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

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

Write the equation represented by the graph shown. Give your answer in the form ๐ฆ โ ๐ = ๐(๐ฅ โ ๐).

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### Video Transcript

Write the equation represented by the graph shown. Give your answer in the form ๐ฆ minus ๐ equals ๐ 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 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 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. 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 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, 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 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 and ๐. So point ๐ 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 is equal to 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. So our slope is equal to a half. Great! So weโve now found ๐.

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. Well, ๐ is going to be the ๐ฆ-coordinate of the point thatโs actually highlighted on the graph, which is point ๐. 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 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 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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