m81350 Graphing with Intercepts By the end of this section, you will be able to: Identify the intercepts on a graph Find the intercepts from an equation of a line Graph a line using the intercepts Choose the most convenient method to graph a line 1a6e2f8c-0399-4192-91b9-bcb8064aeee2 Before you get started, take this readiness quiz. Solve: 3x+4y=−12 for x when y=0. If you missed this problem, review . −4 Is the point (0,−5) on the x-axis or y-axis? If you missed this problem, review . y-axis Which ordered pairs are solutions to the equation 2x−y=6?ⓐ (6,0)ⓑ (0,−6)ⓒ (4,−2). If you missed this problem, review . b

Every linear equation has a unique line that represents all the solutions of the equation. When graphing a line by plotting points, each person who graphs the line can choose any three points, so two people graphing the line might use different sets of points. At first glance, their two lines might appear different since they would have different points labeled. But if all the work was done correctly, the lines will be exactly the same line. One way to recognize that they are indeed the same line is to focus on where the line crosses the axes. Each of these points is called an intercept of the line. Each of the points at which a line crosses the x-axis and the y-axis is called an intercept of the line. Let’s look at the graph of the lines shown in .

First, notice where each of these lines crosses the x- axis: Figure: The line crosses the x-axis at: Ordered pair of this point 42 3 (3,0) 43 4 (4,0) 44 5 (5,0) 45 0 (0,0) Do you see a pattern?For each row, the y- coordinate of the point where the line crosses the x- axis is zero. The point where the line crosses the x- axis has the form (a,0); and is called the x-intercept of the line. The x- intercept occurs when y is zero.Now, let's look at the points where these lines cross the y-axis. Figure: The line crosses the y-axis at: Ordered pair for this point 42 6 (0,6) 43 -3 (0,-3) 44 -5 (0,-5) 45 0 (0,0) The x-intercept is the point, (a, 0), where the graph crosses the x-axis. The x-intercept occurs when y is zero. The y-intercept is the point, (0, b), where the graph crosses the y-axis. The y-intercept occurs when x is zero. Find the x- andy-intercepts of each line: ⓐ x+2y=4 ⓑ 3x−y=6 ⓒ x+y=−5 ⓐ The graph crosses the x-axis at the point (4, 0). The x-intercept is (4, 0). The graph crosses the y-axis at the point (0, 2). The y-intercept is (0, 2). ⓑ The graph crosses the x-axis at the point (2, 0). The x-intercept is (2, 0) The graph crosses the y-axis at the point (0, −6). The y-intercept is (0, −6). ⓒ The graph crosses the x-axis at the point (−5, 0). The x-intercept is (−5, 0). The graph crosses the y-axis at the point (0, −5). The y-intercept is (0, −5). Find the x- and y-intercepts of the graph: x−y=2. x-intercept (2,0): y-intercept (0,−2) Find the x- and y-intercepts of the graph: 2x+3y=6. x-intercept (3,0); y-intercept (0,2)

Recognizing that the x-intercept occurs when y is zero and that the y-intercept occurs when x is zero gives us a method to find the intercepts of a line from its equation. To find the x-intercept, let y=0 and solve for x. To find the y-intercept, let x=0 and solve for y. Use the equation to find: the x-intercept of the line, let y=0 and solve for x. the y-intercept of the line, let x=0 and solve for y. x y 00 Find the intercepts of 2x+y=6 We'll fill in .

To find the x- intercept, let y=0: Substitute 0 for y. Add. Divide by 2. The x-intercept is (3, 0). To find the y- intercept, let x=0: Substitute 0 for x. Multiply. Add. The y-intercept is (0, 6).

The intercepts are the points (3,0) and (0,6). Find the intercepts: 3x+y=12 (4,0) and (0,12) Find the intercepts: x+4y=8 (8,0) and (0,2) Find the intercepts of 4x−3y=12. To find the x-intercept, let y=0. 4x−3y=12 Substitute 0 for y. 4x−3·0=12 Multiply. 4x−0=12 Subtract. 4x=12 Divide by 4. x=3 The x-intercept is (3, 0). To find the y-intercept, let x=0. 4x−3y=12 Substitute 0 for x. 4·0−3y=12 Multiply. 0−3y=12 Simplify. −3y=12 Divide by −3. y=−4 The y-intercept is (0,−4). The intercepts are the points (−3,0) and (0,−4). 4x−3y=12 x y 3 0 0 −4 Find the intercepts of the line: 3x−4y=12. x-intercept (4,0); y-intercept: (0,−3) Find the intercepts of the line: 2x−4y=8. x-intercept (4,0); y-intercept: (0,−2)

