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Expressions, Rules, and Graphing Linear Equations by Lauren McCluskey

Expressions, Rules, and Graphing Linear Equations by Lauren McCluskey. This power point could not have been made without Monica Yuskaitis, whose power point, “Algebra I” formed much of the introduction. Variable:.

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Expressions, Rules, and Graphing Linear Equations by Lauren McCluskey

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  1. Expressions, Rules, and Graphing Linear Equationsby Lauren McCluskey • This power point could not have been made without Monica Yuskaitis, whose power point, “Algebra I” formed much of the introduction.

  2. Variable: • Variable – A variable is a letter or symbol that represents a number (unknown quantity or quantities). • A variable may be any letter in the alphabet. • 8 + n = 12 “Algebra I” by M. Yuskaitis

  3. Algebraic Expression: • Algebraic expression – a group of numbers, symbols, and variables that express an operation or a series of operations with no equal or inequality sign. • There is no way to know what quantity or quantities these variables represent. • m +8 • r –3 “Algebra I” by M. Yuskaitis

  4. Simplify • Simplify – Combine like terms and complete all operations m = 2 • m + 8 + m 2 m + 8 • 3x + (-15) -2x + 5 x -10 “Algebra I” by M. Yuskaitis

  5. Evaluate • Evaluate an algebraic expression – To find the value of an algebraic expression by substituting numbers for variables. • m + 8 m = 2 2 + 8 = 10 • r – 3 r = 5 5 – 3 = 2 “Algebra I” by M. Yuskaitis

  6. Translating Words to Algebraic Expressions • Sum Difference • More than Less than • Plus Minus • Increased Decreased • Altogether “Algebra I” by M. Yuskaitis

  7. Translate these Phrases to Algebraic Expressions n + 10 • Ten more than a number • A number decrease by 4 • 6 less than a number • A number increased by 8 • The sum of a number & 9 • 4 more than a number n - 4 x - 6 n + 8 n + 9 y + 4 “Algebra I” by M. Yuskaitis

  8. Each of these Algebraic Expressions might represent Patterns: • For example: n + 10

  9. Or it might be Geometric (n-4): n=1 13 seats n=2 8 seats

  10. Patterns Patterns may be seen in: • Geometric Figures • Numbers in Tables • Numbers in Real-life Situations • Sequences of Numbers • Linear Graphs Patterns are predictable.

  11. Patterns with Geometric Figures (Triangles) • Jian made some designs using equilateral triangles. He noticed that as he added new triangles, there was a relationship between n, the number of triangles, and p, the outer perimeter of the design. Write a rule for this pattern. P= 4 P=6 P=3 P=5 from the MCAS

  12. How to Write a Rule: 1) Make a table. 2) Find the constant difference. 3) Multiply the constant difference by the term number (x). 4) Add or subtract some number in order to get y.

  13. P = 6 P = 4 P = 3 P = 5 1 ) Make a Table: Let x be the position in the pattern while y is the total perimeter. # of Triangles Rule: Perimeter (x) (y)1 ? 3 2 4 3 5 ... …xy from the MCAS

  14. P = 6 P = 4 2)Find the Constant Difference: How did the output change? P = 3 P = 5 Perimeter (y) 3 4 5 6 … p from the MCAS

  15. P = 6 P = 4 3) Multiply by the Input # (x). 4) Then Add or Subtract some # to get the Output # (y). P = 3 P = 5 # of Triangles Rule: Perimeter (x) (y)1 1x +2 3 2 1x +2 4 3 1x +2 5 ... …xy It Works! from the MCAS

  16. Patterns in Numbers in Tables: • Write a rule for the table below. from the MCAS

  17. 2) Look for the Constant Difference. • What is the change when the input # • increases by 1? • From the 10th tothe 11th the output #s • increase from 21 to 23. So the constant difference is +2.

  18. 3) Multiply x by the Constant Difference. Then… 4) Add or Subtract some #. 2 +1 x Constant Difference Constant Input #

  19. Patterns in Numbers in Real-Life Situations: Write a rule for x number of rides: from the MCAS

  20. 1) Make a Table: 12 14 16

  21. 2) Find the Constant Difference. +$2 +$2 +$2… So the Constant Difference is +2.

