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Analyzing Linear Equations: Rate of Change & Slope

Learn about rate of change, slope, direct variation, and how to write equations in slope-intercept and point-slope forms.

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Analyzing Linear Equations: Rate of Change & Slope

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  1. Chapter 4 Analyzing Linear Equations

  2. 4.1 Rate of Change & Slope • Rate of Change- a ratio that describes, on average, how much one quantity changes compared to another Change in y Rate of change= Change in x x y Ex: Negative slope Positive slope +1 4 +4 = 4 ft/sec Rate of change = +1 +4 1 Undefined slope +1 +4 Zero slope

  3. rise Slope: m = y2 – y1 • Slope- the ratio of the change in y-coordinates over the change in x-coordinates for a line Slope= run x2 – x1 Find the slope (4, 6) (-1, 2) Ex: Ex: x1 y1 x2 y2 Ex: Find the value of r so that the line through (10, r) and (3, 4) has a slope of -2/7 x1 y1 x2 y2 -28 -28 Simplify and solve for r r = 2 -14= -7r Plug in the m and the coordinates for the slope /-7 /-7

  4. 4.2 Slope and Direct Variation • Direct Variation- a proportional relationship y = kx *this represents a constant rate of change and k is the constant of variation Ex: a. Name the constant of variation c. Compare slope and the constant of variation. What do you notice? k = -1/2 They are the same thing b. Find the slope

  5. Graph a direct Variation: • Write the slope as a ratio • Start at the origin (0, 0) • Move up and across according to the slope • Draw a line to connect the ordered pairs a. y = 4x b. y = -1/3x Up 4 and right 1 Down 1 and right 3

  6. 4.3 Slope-Intercept Form y = mx + b slope y-intercept Ex: Write an equation in slope-intercept form given slope and the y-intercept b. y-intercept= (0, 6) m = 2/5 a. Slope = 3 y-intercept = -5 y = 3x - 5 y = 2/5x + 6

  7. Graph from an equation: • write the slope as a ratio and y-intercept as an ordered pair • Graph the y-intercept • Use slope to find other points and connect with a line Ex: Write an equation in slope-intercept form given a graph. Graph an equation given in slope-intercept form Ex: y-intercept= (0, 2) So b=2 a. y = -2/3x + 1 m = -2/3 y-int= (0, 1) y = 5/3x + 2 b. 5x – 3y = 6 5x – 3y = 6 -5x -5x -3y = -5x + 6 /-3 /-3 y = 5/3x - 2 m = 5/3 y-int= (0, -2)

  8. 6.7 Graphing Inequalities with Two Variables • The equation makes the line to define the boundary • The shaded region is the half-plane • Get the equation into slope-intercept form • List the intercept as an ordered-pair and the slope as a ratio • Graph the intercept and use the slope to find at least 2 more points • Draw the line (dotted or solid) • Test an ordered-pair not on the line • If it is true shade that side of the line • If it is false shade the other side of the line

  9. < or > or Dotted Line Solid Line Ex1: y 2x - 3 m = b = -3 = (0, -3) Use a solid line because it is Test: (0, 0) 0 2(0) – 3 0 0 – 3 0 -3 false (shade other side)

  10. Ex2: y – 2x < 4 y – 2x < 4 + 2x +2x y < 2x + 4 m = b = 4 = (0, 4) Use a dotted line because it is < Test: (0, 0) 0 < 2(0) + 4 0 < 0 + 4 0 < 4 true (shade this side)

  11. Ex3: 3y - 2 > -x + 7 3y – 2 > -x + 7 +2 +2 3y > -x + 9 /3 /3 /3 y > - x + 3 m = - b = 3 = (0, 3) Use a dotted line because it is > Test: (0, 0) 0 > - (0) + 3 0 > 0 + 3 0 > 3 false (shade other side)

  12. 4.4 Writing Equations in Slope-Intercept Form y = mx + b slope y-intercept Write an equation given slope and one point Ex. Write the equation of the line that passes through (1, 5) with slope 2 y = mx + b y = 2x + b Replace m with the slope 5 = 2(1) + b Replace the x and y with the ordered pair coordinates 5 = 2 + b Solve for b (the y-intercept) -2 -2 3 = b Replace the numbers for slope and the y-intercept y = 2x + 3

  13. Write an equation given two points Ex. Write an equation for the line that passes through (-3, -1) and (6, -4) y = mx + b Find slope y = -1/3x + b Replace m with the slope -1 = -1/3(-3) + b Replace the x and y with one of the ordered pair coordinates -1 = 1 + b Solve for b (the y-intercept) -1 -1 -2 = b Replace the numbers for slope and the y-intercept y = -1/3x - 2

  14. 4.5 Writing Equations in Point-Slope Form Point-Slope Form: y – y1 = m(x – x1) Ex: Write an equation given slope and one point Write the equation of a line that passes through (6, -2) with slope 5 y – y1 = m(x – x1) y – -2= 5(x – 6) Replace m with the slope and y1 and x1 with the ordered pair y + 2 = 5(x – 6) Simplify Ex: Write an equation of a horizontal line Write the equation for the line that passes through (3, 2) and is horizontal y – y1 = m(x – x1) y – 2 = 0(x – 3) Replace m with the slope and y1 and x1 with the ordered pair y – 2 =0 Simplify y = 2

  15. Ex: Write an equation in Standard Form Ex: Write an equation in Slope-Intercept Form y + 5= -5/4(x -2) y -2= ½(x+ 5) 4[y + 5= -5/4(x -2)] Multiply by 4 2[y -2= ½(x+ 5)] Multiply by 2 Distribute 4y + 20= -5(x -2) 2y -4= x+ 5 Add 4(move it to the other side) 4y + 20= -5x + 10 +4= +4 Add 5x(move it to the other side) +5x +5x 2y = x+ 9 Divide by 2 5x + 4y + 20= 10 /2 /2 Subtract 20(move it to the other side) y = 1/2x+ 9/2 Simplify - 20 -20 5x + 4y = -10 Simplify Ex: Write an equation in Point-Slope Form given two points Write the equation for the line that passes through (2, 1) (6, 4) Find slope y - 1= 3/4(x -2) Replace m with slope and y1 and x1 with one of the ordered pairs or y - 4= 3/4(x -6)

  16. 4.6 Statistics: Scatter Plots and Lines of Fit • Scatter Plot- a graph in which two sets of data are plotted as ordered pairs in a coordinate plane • Used to investigate a relationship between two quantities Positive Correlation Negative Correlation No Correlation

  17. If the data points do not lie in a line, but are close to making a line you can draw a Line of Fit • This line describes the trend of the data (once you have this line you can use ordered pairs from it to write an equation) Ex: a. Make a scatter plot b. Draw a line of fit. What correlation do you find? Positive correlation 500 c. Write an equation in slope-intercept form for the line 400 300 (8, 230) (12, 306) Find slope 200 m= 19 100 y = 19x +78 Plug in an ordered pair and slope to find the y-intercept 1 3 5 7 9 11 13 15

  18. Parallel lines- do not intersect and have the same slope Perpendicular lines- make right angles and have opposite slopes 4.7 Geometry : Parallel and Perpendicular Lines c a Line a has slope 6/5 Line b has slope 6/5 b d Line c has slope 3/2 Line d has slope -2/3

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