To graph a linear equation by plotting points, you can use the intercepts as two of your three points. Find the two intercepts, and then a third point to ensure accuracy, and draw the line. This method is often the quickest way to graph a line. Graph −x+2y=6 using intercepts. First, find the x-intercept. Let y=0, −x+2y=6−x+2(0)=6−x=6x=−6 The x-intercept is (–6, 0). Now find the y-intercept. Let x=0. −x+2y=6−0+2y=62y=6y=3 The y-intercept is (0, 3). Find a third point. We’ll use x=2, −x+2y=6−2+2y=62y=8y=4 A third solution to the equation is (2, 4). Summarize the three points in a table and then plot them on a graph. −x+2y=6 x y (x,y) −6 0 (−6,0) 0 3 (0,3) 2 4 (2,4) Do the points line up? Yes, so draw line through the points. Graph the line using the intercepts: x−2y=4. Graph the line using the intercepts: −x+3y=6. Find the x- and y-intercepts of the line. Let y=0 and solve for x Let x=0 and solve for y. Find a third solution to the equation. Plot the three points and then check that they line up. Draw the line. Graph 4x−3y=12 using intercepts. Find the intercepts and a third point. We list the points and show the graph. 4x−3y=12 x y (x,y) 3 0 (3,0) 0 −4 (0,−4) 6 4 (6,4) Graph the line using the intercepts: 5x−2y=10. Graph the line using the intercepts: 3x−4y=12. Graph y=5x using the intercepts. This line has only one intercept! It is the point (0,0). To ensure accuracy, we need to plot three points. Since the intercepts are the same point, we need two more points to graph the line. As always, we can choose any values for x, so we’ll let x be 1 and −1. Organize the points in a table. y=5x x y (x,y) 0 0 (0,0) 1 5 (1,5) −1 −5 (−1,−5) Plot the three points, check that they line up, and draw the line. Graph using the intercepts: y=4x. Graph using the intercepts: y=−x.

While we could graph any linear equation by plotting points, it may not always be the most convenient method. This table shows six of equations we’ve graphed in this chapter, and the methods we used to graph them. Equation Method #1 y=2x+1 Plotting points #2 y=12x+3 Plotting points #3 x=−7 Vertical line #4 y=4 Horizontal line #5 2x+y=6 Intercepts #6 4x−3y=12 Intercepts What is it about the form of equation that can help us choose the most convenient method to graph its line?Notice that in equations #1 and #2, y is isolated on one side of the equation, and its coefficient is 1. We found points by substituting values for x on the right side of the equation and then simplifying to get the corresponding y- values. Equations #3 and #4 each have just one variable. Remember, in this kind of equation the value of that one variable is constant; it does not depend on the value of the other variable. Equations of this form have graphs that are vertical or horizontal lines.In equations #5 and #6, both x and y are on the same side of the equation. These two equations are of the form Ax+By=C. We substituted y=0 and x=0 to find the x- and y- intercepts, and then found a third point by choosing a value for x or y. This leads to the following strategy for choosing the most convenient method to graph a line. If the equation has only one variable. It is a vertical or horizontal line. x=a is a vertical line passing through the x-axis at a y=b is a horizontal line passing through the y-axis at b. If y is isolated on one side of the equation. Graph by plotting points. Choose any three values for x and then solve for the corresponding y- values. If the equation is of the form Ax+By=C, find the intercepts. Find the x- and y- intercepts and then a third point. Identify the most convenient method to graph each line:ⓐ y=−3ⓑ 4x−6y=12 ⓒ x=2ⓓ y=25x−1 ⓐ y=−3This equation has only one variable, y. Its graph is a horizontal line crossing the y-axis at −3. ⓑ 4x−6y=12This equation is of the form Ax+By=C. Find the intercepts and one more point. ⓒ x=2There is only one variable, x. The graph is a vertical line crossing the x-axis at 2. ⓓ y=25x−1Since y is isolated on the left side of the equation, it will be easiest to graph this line by plotting three points. Identify the most convenient method to graph each line: ⓐ 3x+2y=12ⓑ y=4ⓒ y=15x−4ⓓ x=−7 ⓐ interceptsⓑ horizontal lineⓒ plotting pointsⓓ vertical line Identify the most convenient method to graph each line: ⓐ x=6 ⓑ y=−34x+1 ⓒ y=−8 ⓓ 4x−3y=−1 ⓐ vertical lineⓑ plotting pointsⓒ horizontal lineⓓ intercepts Graph by Finding Intercepts Use Intercepts to Graph State the Intercepts from a Graph

