LINEAR EQUATIONS <ul><li>Basic Coordinate Plane Info </li></ul><ul><li>Review on Plotting Points </li></ul><ul><li>Finding Slopes </li></ul><ul><li>x and y intercepts </li></ul><ul><li>Slope-Intercept Form of a Line </li></ul><ul><li>Graphing Lines </li></ul><ul><li>Determine the equation of a line given two points, slope and one point, or a graph. </li></ul>
COORDINATE PLANE <ul><li>Parts of a plane </li></ul><ul><li>X-axis </li></ul><ul><li>Y-axis </li></ul><ul><li>Origin </li></ul><ul><li>Quadrants I-IV </li></ul>X-axis Y-axis Origin ( 0 , 0 ) QUAD I QUAD II QUAD III QUAD IV
PLOTTING POINTS Remember when plotting points you always start at the origin. Next you go left (if x-coordinate is negative) or right (if x-coordinate is positive. Then you go up (if y-coordinate is positive) or down (if y-coordinate is negative) Plot these 4 points A (3, -4), B (5, 6), C (-4, 5) and D (-7, -5) A B C D
SLOPE Slope is the ratio of the vertical rise to the horizontal run between any two points on a line. Usually referred to as the rise over run. Slope triangle between two points. Notice that the slope triangle can be drawn two different ways. Rise is -10 because we went down Run is -6 because we went to the left Rise is 10 because we went up Run is 6 because we went to the right Another way to find slope
FORMULA FOR FINDING SLOPE The formula is used when you know two points of a line. EXAMPLE
X AND Y INTERCEPTS The x-intercept is the x-coordinate of a point where the graph crosses the x-axis. The y-intercept is the y-coordinate of a point where the graph crosses the y-axis. The x-intercept would be 4 and is located at the point (4, 0). The y-intercept is 3 and is located at the point (0, 3).
SLOPE-INTERCEPT FORM OF A LINE The slope intercept form of a line is y = mx + b, where “m” represents the slope of the line and “b” represents the y-intercept. When an equation is in slope-intercept form the “y” is always on one side by itself. It can not be more than one y either. If a line is not in slope-intercept form, then we must solve for “y” to get it there. Examples
Put y – x = 10 into slope-intercept form Add x to both sides and would get y = x + 10 Put 2y – 8 = 6x into slope-intercept form. Add 8 to both sides then divide by 2 and would get y = 3x + 4 Put y + 4 = 2x into slope-intercept form. Subtract 4 from both sides and would get y = 2x – 4. IN SLOPE-INTERCEPT NOT IN SLOPE-INTERCEPT y = 3x – 5 y – x = 10 y = -2x + 10 2y – 8 = 6x y = -.5x – 2 y + 4 = 2x
GRAPHING LINES BY MAKING A TABLE OR USING THE SLOPE-INTERCEPT FORM I could refer to the table method by input-output table or x-y table. For now I want you to include three values in your table. A negative number, zero, and a positive number. Graph y = 3x + 2 By making a table it gives me three points, in this case (-2, -4) (0, 2) and (1, 5) to plot and draw the line. See the graph. INPUT (X) OUTPUT (Y) -2 -4 0 2 1 5
Plot (-2, -4), (0, 2) and (1, 5) Then draw the line. Make sure your line covers the graph and has arrows on both ends. Be sure to use a ruler. Slope-intercept graphing
Slope-intercept graphing <ul><li>Steps </li></ul><ul><li>Make sure the equation is in slope-intercept form. </li></ul><ul><li>Identify the slope and y-intercept. </li></ul><ul><li>Plot the y-intercept. </li></ul><ul><li>From the y-intercept use the slope to get another point to draw the line. </li></ul><ul><li>y = 3x + 2 </li></ul><ul><li>Slope = 3 (note that this means the fraction or rise over run could be (3/1) or (-3/-1). The y-intercept is 2. </li></ul><ul><li>Plot (0, 2) </li></ul><ul><li>From the y-intercept, we are going rise 3 and run 1 since the slope was 3/1. </li></ul>
FIND EQUATION OF A LINE GIVEN 2 POINTS <ul><li>Find the slope between the two points. </li></ul><ul><li>Plug in the slope in the slope-intercept form. </li></ul><ul><li>Pick one of the given points and plug in numbers for x and y. </li></ul><ul><li>Solve and find b. </li></ul><ul><li>Rewrite final form. </li></ul>Find the equation of the line between (2, 5) and (-2, -3). <ul><li>Slope is 2. </li></ul><ul><li>y = 2x + b </li></ul><ul><li>Picked (2, 5) so (5) = 2(2) + b </li></ul><ul><li>b = 1 </li></ul><ul><li>y = 2x + 1 </li></ul>Two other ways
<ul><li>Steps if given the slope and a point on the line. </li></ul><ul><li>Substitute the slope into the slope-intercept form. </li></ul><ul><li>Use the point to plug in for x and y. </li></ul><ul><li>Find b. </li></ul><ul><li>Rewrite equation. </li></ul>If given a graph there are three ways. One way is to find two points on the line and use the first method we talked about. Another would be to find the slope and pick a point and use the second method. The third method would be to find the slope and y-intercept and plug it directly into y = mx + b.
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