One Point, One Slope, One Line 04:38 minutes

Video Transcript

Transcript One Point, One Slope, One Line

Come one, come all to see the AMAZING Shoddy Mathemagician, Thomas, performing this week only at the Apollo Theater in New York! The Shoddy Mathemagician is world-famous for his shoddy magic tricks like pulling streamers out of his mouth and his specialty, the Disconnected Thumb! Thomas’ll be performing his new magic trick, One Point, One Slope, ONE LINE! The Shoddy Mathemagician claims that, given any one point and any one slope, he’ll be able to tell the audience how many lines run through that point. They say a magician never reveals his secrets, but we’ll clue you into the inner-workings of the Shoddy Mathemagician’s new trick. First, the Shoddy Magician asks the audience for an equation for a line in standard form, ‘y’ equals 'mx' plus ‘b’. The audience responds with 'm' equals 2, and the point (0, 8). If you notice, the point (0, 8) lies on the y-axis. This means that when 'x' is zero, 'y' is equal to 8. We can express the y-intercept of the line from the standard form by setting 'b' equal to 8. Doing this gives us the equation 'y' equals 2x plus 8. Let’s graph what we have so far, so it’s easier to see. We already have our point with the coordinates (0, 8). Since our slope is 2, and because slope is rise over run, we can count to the right one and up two to find the next point on the line. This gives us the point (1, 10). Let's do it again to make sure the points will still be on the same line. One to the right and up two, leaving us with the point (3, 12). After connecting these points, we can see we've drawn a line. How many lines pass through the point (0, 8) with a slope of 2? ...ONE!!! Whenever we’re given one point and one slope, there’s only one line that goes through the point. Pretty cool, huh? The audience is stunned! They want more! But this time, with negative numbers! The audience requests the number of lines that pass through the point (0, -5) with a negative one-half slope. Just like before, notice how our 'x' term is zero. This, of course, means that our y-intercept is negative 5. So we can let 'b' equal negative 5. The audience requested a slope of negative one-half, so 'm' is negative one-half. This gives us the linear equation in standard form of 'y' equals negative one-half 'x' minus 5. Let’s see this on a graph. We'll start with our y-intercept at (0, -5) and, since our ‘m’ value is -1/2, our slope is -1/2. We count to the left two -- since we’re dealing with a negative slope -- and up one, giving us the next point on our line at (-2, -4). What's that? Of COURSE we'll do it again to make sure the points will still be on the same line! Count to the left two and up one, giving us the next point on our line at (-4, -3). Look what happens when we connect these points. How many lines pass through the point (0, -5) with a slope of negative one-half? That’s right!!! ONE!!! You’re getting the hang of it! You’ll be a mathemagician yet! Just like the Shoddy Mathemagician!!! Let’s review. You can select any one point you want, and any one slope, positive or negative. There will always be one line passing through that point with this slope! Let’s see how the show’s going. Hmm...the audience doesn’t sound too happy. I guess it’s curtains for the Shoddy Mathemagician!