InterceptsThe x-intercept is the point, (a,0), where the graph crosses the x-axis. The x-intercept occurs when y is zero. The y-intercept is the point, (0,b), where the graph crosses the y-axis. The y-intercept occurs when x is zero. The x-intercept occurs when y is zero. The y-intercept occurs when x is zero. Find the x and y intercepts from the equation of a lineTo find the x-intercept of the line, let y=0 and solve for x. To find the y-intercept of the line, let x=0 and solve for y. x y 00 Graph a line using the interceptsFind the x- and y- intercepts of the line.Let y=0 and solve for x. Let x=0 and solve for y. Find a third solution to the equation. Plot the three points and then check that they line up. Draw the line. Choose the most convenient method to graph a lineDetermine if the equation has only one variable. Then it is a vertical or horizontal line.x=a is a vertical line passing through the x-axis at a.y=b is a horizontal line passing through the y-axis at b.Determine if y is isolated on one side of the equation. The graph by plotting points. Choose any three values for x and then solve for the corresponding y- values.Determine if the equation is of the form Ax+By=C, find the intercepts. Find the x- and y- intercepts and then a third point.

Identify the Intercepts on a Graph In the following exercises, find the x- and y- intercepts. (3,0),(0,3) (5,0),(0,−5) (−2,0),(0,−2) (−1,0),(0,1) (0,0) Find the x and y Intercepts from an Equation of a Line In the following exercises, find the intercepts. x+y=4 (4,0),(0,4) x+y=3 x+y=−2 (−2,0),(0,−2) x+y=−5 x−y=5 (5,0),(0,−5) x−y=1 x−y=−3 (−3,0),(0,3) x−y=−4 x+2y=8 (8,0),(0,4) x+2y=10 3x+y=6 (2,0),(0,6) 3x+y=9 x−3y=12 (12,0),(0,−4) x−2y=8 4x−y=8 (2,0),(0,−8) 5x−y=5 2x+5y=10 (5,0),(0,2) 2x+3y=6 3x−2y=12 (4,0),(0,−6) 3x−5y=30 y=13x−1 (3,0),(0,−1) y=14x−1 y=15x+2 (−10,0),(0,2) y=13x+4 y=3x (0,0) y=−2x y=−4x (0,0) y=5x Graph a Line Using the Intercepts In the following exercises, graph using the intercepts. −x+5y=10 −x+4y=8 x+2y=4 x+2y=6 x+y=2 x+y=5 x+y=3 x+y=−1 x−y=1 x−y=2 x−y=−4 x−y=−3 4x+y=4 3x+y=3 3x−y=−6 2x−y=−8 2x+4y=12 3x+2y=12 3x−2y=6 5x−2y=10 2x−5y=−20 3x−4y=−12 y=−2x y=−4x y=x y=3x Choose the Most Convenient Method to Graph a Line In the following exercises, identify the most convenient method to graph each line. x=2 vertical line y=4 y=5 horizontal line x=−3 y=−3x+4 plotting points y=−5x+2 x−y=5 intercepts x−y=1 y=23x−1 plotting points y=45x−3 y=−3 horizontal line y=−1 3x−2y=−12 intercepts 2x−5y=−10 y=−14x+3 plotting points y=−13x+5

Road trip Damien is driving from Chicago to Denver, a distance of 1,000 miles. The x-axis on the graph below shows the time in hours since Damien left Chicago. The y-axis represents the distance he has left to drive. ⓐ Find the x- and y- intercepts ⓑ Explain what the x- and y- intercepts mean for Damien. ⓐ (0,1,000),(15,0). ⓑ At (0,1,000) he left Chicago 0 hours ago and has 1,000 miles left to drive. At (15,0) he left Chicago 15 hours ago and has 0 miles left to drive. Road trip Ozzie filled up the gas tank of his truck and went on a road trip. The x-axis on the graph shows the number of miles Ozzie drove since filling up. The y-axis represents the number of gallons of gas in the truck’s gas tank. ⓐ Find the x- and y- intercepts. ⓑ Explain what the x- and y- intercepts mean for Ozzie.

How do you find the x-intercept of the graph of 3x−2y=6? Answers will vary. How do you find the y-intercept of the graph of 5x−y=10? Do you prefer to graph the equation 4x+y=−4 by plotting points or intercepts? Why? Answers will vary. Do you prefer to graph the equation y=23x−2 by plotting points or intercepts? Why?

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?

intercepts of a lineEach of the points at which a line crosses the x-axis and the y-axis is called an intercept of the line.