Does the Graph of the Equation Rise or Fall?Date: 10/22/2002 at 07:31:39 From: Becky Della Noce Subject: Linear relations and function (slopes and intercepts) Hi, We are learning about graphs and how to graph the x and y intercepts, whether it is rise or fall. The question I don't understand is: Determine whether the graph of each equation rises to the right or falls to the right, is horizontal, or is vertical. The example question to figure out is 2x + 12 = 0. Thanks a lot, Becca Date: 10/22/2002 at 11:47:13 From: Doctor Ian Subject: Re: Linear relations and function (slopes and intercepts) Hi Becca, Let's think about the line y = 2x + 12 Here's one way to think about it. Suppose x is zero. Then y = 2(0) + 12 = 12 So the line crosses the y-axis at (0,12). We can label that point p on the graph below: | p | | | --------- --------- | Now, let's consider what happens if we increase x. The 12 stays the same, no matter what value of x we pick. But we're going to add 2x to that, and 2x gets bigger as x gets bigger, right? So as we move to the right - that is, as we choose bigger values of x - the value of y has to increase. So the graph must slope upward as we go to the right: | . more x means more y | . p........... | | | --------- --------- | Does that make sense? Now, what if the equation were y = -2x + 12 We'd still have our point p in the same place, but now every time we make x bigger, we make y _smaller_, because we're subtracting instead of adding: | | p........... | . | . more x means less y | --------- --------- | The only difference between the two equations is the sign of the x term. In the case where the sign is positive (more x means more y), the line slopes upward. In the case where the sign is negative (more x means less y), the line slopes downward. What if the slope is zero? Then we have y = 0x + 12 or just y = 12 which is to say, the value of y is _always_ 12, regardless of what x is. This is a horizontal line - a line that has the same value for y everywhere. And what about a vertical line? That's like a horizontal line, but instead of keeping y constant, we keep x constant, e.g., x = 3 Now, the equation you started with can be rearranged to look like this: 2x + 12 = 0 2x = -12 x = -6 So it's the equation of a vertical line. The value of x is -6, no matter _what_ value you choose for y. Does this help? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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