Video: Determining the Point-Slope Form of the Equation of a Straight-Line Graph

Write the equation represented by the graph shown. Give your answer in the form ๐‘ฆ โˆ’ ๐‘Ž = ๐‘š(๐‘ฅ โˆ’ ๐‘).

04:22

Video Transcript

Write the equation represented by the graph shown. Give your answer in the form ๐‘ฆ minus ๐‘Ž equals ๐‘š ๐‘ฅ minus ๐‘.

Right, so to solve this problem and find the equation thatโ€™s represented by this graph, weโ€™re gonna have to use this form thatโ€™s given here. And this is known as the point-slope form or the point-slope equation. Letโ€™s break down what each part of it actually means.

Well, first of all, we have our ๐‘Ž and ๐‘. And these are actually coordinates from a point that you choose. And in this case, weโ€™ve actually highlighted that point on the graph. Okay, so we now know that ๐‘Ž is going to be our ๐‘ฆ-coordinate and ๐‘ is going to be the ๐‘ฅ-coordinate of that point thatโ€™s highlighted.

Right, next part weโ€™re gonna have a look at is the ๐‘š. And the ๐‘š is the slope of the graph. And this is the slope of the graph between any two points on that straight line. Because itโ€™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, weโ€™re gonna start by finding ๐‘š, so the slope of the graph. To do that, weโ€™ve got an equation, which weโ€™ll put on the right-hand side, which is the ๐‘š is equal to ๐‘ฆ two minus ๐‘ฆ one over ๐‘ฅ two minus ๐‘ฅ one. So this basically means this is the change in ๐‘ฆ divided by the change in ๐‘ฅ. In order to enable us to do this, we need to look at another point on our graph, which Iโ€™ve circled here.

And when youโ€™re choosing which point to use, it doesnโ€™t really matter. But Iโ€™d say to make it easy for yourself always choose a point thatโ€™s very definitely on an ๐‘ฅ-coordinate and a ๐‘ฆ-coordinate, okay? And Iโ€™ve just highlighted onto the graph the change in ๐‘ฆ and the change in ๐‘ฅ. So that means how much it goes up and how much it goes along.

Okay, so weโ€™re now going to put values into the formula. But to do that, we need to know the coordinates of the two points weโ€™ve chosen. I have labelled them ๐‘Ž and ๐‘. So point ๐‘Ž is minus two, six and point ๐‘ is two, eight. Right, and weโ€™re now going to put this into our formula. And to enable us to do that, Iโ€™ve actually just labelled them ๐‘ฅ one, ๐‘ฆ one; ๐‘ฅ two and ๐‘ฆ two. And this gives us the equation ๐‘š is equal to ๐‘ฅ [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 weโ€™re gonna get two divided by four. So we can simplify even further. And it gives us ๐‘š. So our slope is equal to a half. Great! So weโ€™ve now found ๐‘š.

Fantastic! Now that weโ€™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 ๐‘ฆ minus ๐‘Ž. Well, ๐‘Ž is going to be the ๐‘ฆ-coordinate of the point thatโ€™s actually highlighted on the graph, which is point ๐‘Ž. So thatโ€™s going to be equal to six.

Then, weโ€™re gonna open the parentheses. And itโ€™s gonna be ๐‘ฅ minus ๐‘, which in this case is gonna be our ๐‘ฅ-value of the point that weโ€™ve highlighted, which is negative two. So this gives us the equation ๐‘ฆ minus six equals a half ๐‘ฅ minus negative two. Okay, we can still simplify this a little bit further because obviously ๐‘ฅ minus negative two, this will turn positive. So we can say weโ€™ve got the final equation ๐‘ฆ minus six equals a half ๐‘ฅ plus two. And thatโ€™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 ๐‘š is equal to ๐‘ฆ two minus ๐‘ฆ one divided by ๐‘ฅ two minus ๐‘ฅ one. And once we did that, we actually then put that into our point-slope form along with the values for ๐‘Ž and ๐‘, which were the ๐‘ฅ- and ๐‘ฆ-coordinate of the point highlighted on the graph, which gave us our final equation of ๐‘ฆ minus six equals a half ๐‘ฅ plus two.

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