Plotting Points in 2-dimensions. Graphing 2 variable equations. Stuff About Lines

Plotting Points in 2-dimensions Graphing 2 variable equations Stuff About Lines

Plotting Points in 2-dimensions

Plotting Points: 2-dimension Setup of the Cartesian Coordinate System: Draw 2 number lines: 1) The first number line will go left-to-right and is called the x-axis

Plotting Points: 2-dimension Setup of the Cartesian Coordinate System: Draw 2 number lines: 2) The second number line will go down-to-up and is called the y-axis

Plotting Points: 2-dimension Setup of the Cartesian Coordinate System: Draw 2 number lines Each number line has a 0 on it. Draw the number lines in such a way that the two 0 s cross. The point where the cross is called the ORIGIN

Plotting Points: 2-dimension Setup of the Cartesian Coordinate System: Draw 2 number lines Each number line has a 0 on it. Draw the number lines in such a way that the two 0 s cross. The point where the cross is called the ORIGIN Now any location in the Cartesian coordinate system plane can be described using a pair of numbers like (3, -2). This is called an ordered pair. The 1 st number is called the x-coordinate and the second number is called the y-coordinate

Plotting Points: 2-dimension Plotting Points in the Cartesian Coordinate System: 1) The x-coordinate tells you how to move left or right If the x-coordinate is POSITIVE, move to the RIGHT If the x-coordinate is NEGATIVE, move to the LEFT 2) The y-coordinate tells you how to move up or down If the y-coordinate is POSITIVE, move to the UP If the y-coordinate is NEGATIVE, move to the DOWN

Plotting Points: 2-dimension Plotting Points in the Cartesian Coordinate System: 3) Each time you plot a point, start at the origin 4) From the origin, move left or right by the amount the x-coordinate tells you 5) From the point in step (4), move up or down by the amount the y-coordinate tells you 6) Put a dot at the resulting spot and label it

Plotting Points: 2-dimension Ex 1: Plot the following points A(3, 5) B(-2, 4) C(0, -1) D(-4,-7) E(-3, 0) F(4, -2) Now do worksheet problem 1

Graphing 2 variable equations

Graphing 2 variable equations A 2-variable equation is a way of describing infinitely many points To graph the 2-variable equation, you need to plot all of the points that the equation is trying to describe (solutions)

Graphing 2 variable equations Ex 2a: List a few points whose x-coordinate is 7 2 variable equation x = 7 i.e. think of all points whose x-coordinate is 7 To graph x = 7 plot all points whose x-coordinate is 7

Graphing 2 variable equations Ex 2b: List a few points whose x-coordinate and y-coordinate are equal. 2 variable equation x = y i.e. think of all points whose x-coordinate and y-coordinate are equal To graph x = y plot all points whose x-coordinate and y-coordinate are equal

Graphing 2 variable equations Ex 2c: List a few points where the sum of the x-coordinate and y-coordinate is 10. 2 variable equation x + y = 10 i.e. think of all points where the sum of the x-coordinate and y-coordinate is 10 To graph x + y = 10 plot all points where the sum of the x-coordinate and y-coordinate is 10

Graphing 2 variable equations Ex 2d: List a few points where if you square the x-coordinate, then square the y-coordinate, then add the results, you get 25. 2 variable equation x 2 + y 2 = 25 i.e. think of all points where if you square the x-coordinate, then square the y-coordinate, then add the results, you get 25 To graph x 2 + y 2 = 25 plot all points where if you square the x-coordinate, then square the y-coordinate, then add the results, you get 25

Notes: Graphing 2 variable equations A 2-variable equation is a way of describing infinitely many points to you. A 2-variable equation has infinitely many solutions (and infinitely many non-solutions) Every point in the plane is either a solution or nonsolution to the equation To graph an equation in 2 variables, you have to plot all of the solutions

What Does It Mean To Graph An Equation In 2 Variables (x and y)

Ex: Graph the equation 2x+y=5 After a Million More Points Solutions (1, 3) (4, -3) (-2, 9) (4.5, -4) (4.25, -3.5) Non-solutions (0, 3) (1, 1) (2, 7) (-1, 4) (6, -2) (7, 3) (2, -4) (-3, -5)

Q: What does it mean to graph an equation in 2 variables? A: The minute you see an equation in 2 variables, the points in the plane get separated into 2 groups: Solutions & Non-Solutions. To graph an equation in 2 variables, you have to plot ALL of the solutions.

Stuff About Lines

Q: What must an equation in 2 variables look like for its graph to be a straight line? A: You must be able to rewrite the equation in the form y equals a number times x plus a number or x equals a number i.e. y=mx+b where m and b are numbers or x=a where a is a number Examples of equations of lines:

Steps in graphing lines (using 2 points): 1) Recognize that the equation you are graphing is the equation of a straight line (i.e. that it is or can be written in the correct form) 2) Find two points on the line (i.e. find 2 solutions to the given equation) 3) Plot the 2 points you found in step 2 4) Connect the points you found in step 2 with a line Ex 3: Graph the following equations a) y=5x-7 b) 2x-5y=20 c) y=4 d) x=-2 Now do #5 on your worksheet

Slopes of Lines Definition: The slope of a line is a number that tells you how steep the line it. The symbol for slope is m. m is positive m is negative m = 0 m is no slope, undefined slope, or infinite slope

Slopes of Lines Question: How do you find the slope of a line? Answer: If you are given the equation of the line Solve the equation for y The number in front of the x is the slope If the line equation is x = a number, the slope is no slope. Ex 4: Find the slopes of the following lines a) y = 5x 7 b) 2x 5y = 20 c) y = 4 d) x = 2 Now do #6 on your worksheet

Slopes of Lines Question: How do you find the slope of a line? Answer: If you are know 2 points on the line, you can find the slope by using the equation m = y 2 y 1 x 2 x 1 Ex 4: Find the slope of the line that passes through e) (4, -1) and (-2, 3) f) 4, 7 and (4, 3) g) (2, -5) and (0, -5) Now do #7 on your worksheet

x- and y- Intercepts of Lines Definition: The x-intercept of a line is the point where the line crosses the x-axis (if the line crosses the x-axis) The y-intercept of a line is the point where the line crosses the y-axis (if the line crosses the y-axis)

x- and y- Intercepts of Lines Question: How do you find the intercepts of a line? Answer: If the equation is written in the form y = mx + b, the y- intercept is the point (0, b) To find the x-intercept, set y equal to 0 in the equation of the line and solve for x. Write your answer as a point. The x-intercept will have y-coordinate 0. To find the y-intercept, set x equal to 0 in the equation of the line and solve for y. Write your answer as a point. The y-intercept will have x-coordinate 0.

x- and y- Intercepts of Lines Question: How do you find the intercepts of a line? Ex 5: Find the x- and y-intercepts of the following lines... a) y = 5x 7 b) 2x 5y = 20 c) y = 4 d) x = 2 Now do #8 on your worksheet

Finding Equations of Lines Question: How do you find the equation of a line? Answer: You will need to know the slope of the line and a point on the line. Then you can use either of the 2 equations below y = mx + b y y 1 = m(x x 1 )

Finding Equations of Lines Question: How do you find the equation of a line? Ex 6: a) Find the equation of the line that has slope 7 and passes through the point (0, 2) b) Find the equation of the line that has slope -3 and passes through the point (4, -1) c) Find the equation of the line that passes through the points (5, -1) and (2, -2) d) Find the equation of the line that passes through the points (5, 3) and (5, 7) e) Find the equation of the horizontal line that passes through the point (2, 4) Now do #9 on your worksheet