Functions in the Coordinate Plane Linear Functions

Slope The rate of change

Methods to Determine Slope From graph, count

Methods to Determine Slope Rise Run From graph, count

Methods to Determine Slope Rise Run From graph, count If it rises from left to right, it is positive

Methods to Determine Slope Rise Run From graph, count If it rises from left to right, it is positive If it goes down from left to right, it is negative

Methods to Determine Slope Rise Run From graph, count Run Rise

Methods to Determine Slope From a table of data, find the average.

Methods to Determine Slope From a table of data, find the average. y or change in y x change in x

Methods to Determine Slope From two points, find

Methods to Determine Slope From two points, find y or y2 – y1 x x2 – x1

Methods to Determine Slope From an equation y = mx + b, the slope is the m value

Where the line intersects the x- axis or the y-axis Intercepts Where the line intersects the x- axis or the y-axis Y axis X axis

y-intercept (0,y) Where the line hits the y axis (0,y)

The y-intercept is the b value y-intercept (0,y) Where the line hits the y axis From the equation: y = mx + b The y-intercept is the b value (0,y)

The y-intercept is the b value y-intercept (0,y) Where the line hits the y axis From the equation: y = mx + b The y-intercept is the b value (0,y) 3 2 1 From the equation: y = x+3 The y-intercept is at 3

x-intercept (x,0) Where the line hits the x axis (x,0)

Plug in 0 for y and solve for x x-intercept (x,0) Where the line hits the x axis From the equation: y = mx + b Plug in 0 for y and solve for x (x,0)

Plug in 0 for y and solve for x x-intercept (x,0) Where the line hits the x axis From the equation: y = mx + b Plug in 0 for y and solve for x (x,0) From the equation: y = x + 2 0 = x + 2 1 2

x-intercept (x,0) x-intercept (x,0) (x,0) (x,0) From the equation: Where the line hits the x axis Where the line hits the x axis From the equation: y = mx + b Plug in 0 for y and solve for x From the equation: y = mx + b Plug in 0 for y and solve for x (x,0) (x,0) From the equation: y = x + 2 0 = x + 2 -2 -2 From the equation: y = x + 2 0 = x + 2 1 2

x-intercept (x,0) x-intercept (x,0) (x,0) (x,0) From the equation: Where the line hits the x axis Where the line hits the x axis From the equation: y = mx + b Plug in 0 for y and solve for x From the equation: y = mx + b Plug in 0 for y and solve for x (x,0) (x,0) From the equation: y = x + 2 0 = x + 2 -2 -2 -2 = x From the equation: y = x + 2 0 = x + 2 1 2

Example 1 For the given graph: A. set up a table of values for at least four points B. find the slope C. determine the equation of the line D. identify the x- intercept and y intercept

Example 1 X Y For the given graph: A. set up a table of values for at least four points X Y

Example 1 X Y For the given graph: A. set up a table of values for at least four points X Y -1 -7 -5 1 -3 2

Example 1 Rise Run Pick two points For the given graph: B. find the slope Pick two points

Example 1 Rise Run Pick two points Rise: up or down: COUNT Up + Down - For the given graph: B. find the slope Pick two points Rise: up or down: COUNT Up + Down - Rise =

Example 1 Rise Run Pick two points Rise: up or down: COUNT Up + Down - For the given graph: B. find the slope Pick two points Rise: up or down: COUNT Up + Down - Rise = 2 Run: right or left Right + Left – Run = 2

Example 1 Rise Run Pick two points Rise: up or down: COUNT Up + Down - For the given graph: B. find the slope 1 Pick two points Rise: up or down: COUNT Up + Down - Rise = 2 Run: right or left Right + Left – Run = 1 2

Example 1 Rise Run Rise Run 2 1 Slope = = = 2 Pick two points For the given graph: B. find the slope 1 Pick two points Rise: up or down: COUNT Up + Down - Rise = 2 Run: right or left Right + Left – Run = 1 2 Rise Run 2 1 Slope = = = 2

Example 1 Equation of a Line: y = mx + b m = slope = b = y int = For the given graph: C. determine the equation of the line Equation of a Line: y = mx + b m = slope = b = y int =

Example 1 Equation of a Line: y = mx + b m = slope = 2 For the given graph: C. determine the equation of the line Equation of a Line: y = mx + b m = slope = 2 b = y int = (0, )

Example 1 Equation of a Line: y = mx + b m = slope = 2 For the given graph: C. determine the equation of the line Equation of a Line: y = mx + b m = slope = 2 b = y int = (0, -5)

