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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: • 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 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 Simplify • Simplify – Combine like terms and complete all operations m=2 •m+8+m • 3x + (-15) -2x + 5 2m+8 x -10 “Algebra I” by M. Yuskaitis Evaluate • Evaluate an algebraic expression – To find the value of an algebraic expression by substituting numbers for variables. •m+8 •r–3 m = 2 2 + 8 = 10 r=5 5–3=2 “Algebra I” by M. Yuskaitis Translating Words to Algebraic Expressions • • • • • Sum More than Plus Increased Altogether Difference Less than Minus Decreased “Algebra I” by M. Yuskaitis Translate these Phrases to Algebraic Expressions • 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 + 10 n-4 x-6 n+8 n+9 y+4 “Algebra I” by M. Yuskaitis Each of these Algebraic Expressions might represent Patterns: • For example: n + 10 (x) (y) 1 11 2 12 3 13 Or it might be Geometric (n-4): n=1 13 seats n=2 8 seats Patterns Patterns are predictable. Patterns may be seen in: • Geometric Figures • Numbers in Tables • Numbers in Real-life Situations • Sequences of Numbers • Linear Graphs 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=3 P=6 P=5 from the MCAS 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. 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 2 3 ... x ? 3 4 5 … y from the MCAS P=4 P=6 2)Find the Constant Difference: How did the P=3 P=5 output change? Perimeter (y) 3 4 5 6 … p +1 +1 +1 from the MCAS P=4 3) Multiply by the Input # (x). P=3 P=5 4) Then Add or Subtract some # to get the Output # (y). # of Triangles Rule: (x) 1 2 3 ... x It Works! 1x +2 1x +2 1x +2 P=6 Perimeter (y) 3 4 5 … y from the MCAS Patterns in Numbers in Tables: • Write a rule for the table below. Input (x) 2 Output (y) 5 5 10 11 11 21 23 from the MCAS 2) Look for the Constant Difference. Input (x) 2 5 10 11 Output (y) 5 11 21 23 •What is the change when the input # increases by 1? •From the 10th to the 11th the output #s increase from 21 to 23. So the constant difference is +2. 3) Multiply x by the Constant Difference. Then… 4) Add or Subtract some #. Input (x) 2 5 10 11 Output (y) 5 11 21 23 Constant Difference 2x +1 Constant Input # Patterns in Numbers in Real-Life Situations: Write a rule for x number of rides: from the MCAS 1) Make a Table: In (x) # of Rides 1 Out (y) Cost $ 12 2 $ 14 3 $ 16 2) Find the Constant Difference. In (x) Out (y) # of Rides Cost $12 1 2 3 $14 $16 +$2 +$2 +$2… So the Constant Difference is +2. 3) Multiply x by the Constant Difference. Then… 4) Add or Subtract some #. In (x) Out (y) # of Rides Cost 1 2 3 $12 $14 $16 Constant Difference 2 x +10 Input # Constant 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 #. 1) Make a Table: (x) 1 2 3 (y) 12 16 20 +4 +4 2) Find the Constant Difference. The Constant Difference is +4. 3) Multiply x by the Constant Difference. Then… 4) Add or Subtract some #. (x) 1 2 3 (y) 12 16 20 Constant Difference 4 x +8 Input # Constant Patterns in Linear Graphs “Linear” means it makes a straight line. Remember: 1. Make a Table. 2. Find the Constant Difference. 3. Multiply x by the Constant Difference. 4. Add or Subtract some #. To Make a Table from a Graph: (x) (y) -1 -3 0 -1 +2 +2 1 1 Find the Constant Difference. 3) Multiply x by the Constant Difference. Then… 4) Add or Subtract some #. (x) -1 0 1 (y) -3 -1 1 Constant Difference 2 x -1 Input Constant 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. 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! 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! 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. Graphing Linear Patterns There are 3 forms of equations that can be graphed: 1) Slope-intercept form 2) Standard form 3) Point-slope form Slope-Intercept Form (Slope) • The “slope” of a line is the measure of its steepness. rise run Or: Rise over Run Y-Intercept: • The y-intercept is the point where a line crosses the y-axis. -1 • Hint: Think of the word, ‘intersection’, where 2 streets cross, in order to remember ‘intercept’. 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 = 2 2 = Change in x = 1 1 So the slope is +2. 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). Name the Type of Slope: Slope-Intercept Form: You can see both the slope and the y-intercept slope on the graph: 2 x -1 y-intercept 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: 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. 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. 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. 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 Slopes of Parallel Lines: Two lines on the same plane that have the same slope will be parallel. Slope is 0. Slope is undefined. 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. Are they Parallel or Perpendicular? y = 2x + 10 y = 2x -5 y = -3x + 2 y = 1/3x + 1