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

2. 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
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3. 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
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4. 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

5. 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.
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6. Example Graph Solution Note that for any choice of y, x must be –3. x1 2
Note that for any choice of y, x must be –3.
Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

7. 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).
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8. 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.
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9. 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.
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10. Example Graph the equation 2x – 3y = 6 by using intercepts. Solution -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

11. 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).
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12. Example Solve 2x + 1 = 3. Solutiony -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.
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13. 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.
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14. Example Solution Determine whether the equation is linear.
(0,-1) x y
(1,0)
(-1,0 )
(2,3)
(-2,3) 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