Example 1 Equation of a Line: y = mx + b m = slope = 2 For the given graph: C. determine the equation of the line Equation of a Line: y = mx + b m = slope = 2 b = y int = (0, -5) SO, y = 2x -5

Example 1 y int = x int = For the given graph: D. identify the x- intercept and y intercept y int = x int =

Example 1 y int = (0, -5) x int = For the given graph: D. identify the x- intercept and y intercept y int = (0, -5) x int =

x int = (x, 0) since we have our equation, plug in 0 for y Example 1 For the given graph: D. identify the x- intercept and y intercept y int = (0, -5) x int = x int = (x, 0) since we have our equation, plug in 0 for y

Example 1 y int = (0, -5) x int = y = 2x -5 For the given graph: D. identify the x- intercept and y intercept y int = (0, -5) x int = x int = (x, 0) since we have our equation, plug in 0 for y y = 2x -5

Example 1 y int = (0, -5) x int = y = 2x -5 0 = 2x-5 For the given graph: D. identify the x- intercept and y intercept y int = (0, -5) x int = x int = (x, 0) since we have our equation, plug in 0 for y y = 2x -5 0 = 2x-5

Example 1 y int = (0, -5) x int = y = 2x -5 0 = 2x-5 5=2x For the given graph: D. identify the x- intercept and y intercept y int = (0, -5) x int = x int = (x, 0) since we have our equation, plug in 0 for y y = 2x -5 0 = 2x-5 5=2x

Example 1 y int = (0, -5) x int = y = 2x -5 0 = 2x-5 5=2x For the given graph: D. identify the x- intercept and y intercept y int = (0, -5) x int = x int = (x, 0) since we have our equation, plug in 0 for y y = 2x -5 0 = 2x-5 5=2x X = 5/2 = 2.5

Example 1 y int = (0, -5) x int = (2.5, 0) y = 2x -5 0 = 2x-5 5=2x For the given graph: D. identify the x- intercept and y intercept y int = (0, -5) x int = (2.5, 0) x int = (x, 0) since we have our equation, plug in 0 for y y = 2x -5 0 = 2x-5 5=2x X = 5/2 = 2.5

Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: A. find the slope B. determine the equation of the line C. Graph the function D. identify the x- intercept and y intercept

y or change in y x change in x y2 – y1 x2 – x1 = Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: A. find the slope y or change in y x change in x y2 – y1 x2 – x1 = Choose two points:

y or change in y x change in x y2 – y1 x2 – x1 = Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: A. find the slope y or change in y x change in x y2 – y1 x2 – x1 = (x2, y2) (x1, y1) Choose two points: (1,4) and (4,-5), plug into equation

y or change in y x change in x y2 – y1 x2 – x1 = Example 2 y2 – y1 -2 13 1 4 -5 8 -17 For the given table of values: A. find the slope y or change in y x change in x y2 – y1 x2 – x1 = Choose two points: (1,4) and (4,-5), plug into equation y2 – y1 x2 – x1 -5 – 4 4 – 1 =

y or change in y x change in x y2 – y1 x2 – x1 = Example 2 y2 – y1 -2 13 1 4 -5 8 -17 For the given table of values: A. find the slope y or change in y x change in x y2 – y1 x2 – x1 = Choose two points: (1,4) and (4,-5), plug into equation y2 – y1 x2 – x1 -5 – 4 4 – 1 -9 3 = =

y or change in y x change in x y2 – y1 x2 – x1 = Example 2 y2 – y1 -2 13 1 4 -5 8 -17 For the given table of values: A. find the slope y or change in y x change in x y2 – y1 x2 – x1 = Choose two points: (1,4) and (4,-5), plug into equation y2 – y1 x2 – x1 -5 – 4 4 – 1 -9 3 = = = -3

Example 2 y– y1 = m ( x – x1) x y -2 13 1 4 -5 8 -17 For the given table of values: B. determine the equation of the line y– y1 = m ( x – x1)

Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) -2 13 1 4 -5 8 -17 For the given table of values: B. determine the equation of the line When we have our slope, we plug in one point into x1 and y1 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) y – 4 = -3 ( x – 1)

Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) -2 13 1 4 -5 8 -17 For the given table of values: B. determine the equation of the line Now we solve for y y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) y – 4 = -3 ( x – 1)

Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) -2 13 1 4 -5 8 -17 For the given table of values: B. determine the equation of the line Now we solve for y y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) y – 4 = -3 ( x – 1) y – 4 = -3x +3

Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) -2 13 1 4 -5 8 -17 For the given table of values: B. determine the equation of the line Now we solve for y y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) y – 4 = -3 ( x – 1) y – 4 = -3x +3 y = -3x +3 +4

Example 2 y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) -2 13 1 4 -5 8 -17 For the given table of values: B. determine the equation of the line Now we solve for y y– y1 = m ( x – x1) y – y1 = -3 ( x – x1) y – 4 = -3 ( x – 1) y – 4 = -3x +3 y = -3x +7

Plot your points Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: C. Graph the function x y -2 13 1 4 -5 8 -17 Plot your points

Plot your points Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: C. Graph the function x y -2 13 1 4 -5 8 -17 Plot your points

Connect Points Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: C. Graph the function x y -2 13 1 4 -5 8 -17 Connect Points

Connect Points Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: C. Graph the function x y -2 13 1 4 -5 8 -17 Connect Points

Example 2 y int = x int = x y -2 13 1 4 -5 8 -17 For the given table of values: D. identify the x- intercept and y intercept y int = x int =

Since we have our equation, y int = b Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: D. identify the x- intercept and y intercept y int = x int = y = -3x +7 Since we have our equation, y int = b

We plug in 0 for y to solve for the x int Example 2 x y -2 13 1 4 -5 8 -17 For the given table of values: D. identify the x- intercept and y intercept y int = (0,7) x int = y = -3x +7 We plug in 0 for y to solve for the x int

Example 2 y int = (0,7) x int = y = -3x +7 0 = -3x +7 x y -2 13 1 4 -5 8 -17 For the given table of values: D. identify the x- intercept and y intercept y int = (0,7) x int = y = -3x +7 0 = -3x +7

Example 2 y int = (0,7) x int = y = -3x +7 0 = -3x +7 3x = 7 x y -2 13 4 -5 8 -17 For the given table of values: D. identify the x- intercept and y intercept y int = (0,7) x int = y = -3x +7 0 = -3x +7 3x = 7

Example 2 y int = (0,7) x int = y = -3x +7 0 = -3x +7 3x = 7 x = 7/3 x -2 13 1 4 -5 8 -17 For the given table of values: D. identify the x- intercept and y intercept y int = (0,7) x int = y = -3x +7 0 = -3x +7 3x = 7 x = 7/3

Example 2 y int = (0,7) x int = (7/3, 0) y = -3x +7 0 = -3x +7 3x = 7 -2 13 1 4 -5 8 -17 For the given table of values: D. identify the x- intercept and y intercept y int = (0,7) x int = (7/3, 0) y = -3x +7 0 = -3x +7 3x = 7 x = 7/3

Example 3 For the given function: y = -2/3x +4 A. set up a table of values B. graph the function C. find the slope D. identify the x- intercept and y intercept

Example 3 For the given function: y = -2/3x +4 A. set up a table of values x y Plug in values for x into the equations to find their corresponding y value

Example 3 For the given function: y = -2/3x +4 A. set up a table of values x y -3 6 4 3 2 Plug in values for x into the equations to find their corresponding y value

Example 3 For the given function: y = -2/3x +4 B. graph the function

Example 3 For the given function: y = -2/3x +4 B. graph the function You have the equation, y=mx+b, so m is the slope and b is the y int. Plot your y int and count your slope

Example 3 For the given function: y = -2/3x +4 B. graph the function You have the equation, y=mx+b, so m is the slope and b is the y int. Plot your y int and count your slope

Example 3 For the given function: y = -2/3x +4 B. graph the function You have the equation, y=mx+b, so m is the slope and b is the y int. Plot your y int and count your slope Note: you could also plot your points from the table

Example 3 Connect the points For the given function: y = -2/3x +4 B. graph the function Connect the points

Example 3 For the given function: y = -2/3x +4 C. find the slope

You have your equation, so your slope is the m value in y=mx+b. Example 3 For the given function: y = -2/3x +4 C. find the slope You have your equation, so your slope is the m value in y=mx+b. What is the slope?

You have your equation, so your slope is the m value in y=mx+b. Example 3 For the given function: y = -2/3x +4 C. find the slope You have your equation, so your slope is the m value in y=mx+b. What is the slope? m=-2/3

Example 3 For the given function: y = -2/3x +4 D. identify the x- intercept and y intercept

Example 3 For the given function: y = -2/3x +4 D. identify the x- intercept and y intercept We have the equation where b in y=mx+b is the y int. What is the y int?

