1. Today we will explore the Essential Question, “What is the method for graphing a linear equation in standard form form using the slope. the y-intercept and the x- intercepts?” A linear equation in two variables is expressed in standard form Ax + By = C, where A, B, and C are real numbers, and A and B are not both zero. If the equation is expressed in this form, the slope is, y-intercept when x = 0 is and the x-intercept when y = 0 is. Standard form of an Equation of a Line: Ax + By = C Where slope =, y-intercept =, x-intercept =.

2. The y-intercept of a line is the point where the line intersects the y -axis. Every point on the y-axis has an x-coordinate of 0. For example, if the y -intercept of a line is 3, then the line intersects the y -axis at the point (0,3). The line shown below has a y -intercept of 3. x y 2 4 6 8 10 -6 -8 -10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2

3. The x-intercept of a line is the point where the line intersects the x -axis. Every point on the x-axis has a y-coordinate of 0. For example, if the x -intercept of a line is 1, then the line intersects the x -axis at the point (1,0). The line shown below has a x-intercept of 1. x y 2 4 6 8 10 -6 -8 -10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 (1, 0)

4. Given the linear equation in standard form 2x – 3y = 12, find the slope, y- intercept and x-intercept. Also graph the function. Lets use the x-intercept and y-intercept to graph the linear equation. The x-intercept is (6, 0) and the y-intercept is (0, -4) x y 2 4 6 8 10 -6 -8 -10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 If the equation is expressed in this form, the slope is, y-intercept when x = 0 is and the x-intercept when y = 0 is. The slope =, then, the y-intercept is,then, and the x-intercept =, then (6, 0) (0, -4)

5. x y 2 4 6 8 10 -6 -8 -10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 The line shown in the graph has a y- intercept of (0, 8) and a x-intercept of (-4, 0) Once we have the two intercepts lets find the slope of the line. Recall that the slope-intercept form of the equation of a line is where m denotes the slope and b denotes the y-intercept. We know that m = 2 and b = 8.2 8 NOTE: There is an interactive website for graphing a line using the slope and the y-intercept at http://www.shodor.org/interactivate/activities/SlopeSlider/. Substituting these values into the formula, we get the equation of the line Now we are going to write the equation in standard form. Lets get 8 by itself now, we can multiple both sides by -1, we get Since we have two points lets use the slope formula. m = =

6. Modeled Examples: Example 1: Use the slope and the y-intercept and x-intercept to write an equation in standard form which could represent the graph of the line shown in the diagram. x y 2 4 6 8 10 -6 -8 -10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 Solution: The x-intercept is (5, 0) and the y-intercept (0, 4) Since we have two points lets use the slope formula. m = = Substituting these into the formula y = mx + b, we get the equation of the line. Now we are going to write the equation in standard form. Lets get 4 by itself now, we can multiple both sides by 5, we get. Now we have to isolate the 20. we will add 4x to both sides,

7. Example 2. Given the linear equation in standard form 6x – 5y = 30, find the slope, y- intercept and x-intercept. Also graph the function. Lets use the x-intercept and y-intercept to graph the linear equation. The x-intercept is (5, 0) and the y-intercept is (0, -6) x y 2 4 6 8 10 -6 -8 -10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 If the equation is expressed in this form, the slope is, y-intercept when x = 0 is and the x-intercept when y = 0 is. The slope =, then, the y-intercept is,then, and the x-intercept =, then (5, 0) (0, -6)

8. Guided Practice Problems: 1. Roger went to a garage sale where hardback books sold for $5 each and paperback books sold for $2.50 each. He has $20 to spend. The equation below can be used to find how many books of each type Roger can buy, where x is the number of hardback books and y is the number of paperback books. 5x + 2.5y = 20 Find the x-intercept, y-intercepts and slope, then graph the equation If the equation is expressed in this form, the slope is, y-intercept when x = 0 is and the x-intercept when y = 0 is. The slope =, then, the y-intercept is,then, and the x-intercept =, then x y 2 4 6 8 10 -6 -8 -10 -4 -10 -8 -6 -4 -2 0 2 4 6 8 10 -2 (0, 8) (4, 0)

9. 1 2 -4 -3 -2 -1 1 2 3 4 Example 2. An engineer needs to determine the slope between two points on a gondola ride in order to evaluate the power requirements when the gondola is full of passengers. A coordinate grid has been placed over a diagram between the two points, as shown below. For estimation purposes, a straight line between the two points can be used to find the slope.

10. 3. Use the slope and the y-intercept to graph the line whose equation is. x y 1 2 3 4 5 -3-3 -4-4 -5 -2-2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 Solution: The equation of the line is written in the form as shown, so and. Since b denotes the y-intercept and, the line passes through the point. Plot that point on the graph. We can use the definition of slope to find another point on the line. First we must write the slope, m, as a ratio. How can -3 be written as a ratio? We can use or any other ratio which denotes -3. Begin at the y-intercept and move down 3and to the right 1.Plot that point. Draw the line through the two points plotted. This line is the graph of the given equation.

11. 4. Use the slope and the y-intercept to graph the line whose equation is. x y 1 2 3 4 5 -3-3 -4-4 -5 -2-2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 Solution: The equation of the line is written in the form as shown, so and. Since b denotes the y-intercept and, the line passes through the point. Plot that point on the graph. We can use the definition of slope to find another point on the line. This slope, m, is already written as a ratio. Begin at the y-intercept and move up 2and to the right 3.Plot that point. Draw the line through the two points plotted. This line is the graph of the given equation.

12. Independent Practice Problems: 1. Use the slope and the y-intercept to graph the line whose equation is. 2. Write the equation of the line whose graph is shown in slope-intercept form. x y -10 -8 -6 -4 -2 0 2 4 6 8 10 2 4 6 8 10 -6 -10 -2 -4 -8 x y 1 2 3 4 5 -3-3 -4-4 -5 -2-2 -5 -4 -3 -2 -1 0 1 2 3 4 5 -1 +3 +2