Video Transcript
Write the equation represented by the graph shown. Give your answer in the form ๐ฆ minus ๐ equals ๐ ๐ฅ minus ๐.
Right, so to solve this problem and find the equation thatโs represented by this graph, weโre gonna have to use this form thatโs given here. And this is known as the point-slope form or the point-slope equation. Letโs break down what each part of it actually means.
Well, first of all, we have our ๐ and ๐. And these are actually coordinates from a point that you choose. And in this case, weโve actually highlighted that point on the graph. Okay, so we now know that ๐ is going to be our ๐ฆ-coordinate and ๐ is going to be the ๐ฅ-coordinate of that point thatโs highlighted.
Right, next part weโre gonna have a look at is the ๐. And the ๐ is the slope of the graph. And this is the slope of the graph between any two points on that straight line. Because itโs 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, weโre gonna start by finding ๐, so the slope of the graph. To do that, weโve got an equation, which weโll put on the right-hand side, which is the ๐ is equal to ๐ฆ two minus ๐ฆ one over ๐ฅ two minus ๐ฅ one. So this basically means this is the change in ๐ฆ divided by the change in ๐ฅ. In order to enable us to do this, we need to look at another point on our graph, which Iโve circled here.
And when youโre choosing which point to use, it doesnโt really matter. But Iโd say to make it easy for yourself always choose a point thatโs very definitely on an ๐ฅ-coordinate and a ๐ฆ-coordinate, okay? And Iโve just highlighted onto the graph the change in ๐ฆ and the change in ๐ฅ. So that means how much it goes up and how much it goes along.
Okay, so weโre now going to put values into the formula. But to do that, we need to know the coordinates of the two points weโve chosen. I have labelled them ๐ and ๐. So point ๐ is minus two, six and point ๐ is two, eight. Right, and weโre now going to put this into our formula. And to enable us to do that, Iโve actually just labelled them ๐ฅ one, ๐ฆ one; ๐ฅ two and ๐ฆ two. And this gives us the equation ๐ is equal to ๐ฅ [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 weโre gonna get two divided by four. So we can simplify even further. And it gives us ๐. So our slope is equal to a half. Great! So weโve now found ๐.
Fantastic! Now that weโve 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 ๐ฆ minus ๐. Well, ๐ is going to be the ๐ฆ-coordinate of the point thatโs actually highlighted on the graph, which is point ๐. So thatโs going to be equal to six.
Then, weโre gonna open the parentheses. And itโs gonna be ๐ฅ minus ๐, which in this case is gonna be our ๐ฅ-value of the point that weโve highlighted, which is negative two. So this gives us the equation ๐ฆ minus six equals a half ๐ฅ minus negative two. Okay, we can still simplify this a little bit further because obviously ๐ฅ minus negative two, this will turn positive. So we can say weโve got the final equation ๐ฆ minus six equals a half ๐ฅ plus two. And thatโs in the point-slope form.
Great! So a quick recap of what we did, first of all, we found the slope using the equation ๐ is equal to ๐ฆ two minus ๐ฆ one divided by ๐ฅ two minus ๐ฅ one. And once we did that, we actually then put that into our point-slope form along with the values for ๐ and ๐, which were the ๐ฅ- and ๐ฆ-coordinate of the point highlighted on the graph, which gave us our final equation of ๐ฆ minus six equals a half ๐ฅ plus two.