  22. 3) Multiply x by the Constant Difference.Then…4) Add or Subtract some #. x +10 2 Constant Difference Constant Input #

  23. Patterns in Sequences of Numbers 12, 16, 20, 24… What’s my rule? Remember: 1) Make a Table. 2) Find the Constant Difference. 3) Multiply x by the Constant Difference. 4) Add or Subtract some #.

  24. 1) Make a Table: +4 +4 2) Find the Constant Difference. The Constant Difference is +4.

  25. 3) Multiply x by the Constant Difference. Then…4) Add or Subtract some #. x +8 4 Constant Difference Constant Input #

  26. Patterns in Linear Graphs “Linear” means it makes a straight line. • Remember: • Make a Table. • Find the Constant Difference. • Multiply x by the Constant Difference. • Add or Subtract some #.

  27. To Make a Table from a Graph: +2 +2 Find the Constant Difference.

  28. 3) Multiply x by the Constant Difference. Then…4) Add or Subtract some #. x -1 2 Constant Difference Input Constant

  29. How to find the 10th or 100th term: • Now that we have a rule we can find any term we want by evaluating for that term #. • Just substitute the term number for x, then simplify.

  30. What would ‘y’ be if x = 10? The rule for the last graph was: 2x -1 Substitute 10 for x and we get: (2)(10) – 1 or 20 -1 = 19. So (10, 19) are solutions for this rule, AND (10, 19) would be a point on this line!

  31. What would ‘y’ be if x = 100? 2x – 1 was the rule for the graph. Substitute 100 for x: (2)(100) – 1 or 200 -1 = 199 So (100, 199) would be a solution for this rule, AND (100, 199) would be on this line!

  32. Review So here we have come full circle, we have: • Written algebraic expressions; • Evaluated these expressions; • Written expressions (rules) for patterns; • Evaluated these rules for specific terms.

  33. Graphing Linear Patterns There are 3 forms of equations that can be graphed: 1) Slope-intercept form 2) Standard form 3) Point-slope form

  34. Slope-Intercept Form (Slope) • The “slope” of a line is the measure of its steepness. Or: Rise over Run

  35. Y-Intercept: • The y-intercept is the point where a line crosses the y-axis. • Hint: Think of the word, ‘intersection’, where 2 streets cross, in order to remember ‘intercept’. -1

  36. Finding the Slope on a Graph: The slope of the line is rise run. Or: the change in y the change in x. Change in y = 22 Change in x = 1 1 So the slope is +2. =

  37. Kinds of Slopes: • Slopes may be positive (y increases as x increases); • Slopes may be negative (y decreases as x increases); • Slopes may be zero (y doesn’t change at all); • Or Slopes may be undefined (x doesn’t change at all).

  38. Name the Type of Slope:

  39. Slope-Intercept Form: You can see both the slope and the y-intercept on the graph: 2 x -1

  40. Standard Form: • It’s easy to find the x- and y-intercept with the standard form (Ax + By = C). • All you need to do is substitute “0” for x and solve for y; then substitute “0” for y and solve for x. Try it:

  41. Write y = 2x -1 in standard form: y = 2x - 1 -2x -2x y - 2x = -1 y - (2) (0) = -1 y = -1 So the y-intercept is -1. 0 - (2) x = -1 -2 -2 x = 1/2 So the x-intercept is 0.5.

  42. Point-Slope Form: The point slope form (y - y1) = m(x - x1) is easiest to use if you are given one point and the slope of the line. Just substitute the coordinates into the equation. Then rewrite the equation in slope-intercept form.

  43. Point-Slope Form • Suppose you did not have the graph, • but you were told that the point (2, 3) is on • the line and the slope is +2… • You could write the equation: y - 3 = 2(x - 2), • then rewrite it in slope-intercept form.

  44. Point-Slope Form: You could rewrite y - 3 = 2(x - 2) to the slope-intercept form: y - 3 = 2(x - 2) y - 3= 2x - 4 +3 +3 y = 2x -1

  45. Slopes of Parallel Lines: Two lines on the same plane that have the same slope will be parallel. Slope is 0. Slope is undefined.

  46. Slopes of Perpendicular Lines: Note: Perpendicular lines form right angles at their intersection. Two lines whose slopes are negative reciprocals are perpendicular. The product of their slopes will equal -1.

  47. Are they Parallel or Perpendicular? y = 2x + 10 y = 2x -5 y = -3x + 2 y = 1/3x + 1

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