Copyright © 2010 Pearson Education, Inc Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Another Look at Linear Graphs 2.4 Another Look at Linear Graphs Graphing Horizontal and Vertical Lines Graphing Using Intercepts Solving Equations Graphically Recognizing Linear Equations Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Horizontal Lines The slope of a horizontal line is 0. The graph of any function of the form f(x) = b or y = b is a horizontal line that crosses the y-axis at (0, b). If two different points (x1, y1) and (x2, y2) are on a horizontal line, then they must have the same second coordinate. In this case we have y1 = y2, so Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Graph Solution Note that for any choice of x, f (x) must be 2. 1 –1 2 Note that for any choice of x, f (x) must be 2. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Vertical Lines The slope of a vertical line is undefined. The graph of any equation of the form x = a is a vertical line that crosses the x-axis at (a, 0). If two different points (x1, y1) and (x2, y2) are on a vertical line, then they must have the same first coordinate. In this case we have x1 = x2, so Since we cannot divide by 0, this is undefined. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Graph Solution Note that for any choice of y, x must be –3. x 1 2 Note that for any choice of y, x must be –3. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

The graph of any function of the form f (x) = b or y = b is a horizontal line that crosses the y-axis at (0, b). The graph of any equation of the form x = a is a vertical line that crosses the x-axis at (a, 0). Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Graphing Using Intercepts Any line that is not horizontal or vertical will cross both the x- and y-axes. The point at which the line crosses the y-axis is called the y-intercept. Similarly, the point at which the line crosses the x-axis is called the x-intercept. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

To Determine Intercepts The x-intercept is of the form (a, 0). To find a, let y = 0 and solve for x. The y-intercept is of the form (0, b). To find b, let x = 0 and solve for y. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Graph the equation 2x – 3y = 6 by using intercepts. Solution -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 -1 1 6 5 4 (3,0) (0,–2 ) x y 3 –2 Plot the intercepts and draw the line. A third point could be calculated and used as a check. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Solving Equations Graphically We can solve 2x + 1 = 3 by finding the x- coordinate of the point where the graphs of f (x) = 2x + 1 and g(x) = 3 intersect (see next slide). Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solve 2x + 1 = 3. Solution y -5 -4 -3 -2 -1 1 2 3 4 5 -3 2 -2 3 -1 1 6 5 4 Careful inspection suggests that x = 1 is the x–value where they intersect. To check, note that f (1) = 2(1) + 1 = 3. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Recognizing Linear Equations Standard Form of a Linear Equation Any equation of the form Ax + By = C, where A, B, and C are real numbers and A and B are not both 0, is a linear equation in standard form and has a graph that is a straight line. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Solution Determine whether the equation is linear. (0,-1) x y (1,0) (-1,0 ) (2,3) (-2,3) -5 -4 -3 -2 -1 1 2 3 4 5 4 3 6 2 5 1 -3 -1 Try to put the equation in standard form: The last equation is not linear because it has an x2-term. The graph is to the right. Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley