15 = b FINAL EQUATION y = mx + b y = 2 x + 15

  28. Find a lineThat has x-intercept = (5,0)and y-intercept (0,-3) Slope intercept • First find slope • m • m = 3/5 • Use y = mx + b • y = 3/5 x + b • Plug in point either one ! • 0 = 3/5 (5) + b • 0 = 3 + b • -3 = b FINAL EQUATION y = mx + b y = 3/5 x - 3

  29. Graph x = 3

  30. Graph y = -3

  31. Graph Y = -3/4 x + 2 Y = mx + b m = slope b = y-intercept • Plot the y-intercept first • Starting from the y-intercept, go up if + or down if – Then go right. 3) Plot point and draw your line. m = -3 / 4 b = 2

  32. Graph 5x + 6y < 30 • 5x + 6y < 30 • 6y< -5x + 30 • Y < -5/6 x +5

  33. Graph 2x + 4y ≥ 16 • 2x + 4y ≥ 16 • 4y ≥ -2x + 16 • Y ≥ -2/4 x + 4 • Y ≥ -1/2 x + 4

  34. Graph 5x - y < 8 • 5x – y < 8 • – y < -5x + 8 • y > 5x – 8

  35. Graph y = |x| (0,0) WHERE IS THE VERTEX?

  36. Graph y = |x - 3| (3,0) WHERE IS THE VERTEX?

  37. Graph y = |x + 3| (-3,0) WHERE IS THE VERTEX?

  38. Graph y = |x + 3| – 2 (-3,-2) WHERE IS THE VERTEX?

  39. Is there a formula for graphing absolute value equations??? • Y = |x + 2| – 3 • Y = |x + 5| + 8 • Y = |x – 8| – 6 • Y = |x – 7| + 4 • Y = |x – 4| – 5 • Y = |2x – 8| + 2 • Y = |3x + 6| – 3 • Y = |mx + b| + c • (-2 , -3) • (-5 , 8) • (8 , -6) • (7 , 4) • (4 , -5) • (4 , 2) • (-2 , -3) • (- b/m , c)

  40. Correlationspositive Outlier As x increases Then y increases POSITIVE SLOPE “Trend line” or “regression line”

  41. Correlationsnegative As x increases Then y decreases negativeSLOPE

  42. Correlationsnone No real trend line

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