LINEAR EQUATIONS PART I Basic Coordinate Plane Info Review on Plotting Points Finding Slopes x and y intercepts Slope-Intercept Form of a Line Graphing Lines Determine the equation of a line given two points, slope and one point, or a graph. Assignments

COORDINATE PLANE QUAD II QUAD I QUAD III QUAD IV Y-axis Parts of a plane X-axis Y-axis Origin Quadrants I-IV QUAD II QUAD I Origin ( 0 , 0 ) X-axis 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) B C A D Plot these 4 points A (3, -4), B (5, 6), C (-4, 5) and D (-7, -5)

Graphing Linear Equations In Slope-Intercept Form

We have already seen that linear equations have two variables and when we plot all the (x,y) pairs that make the equation true we get a line. In this section, instead of making a table, evaluating y for each x, plotting the points and making a line, we will use The Slope-Intercept Form of the equation to graph the line.

Notice that these equations are all solved for y. These equations are all in Slope-Intercept Form: Notice that these equations are all solved for y.

Just by looking at an equation in this form, we can draw the line (no tables). The constant is the y-intercept. The coefficient is the slope. Constant = 1, y-intercept = 1. Coefficient = 2, slope = 2. Constant = -4, y-intercept = -4. Coefficient = -1, slope = -1. Constant = -2, y-intercept = -2. Coefficient = 3/2, slope = 3/2.

The formula for Slope-Intercept Form is: ‘b’ is the y-intercept. ‘m’ is the slope. On the next three slides we will graph the three equations: using their y-intercepts and slopes.

1) Plot the y-intercept as a point on the y-axis 1) Plot the y-intercept as a point on the y-axis. The constant, b = 1, so the y-intercept = 1. right 1 up 2 right 1 up 2 2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = 2, so the slope = 2/1.

1) Plot the y-intercept as a point on the y-axis 1) Plot the y-intercept as a point on the y-axis. The constant, b = -4, so the y-intercept = -4. 2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = -1, so the slope = -1/1. down 1 down 1 right 1 right 1

1) Plot the y-intercept as a point on the y-axis 1) Plot the y-intercept as a point on the y-axis. The constant, b = -2, so the y-intercept = -2. right 2 up 3 right 2 2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = 3/2, so the slope = 3/2. up 3

Sometimes we must solve the equation for y before we can graph it. The constant, b = 3 is the y-intercept. The coefficient, m = -2 is the slope.

1) Plot the y-intercept as a point on the y-axis 1) Plot the y-intercept as a point on the y-axis. The constant, b = 3, so the y-intercept = 3. down 2 2) Plot more points by counting the slope up the numerator (down if negative) and right the denominator. The coefficient, m = -2, so the slope = -2/1. right 1 down 2 right 1

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. Run is 6 because we went to the right Slope triangle between two points. Notice that the slope triangle can be drawn two different ways. Rise is 10 because we went up Rise is -10 because we went down Run is -6 because we went to the left Another way to find slope

FORMULA FOR FINDING SLOPE The formula is used when you know two points of a line. EXAMPLE

Find the slope of the line between the two points (-4, 8) and (10, -4) If it helps label the points. Then use the formula

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

NOT IN SLOPE-INTERCEPT y = 3x – 5 y – x = 10 y = -2x + 10 2y – 8 = 6x y = -.5x – 2 y + 4 = 2x 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.

BY MAKING A TABLE OR USING THE SLOPE-INTERCEPT FORM 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 INPUT (X) OUTPUT (Y) -2 -4 2 1 5 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.

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 Steps Make sure the equation is in slope-intercept form. Identify the slope and y-intercept. Plot the y-intercept. From the y-intercept use the slope to get another point to draw the line. y = 3x + 2 Slope = 3 (note that this means the fraction or rise over run could be (3/1) or (-3/-1). The y-intercept is 2. Plot (0, 2) From the y-intercept, we are going rise 3 and run 1 since the slope was 3/1.

FIND EQUATION OF A LINE GIVEN 2 POINTS Find the equation of the line between (2, 5) and (-2, -3). Find the slope between the two points. Plug in the slope in the slope-intercept form. Pick one of the given points and plug in numbers for x and y. Solve and find b. Rewrite final form. Slope is 2. y = 2x + b Picked (2, 5) so (5) = 2(2) + b b = 1 y = 2x + 1 Two other ways

Steps if given the slope and a point on the line. Substitute the slope into the slope-intercept form. Use the point to plug in for x and y. Find b. Rewrite equation. 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.

Assignments Pages 206-208 #’s 1-31, 33-39