GRAPHING LINEAR EQUATIONS COMMON MISTAKES

Transcription

1 GRAPHING LINEAR EQUATIONS COMMON MISTAKES 1

2 Graphing-Coordinate System and Plotting Points How to Plot Points The grid containing the x and y axes is called the Cartesian Coordinate Plane. Points are plotted by using horizontal and vertical distances from the starting point called the origin (where the x and y axes intersecthas the coordinates (0,0) ). A point s coordinates are labeled (x, y) where x = distance right or left on the x-axis and y = distance up or down from the x-axis. To graph the point, start at the origin, (0,0), go the x distance on the x-axis and then from that location, go the y distance above or below your mark. Confusing the x- and y- distances/directions. Plotting A(, ), (-1, 5), C(-4, 0), D(, -) and E(5, -) would look like E C D A

3 Graphing-Understanding Slope How Slope affects the graph s direction Recall: Slope, m, relates the slant of a line m > 0 slant: upward m = 0 slant: horizontal m < 0 slant: downward m = undefined : vertical Equations use slope, m, in their formats (i.e. Slope- Intercept, and Point-Slope) Incorrectly identifying slope or graphing it. Not realizing slope, m, really is m = vertical change horizontal change Example 1: y = x + The slope is NOT x. Example : 8 x + y = 11 Solving for y finds the slope is 8.

4 Graphing-Slope (continued) Identifying Slope to Graph a Linear Function Slope has many definitions. y y 1 m = x x for points 1 m = rise = dy where the run dx d means change in the x distance or change in the y distances. Graphing requires at least one point and the slope or two points. Incorrectly identifying slope or graphing it. Not realizing slope, m, really is m = vertical change horizontal change Example 1: Identify the slope in Incorrect: The slope is x. Correct: The slope is. y = x Example : Find the slope: 8 x + y = 11 Incorrect: The slope is 8 Correct: Solving for y gives the slope, m, as 8 m = 4

5 Graphing-Slope (continued) Graphing a Linear Function using it s Slope The form y= mx + b, known as the Slope-Intercept Form, is easily graphed. Step 1: Start by solving the equation into y=mx+b form, where m= slope (put into fraction form and b=y-intercept.) Step : Plot (0, b) Step : Use the slope m to find another point.. m>0, m=0, m<0, or m is undefined. The numerator is the up/down (vertical distance) and the denominator directs the distance left/right (Always associate the negative numbers with the numerator). Step 4: Connect the points and complete the line. Incorrectly identifying slope or graphing it. Not realizing slope, m, really is m = vertical change horizontal change Example 1: Identify the slope in Incorrect: The slope is x. Correct: The slope is. y = x Example : Find the slope: 8 x + y = 11 Incorrect: The slope is 8 Correct: Solving for y gives the slope, m, as 8 m = 5

6 Writing the Equation of a Line How to Write the Equation of a Line Lines are written in three basic equation forms Slope-Intercept Form: y = mx +b Standard Form: Ax + y = C Point-Slope Form: y - y 1 = m(x- x 1 ) for some point ( x, y ) and slope 1 1 m. To Write the Equation of the line Step 1: Note the Form of the Equation. Step : Calculate the Slope (using either the definition or applying the given value). Step : Substitute values into the equation appropriately. Miscalculating slope by incorrectly substituting into the defined expression. Not correctly graphing the line given by the newly found equation. Example: Write the equation of the line, in slope-intercept form, for point (, -5) and m= 7. Solution: Using y=mx + b, we substitute m = 7 and calculate b. -5 = 7() + b -19 = b so y= 7x

7 Graphing Horizontal and Vertical Lines How to Graph Horizontal and Vertical Lines Horizontal Lines have the form y = a, with a slope of 0, or m = 0. Graphing Horizontal Lines: Step 1: Start at the origin (0, 0) and move up/ down to(0, a). Step : Plot (0, a) and draw a horizontal line through that point. Vertical Lines have the form x = b, with an undefined slope. Graphing Vertical Lines Step 1: Start at the origin (0, 0) and move right/left to (b, 0) Step : Plot (b, 0) and draw a vertical line through that point. Confusing which form is vertical and which one is horizontal. Graphs: y=5 and x= - x= - Slope is undefined y=5 Slope =0 7 Complete Manual:..\Linear Function Review.docx

8 Graphing Parallel and Perpendicular Lines How to Graph Parallel and Perpendicular Lines Parallel Lines are lines with the same slope. Perpendicular Lines intersect lines at 90 (Right Angles) and have slopes that are negative reciprocals. When graphing, use the techniques of plotting the y- intercept and then using the slope from y=mx + b, where m= slope and b= y-intercept (0,b) Forgetting or confusing the slope relationships between parallel and perpendicular Example: Write the equation of the lines parallel and perpendicular to the line y = x +1 through (, 4). Solution: The point has values x= and y=4. Parallel: same slope, m= /. Solving y=mx +b, where 4= () + b gives b = 1 so y= x + 1. Perpendicular: slopes that are negative reciprocals, m= -/ Solving y = mx + b, where 4= () +b gives b= 16 so y = x

9 Graphing- Plotting Points using the Standard Form of an Equation How to Graph an Equation in Standard Form Recall: Standard Form of a Linear Equation is Ax + y =C. A Solving for y gives: y = x + C When compared to y =mx + b, Slope m= A, y-intercept b = C. Graphing the line is done by first plotting the y-intercept, or b, as the coordinate (0, b); then use the slope to plot a second point. Connect the two points and complete the line. Incorrectly solving for slope. (Apply the formula: A C ). y = x + A= Example: Graph : x + 5y = 10 =5 C=10 Solution: Where m = and b =, 5 Plot (0, ) and then use the slope by going DOWN and RIGHT 5 to find a second point. Connect the line joining the points.. x+5y=10 9

10 Graphing-Applying the Y-Intercept Form How to calculate and make use of Intercepts The y-intercept (0, b) can be calculated by setting x=0 and solving the equation y=mx+b for y. The x-intercept (x, 0) can be calculated by letting y=0 and solving y = mx + b for x. Using intercepts to graph lines in the Standard Form Ax + y =C can be done easily by a pattern: C x-intercept: if y=0, then x =, y-intercept: if x =0, then y = C, Plot and connect the two Intercepts (x, 0) and (0,y); then complete the line. A Incorrectly plotting the intercepts when graphing: plotting a point(s) with the coordinates for x and y are backwards. Example: Find the intercepts that would be used to graph x y = 1 A=, =-, C=1 1 Incorrect: (0, 1 ) and (, 0) are NOT the intercepts. Correct: ( 1,0) and (0, 1 ). 10