Example 3 For the given function: y = -2/3x +4 D. identify the x- intercept and y intercept We have the equation where b in y=mx+b is the y int. What is the y int? (0 , 4) For the x int, check your table and graph. What is the x int?

Example 3 For the given function: y = -2/3x +4 D. identify the x- intercept and y intercept We have the equation where b in y=mx+b is the y int. What is the y int? (0 , 4) For the x int, check your table and graph. What is the x int? (6 , 0)

Slope Intercept Form y = mx +b Standard Form Ax + By = C Given information, find the equation in slope intercept form and in standard form Slope Intercept Form y = mx +b Standard Form Ax + By = C

Example 4: find equation in slope intercept form and in standard form Contains point (-2, 5)

Example 4: find equation in slope intercept form and in standard form Contains point (-2, 5) Plug in: y– y1 = m ( x – x1)

Example 4: find equation in slope intercept form and in standard form Contains point (-2, 5) Plug in: y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2)

Example 4: find equation in slope intercept form and in standard form Contains point (-2, 5) Plug in: y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1

Example 4: find equation in slope intercept form and in standard form Contains point (-2, 5) Plug in: y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1 y = (-1/2)x + 4

Example 4: find equation in slope intercept form and in standard form Contains point (-2, 5) Plug in: y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1 y = (-1/2)x + 4 Slope Intercept

Example 4: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1 y = (-1/2)x + 4 Slope Intercept Slope = -1/2 Contains point (-2, 5) For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive y=(-1/2)x + 4

Example 4: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1 y = (-1/2)x + 4 Slope Intercept Slope = -1/2 Contains point (-2, 5) For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive 2[y=(-1/2)x + 4]

Example 4: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1 y = (-1/2)x + 4 Slope Intercept Slope = -1/2 Contains point (-2, 5) For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive 2[y=(-1/2)x + 4] 2y = -x +8

Example 4: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1 y = (-1/2)x + 4 Slope Intercept Slope = -1/2 Contains point (-2, 5) For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive 2[y=(-1/2)x + 4] 2y = -x +8 x + 2y = 8

Example 4: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 5 = (-1/2) (x +2) y – 5 = (-1/2)x -1 y = (-1/2)x + 4 Slope Intercept Slope = -1/2 Contains point (-2, 5) For standard form: get rid of fractions, get x and y on one side, make sure x’s coefficient is positive 2[y=(-1/2)x + 4] 2y = -x +8 x + 2y = 8 Standard Form

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1)

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 =

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 = =

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 -1 2 = = =

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 -1 2 = = = Now plug in: y– y1 = m ( x – x1)

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 -1 2 = = = Now plug in: y– y1 = m ( x – x1) y – 1 = (-1/2) (x - -6)

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 -1 2 = = = Now plug in: y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6)

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 -1 2 = = = Now plug in: y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x - 3

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 -1 2 = = = Now plug in: y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x – 3 y = (-1/2)x - 2

Example 5: find equation in slope intercept form and in standard form Contains points (2, -3) and (-6, 1) First find the slope: y2 – y1 x2 – x1 1 – -3 -6– 2 4 -8 -1 2 = = = Now plug in: y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x – 3 y = (-1/2)x – 2 Slope Intercept Form

Example 5: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x – 3 y = (-1/2)x – 2 Slope Intercept Form Contains points (2, -3) and (-6, 1) Now rearrange into standard form y = (-1/2)x – 2

Example 5: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x – 3 y = (-1/2)x – 2 Slope Intercept Form Contains points (2, -3) and (-6, 1) Now rearrange into standard form y = (-1/2)x – 2 2[y = (-1/2)x – 2]

Example 5: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x – 3 y = (-1/2)x – 2 Slope Intercept Form Contains points (2, -3) and (-6, 1) Now rearrange into standard form y = (-1/2)x – 2 2[y = (-1/2)x – 2] 2y = -x - 4

Example 5: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x – 3 y = (-1/2)x – 2 Slope Intercept Form Contains points (2, -3) and (-6, 1) Now rearrange into standard form y = (-1/2)x – 2 2[y = (-1/2)x – 2] 2y = -x – 4 x + 2y = -4

Example 5: find equation in slope intercept form and in standard form y– y1 = m ( x – x1) y – 1 = (-1/2) (x + 6) y – 1 = (-1/2)x – 3 y = (-1/2)x – 2 Slope Intercept Form Contains points (2, -3) and (-6, 1) Now rearrange into standard form y = (-1/2)x – 2 2[y = (-1/2)x – 2] 2y = -x – 4 x + 2y = -4 Standard Form