UNIT 4 NOTES. 4-1 and 4-2 Coordinate Plane

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1 UNIT 4 NOTES 4-1 and 4-2 Coordinate Plane y Ordered pairs on a graph have several names. (X coordinate, Y coordinate) (Domain, Range) (Input,Output) Plot these points and label them: a. (3,-4) b. (-5,2) c. (0,4) d. (2,0) e. (-4,-3) 0 x A function is where each x coordinate is paired with exactly one y coordinate. The X VALUES CANNOT REPEAT, but the Y VALUES CAN. EX: Is the relation a function? {(2,1), (3,1), (5,2)} Domain Range Why? EX: Is the relation a function? {(1,2), (3,1), (1,5)} Domain Range Why? The VERTICAL LINE TEST tells you whether a relation is a FUNCTION if the graph hits a vertical line (your pencil) at only ONE POINT (in other words, the X value does not repeat!) y y 0 x 0 x 1

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3 You can write a rule for a function by analyzing a table of values. Find the pattern! 4-4 Writing a Function Rule Write a function rule for each table of values. 1. x f(x) x f(x)

4 3. x f(x) Write a function rule to describe each statement. 4. the amount of money you earn mowing lawns m(n) at $15 per lawn 5. the cost in dollars of printing dollar bills c(d) when it costs 3.8 cents to print a dollar bill. 4

5 4-5 Slope Slope (m) - The steepness of a line Positive Slope Uphill Negative Slope Downhill Zero Slope Flat (horizontal) No Slope (undefined) Vertical 5

6 Ways to find slope 1. Find the slope of: Graph: Rise Run 2. 2 Points Formula: ( x1, y1)( x2, y 2) Find the slope of the line that passes through: 3. ( 4, 2) and (4, 4) y y m x x

7 4. (2 1 2, ) and ( 1 2, 1 2 ) 5. ( 5, 3) and ( 5, 1) 6. ( 7, 4) and (2, 4) 7

8 4-6 Slope Intercept Form y-intercept: where the graph crosses the y-axis Slope Intercept Form: y = mx + b Find the slope & y-intercept of each: where m is the slope & b is the y-intercept 1. y 3x y x y 2 3 x 4. y = 5 8

9 5. 3y 2x x y 2 3 Write an equation for each line: 7. 1 m and b =

10 Use the slope & y-intercept to graph each line: 9. 1 y x y 5x 2 Ordered Pair Solution A point lies on the line of a graph if it is a solution to the equation. Plug in the x- & y-coordinates 11. Does ( 3, 4) lie on the graph of y = 2x + 1? (1, 1)? 10

11 4-7 Standard Form Standard Form: Ax + By = C Write in standard form: 1. y 3x 2 where A, B, and C are integers (so they can t be decimals or fractions) 2. 1 y x 5 2 x-intercept: where the graph crosses the x-axis To find the x-intercept make y = 0 y-intercept: where the graph crosses the y-axis 3. Find the x- and y-ints of: 5x 3y = 12 x-int y-int y = 0 x = 0 To find the y-intercept make x = 0 x-int y-int y = 0 x = 0 Ways to graph linear Equations: - x- and y-ints 4. 2x 5y 10 - slope & y-int 11

12 5. y 4x x 2y 6 Special Equations: x = -no y in the equation -vertical line no slope y = -no x in the equation -horizontal line m = 0 7. x 2 8. y 2 12

13 4-8 Point-Slope Form and Writing Linear Equations Point-Slope Form: y y m x x Where m is the slope & 1 1 x y 1, 1 is a point. Graph: 1. y - 2 = 2 (x-3) 2. y 5 x 2 13

14 Finding an equation when given two points: 1. Find the slope 2. Use one of the points & the slope point slope form 3. Rewrite Equation Write an equation in point-slope form: 3. (2, 5) m = ( 5, 6) m = undefined Write an equation for the line in point slope & slope-intercept form: 4. (3, 5) & (0, 0) 5. ( 3, 12) & (6, 1) 14

15 4-9 Point-Slope Form and Writing Linear Equations For each pair of points find the: a. Slope d. Equation in Standard Form b. Equation in Point-Slope Form e. x and y intercepts c. Equation in Slope-Intercept Form f. Graph of the line 1. (1, 5) & ( 4, 2) 2. (2, 1) & (8, 4) 15

16 3. (2, 7) & (2, 4) 16

17 4-10 Parallel & Perpendicular Lines (part 1) Parallel Lines: -never intersect -have the same slope Find the slope of the line that is parallel and perpendicular to each line. 1. y 3x x 4y 8 Ex: m = 4 & m = 4 Perpendicular Lines: -Intersect at right angles (90 ) -Slopes are negative reciprocals 3. x 7 Ex: m = ½ & m = 2 Determine whether the graphs of each pair of equations are parallel, perpendicular or neither. 4. x 2y = 4 2x + y = x + y = 3 6x + 2y = 10 17

18 4-10 Parallel & Perpendicular Lines (part 2) Finding Parallel or Perpendicular Lines 1. Find the slope of the original equation Write an equation in slope-intercept form of the line that passes through the given point & is parallel to the graph of each equation: 1. 5x + y = 2, (2,3) 2. Find the parallel/perpendicular slope 3. Use the point & new slope in point-slope form 4. Change to standard form 2. 3x 2y = 7, ( 3,1) Write an equation in slope-intercept form of the line that passes through the given point & is perpendicular to the graph of each equation: 3. 7x 2y = 3, (4, 1) 18

19 4. 2x + 5y = 3, (2, 3) 4-11 Fitting Equations to Data Steps to fitting equations to data. 1) Identify two points 2) Find slope 3) Use a point and slope to write a linear equation. Use point-slope formula. 4) Solve for the unknown by substitution. 1. To produce 50 copies of a school newspaper, the cost per paper is 26 cents. To produce 200 newspapers, the cost per paper is 20 cents. Let n be the number of copies of a school newspaper, and let c be the cost per paper. Assume that a linear relationship fits these data with ordered pairs (n, c) (1) Find the linear equation that fits these data. (2) Use the linear equation to predict what it would cost per paper to produce 300 copies. 19

20 (2) To find the cost per paper for 300 papers, substitute 300 for n and solve for c. 2. A college record in the 100-m dash in 1960(t) was 10.5 seconds(r). In 1990 the new record was 10.2 seconds. Assume a linear relationship fit these data with ordered pairs (t, r). (1) Find a linear equation to fit the data points. (2) Use the linear equation to predict the record in