Use the vertical line test to identify functions As we have seen in some examples above, we can represent a function using a graph. Figure How To: Given a graph, use the vertical line test to determine if the graph represents a function. Inspect the graph to see if any vertical line drawn would intersect the curve more than once. If there is any such line, determine that the graph does not represent a function.
Solution If any vertical line intersects a graph more than once, the relation represented by the graph is not a function. Try It 8 Does the graph in Figure 15 represent a function? How To: Given a graph of a function, use the horizontal line test to determine if the graph represents a one-to-one function.
Inspect the graph to see if any horizontal line drawn would intersect the curve more than once. If there is any such line, determine that the function is not one-to-one. Solution The function in a is not one-to-one. While all linear equations produce straight lines when graphed, not all linear equations produce linear functions. In order to be a linear function, a graph must be both linear a straight line and a function matching each x -value to only one y -value. It must also pass a polygraph test, complete an obstacle course, and provide at least three references.
The qualifications are stringent. Remembering the absolute nonsense words "yunction" and "xquation" should help you keep things straight. Saying them out loud on the subway should help free up a seat. Since a linear equation is just a particular kind of relation, we already know how to graph linear equations. We find some dots, then connect them. If Pee Wee can do it, so can we. Between any two points, there's only one way to draw a straight line. Try it yourself: draw two points, and connect them with a straight line.
Can't get too creative with it, can you? No bending the paper, by the way. You don't even want to open that door. What this rule means is that we should be able to graph any linear equation by figuring out two points and drawing the line between them. In practice, it's a good idea to graph at least three points. If we graph three points of a linear equation and they don't all lie on the same line, we know we did something wrong. As much as that might rattle our delicate egos, at least we can go back and fix what we fouled up.
It's better than remaining blissfully ignorant, no matter what that old poet Thomas Gray might have said. The intercepts of a linear equation are the places where the axes catch the pass thrown by the linear equation. This is our effort to make linear equations seem remotely athletic. In reality, they have about as much physical ability as Tim Tebow. Oh, snap. In non-sports-analogy terms, the intercepts are the spots at which the axes and the graph of the linear equation overlap one another.
The x -intercept is the place where the graph hits the x -axis, and the y -intercept is the place where the graph hits the y -axis. It would be awfully confusing if it were the other way around. A linear equation may have one or two intercepts. Sometimes either the x -intercept or the y -intercept doesn't exist, or so intercept atheists would have you believe.
Knowing both intercepts for a linear equation is enough information to draw the graph, provided the intercepts aren't 0.
If they are 0, then our graph could be drawn any which way. If the graph goes through the origin 0, 0 , then both of the intercepts are 0 and we don't have enough information to draw the graph. We even tried calling , but they acted as if they had no idea what we were talking about.
The slope of a linear equation is a number that tells how steeply the line on our graph is climbing up or down. If we pretend the line is a mountain, it's like we're talking about the slope of a mountain. If it helps you, draw a snowcap at the top. Some mountain climbers. A ski lift. Nothing too elaborate though. We move from left to right on the x -axis, the same way that we read.
If the line gets higher as we move right, then we're climbing the mountain, so the line has a positive slope. If the line gets lower as we move right, then we're descending the mountain, so the line has a negative slope. If we stay at the same height, then the slope is zero because we're not going up and we're not going down.
Pretty boring mountain, if you ask us. The slope of a line can be positive, negative, zero, or undefined. A horizontal line has slope zero since it does not rise vertically i. As stated above, horizontal lines have slope equal to zero. This does not mean that horizontal lines have no slope.
Functions represented by horizontal lines are often called constant functions. Vertical lines have undefined slope. Since any two points on a vertical line have the same x -coordinate, slope cannot be computed as a finite number according to the formula,.
This means for each unit increase in x , there is a corresponding m unit increase in y i. Lines with positive slope rise to the right on a graph as shown in the following picture,. Lines with greater slopes rise more steeply.
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