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Now that we’ve talked about slope and intercept, we can talk about the slope-intercept form of a line. I’ll start by saying the equation of any line can be stated in several different algebraically equivalent forms. We could have x and y on the same side of the equation, or we could solve for x, or solve for y.
These algebraic changes don’t change the fu, the underlying mathematical information. No matter how we rearrange the algebra, it’s the same line that’s represented. As long as we do the same thing to both sides of the equation, we’re not actually changing the mathematical information in the equation. Of all the possible algebraic representations, mathematicians have selected one as particularly useful.
If we solve the equation of a line for y, getting y on one side, by itself, equal to everything else, then we automatically put the, the equation into what is called, slope-intercept form. Now, slope-intercept form, you may know this by the name y equals mx plus b. Y equals mx plus b an, is another way of saying slope-intercept form. In this equation, y equals mx plus b, m is the slope of the line, and b is the y-intercept of the line.
And that’s why we call this slope intercept form, because two of the numbers in the equation are the slope and the intercept. If we simply solve the algebra for y we automatically get these two very important quantities, the slope and the y-intercept. We can see that m must be the slope because whenever x increases by 1, y has to increase by m.
We can see that b must be the y-intercept, because if x equals 0, then y equals b. It’s easy to put any equation into slope-intercept form, and once it’s in that form, it’s easy to understand where the graph will go. Starting from the y-intercept, we can take the slope steps that we discussed in the Slope lesson: 1 unit to the right, m units up, 1 unit to the left, m units down, etc.
Here’s a practice question. Pause the video and then we’ll talk about this. Okay, find all the points a, b on the line y equals negative 4x plus 2, four-thirds x plus 2, such that a and b are both integers with absolute values less than or equal to 10.
Okay, well certainly one point that we know is on the line is the y-intercept. We could see the y intercept is 2 so we’ll just write this as, in point form, 0, 2, that’s the y-intercept. Now that slope tells us that we can, if we start moving left, we can move over with a rise of negative 4. In other words, a drop of 4, and a run of 3.
So that’s over 3, 3 to the right, down 4. And so we go 3 to the right. That puts us at, at x equals 3 and down 4, that puts us at y equals negative 2. We can go 3 to the right and down 4 again. That will put us at 6, negative 6. And then 3 to the right, down 4, bring us to 9, negative 10.
We have to stop there because we’ve reached an absolute value as large as 10. Now we can go the other way. Again, start at the y-intercept, now we’re gonna go to the left 3. So in other words the y-intersect, the x-intersect is gonna go down by 3, the x-coordinate, and, the y is gonna go up by 4. So, now we’re gonna go backwards, go left to negative 3 and the y-intercept is gonna go up by 4, so that’s negative 3, 6.
And then take another step to the left. That would bring the x down to negative 6 and the y would go up to 10. And again, we have to stop there. And so those are the points, 1, 2, 3, 4, 5, 6, those six points are the points that satisfy this condition. And we can see these six points on this line.
Here’s another practice question. Pause the video, and then we’ll talk about this. Okay. Let me point out that this question is a bit easy.
This is really too easy to be a, a test question. But the important thing to appreciate is, the skill here is something that could very well be part of a test question. In other words, you might have to get the equation, find the slope, and then do something else with the slope. That would be what the test would ask you to do.
But certainly, one absolutely non-negotiable skill you need to have is, given an equation, you need to be able to find the slope. And of course, what we’re gonna do is we’re just gonna solve for y and put this into slope intercept form. So there’s the equation. We will subtract 3x from both sides so that moves to the other side.
Now we’re gonna divide both sides by 5 and we get y equals negative three-fifth x plus eight-fifths. So we can see that the slope is negative three-fifths. We can also see that the y-intercept would be positive eight-fifths, we get that information kind of as a bonus. But we know that the slope is negative three-fifths.
Notice that horizontal lines have a slope of zero, because they are all run with no rise. If a horizontal line has a ek, has a y-intercept of 4, then m equals 0 and b equals 4. If we put it into slope-intercept form it would be y equals 0 times x plus 4.
And of course, that just becomes y equals 4, which is the standard form of a horizontal line. So notice that the default form for our horizontal line can be interpreted as a kind of slope-intercept form. We’re getting y equals the y0intercept, because the slope is 0. Vertical lines are very different.
First of all, vertical lines have what is known as an undefined slope. Because the slope fraction is always something divided by zero. And of course, as we know, you’re not allowed to divide by zero. If we divide by zero we leave the world in which mathematics makes any sense whatsoever. And so whatever the slope is, it’s something that does not make any sense mathematically.
And so the word we use for this is undefined, not defined by anything within mathematics. That’s the slope of a vertical line. Vertical lines run parallel to the y-axis and never intersect it. Except of course, the only the only vertical line that intersects the y-axis is of course the y-axis itself.
But other than that, no vertical line even has a y-intercept because they run parallel to the y-axis. Therefore the standard form for vertical lines, x equals K. In other words, we’re specifying the x-intercept, and that specifies the entire line. That’s completely unrelated to slope-intercept form.
So vertical lines are very different, but the standard form of horizontal lines is really one special case of slope-intercept form. In summary, if we solve the equation of any line for y, we automatically put the equation into slope-intercept form. In other words y equals mx plus b, where m is the slope of the line and b is the y-intercept.
This form makes it easy to graph the line and to understand where it goes.
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