Write the equation represented by the graph shown. Give your answer in the form 𝑦 minus 𝑎 equals 𝑚 𝑥 minus 𝑏.
Right, so to solve this problem and find the equation that’s represented by this graph, we’re gonna have to use this form that’s given here. And this is known as the point-slope form or the point-slope equation. Let’s break down what each part of it actually means.
Well, first of all, we have our 𝑎 and 𝑏. And these are actually coordinates from a point that you choose. And in this case, we’ve actually highlighted that point on the graph. Okay, so we now know that 𝑎 is going to be our 𝑦-coordinate and 𝑏 is going to be the 𝑥-coordinate of that point that’s highlighted.
Right, next part we’re gonna have a look at is the 𝑚. And the 𝑚 is the slope of the graph. And this is the slope of the graph between any two points on that straight line. Because it’s a straight line, the slope will remain constant. Great! Now we can get on with solving the problem and finding the equation.
So in order to do that, we’re gonna start by finding 𝑚, so the slope of the graph. To do that, we’ve got an equation, which we’ll put on the right-hand side, which is the 𝑚 is equal to 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. So this basically means this is the change in 𝑦 divided by the change in 𝑥. In order to enable us to do this, we need to look at another point on our graph, which I’ve circled here.
And when you’re choosing which point to use, it doesn’t really matter. But I’d say to make it easy for yourself always choose a point that’s very definitely on an 𝑥-coordinate and a 𝑦-coordinate, okay? And I’ve just highlighted onto the graph the change in 𝑦 and the change in 𝑥. So that means how much it goes up and how much it goes along.
Okay, so we’re now going to put values into the formula. But to do that, we need to know the coordinates of the two points we’ve chosen. I have labelled them 𝑎 and 𝑏. So point 𝑎 is minus two, six and point 𝑏 is two, eight. Right, and we’re now going to put this into our formula. And to enable us to do that, I’ve actually just labelled them 𝑥 one, 𝑦 one; 𝑥 two and 𝑦 two. And this gives us the equation 𝑚 is equal to
𝑥 [eight] minus six divided by two minus negative two.
Right, we now simplify this, which will give us two divided by- now be careful here again with negative numbers. Two minus negative two, this turns to a plus when add. So we’re gonna get two divided by four. So we can simplify even further. And it gives us 𝑚. So our slope is equal to a half. Great! So we’ve now found 𝑚.
Fantastic! Now that we’ve found the slope, we can actually use this to write the equation of this graph in the point-slope form. First of all, we start with 𝑦 minus 𝑎. Well, 𝑎 is going to be the 𝑦-coordinate of the point that’s actually highlighted on the graph, which is point 𝑎. So that’s going to be equal to six.
Then, we’re gonna open the parentheses. And it’s gonna be 𝑥 minus 𝑏, which in this case is gonna be our 𝑥-value of the point that we’ve highlighted, which is negative two. So this gives us the equation 𝑦 minus six equals a half 𝑥 minus negative two. Okay, we can still simplify this a little bit further because obviously 𝑥 minus negative two, this will turn positive. So we can say we’ve got the final equation 𝑦 minus six equals a half 𝑥 plus two. And that’s in the point-slope form.
Great! So a quick recap of what we did, first of all, we found the slope using the equation 𝑚 is equal to 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one. And once we did that, we actually then put that into our point-slope form along with the values for 𝑎 and 𝑏, which were the 𝑥- and 𝑦-coordinate of the point highlighted on the graph, which gave us our final equation of 𝑦 minus six equals a half 𝑥 plus two.