Estimating values using linear graphs

​​In a nutshell

Since linear graphs continue in a straight line, you can use them to estimate future values of some variable (if you know how the line begins). By using two points to start off a line, you will be able to estimate other coordinates of points on the line.

Linear relationships

A linear relationship between two variables means that as one variable changes, the other changes proportionally. 

For example, if you are buying sweets from a shop, the more you buy, the higher the cost becomes. Importantly, the increase in cost goes up steadily with the increase of sweets. Your graph to represent this relationship could look like the following:

You can read off values from this graph. For example, you can tell that $$2$$ sweets cost $$10\text{p}$$, and that $$4$$ sweets cost $$40\text{p}$$.

Extrapolating

"Extrapolating" means using information you have to predict something outside of your data set. For example, on the graph above, if you extend the graph so that the $$x$$-axis reaches $$12$$ and you extend the line, you can make an estimate for the price of $$12$$ sweets:

You can now predict that $$12$$ sweets will cost $$60\text{p}$$. This is extrapolating from your original data to estimate a value on a linear graph.​

Example

The relationship between the distance travelled by a car and the time taken is linear. One hour after leaving, the car has travelled $$50$$ miles. Four hours after leaving, the car has travelled $$200$$ miles. Use this information to plot a graph and estimate the distance travelled seven hours after leaving.

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Start by deciding axes' labels and scales: on the $$x$$-axis, time in hours, and on the $$y$$-axis, distance in miles. The $$x$$-axis starts at zero hours after leaving and goes to at least to seven hours after leaving. You don't yet know how tall the $$y$$-axis needs to be, but you do know that it should start at $$0$$ miles travelled.

Next plot the two points you have, $$(1,50)$$ and $$(4,200)$$, and join them up with a ruler and extend the line to the right, past $$(4,200)$$. It will look like this:

To estimate the distance travelled after $$7$$ hours, go to $$7$$ on the $$x$$-axis, then go up until you meet the line. Go across to the $$y$$-axis to see the $$y$$-coordinate. This will be the estimate for the number of miles travelled in the $$7$$ hours since leaving:

About $$\underline{350 \ miles }$$ will have been travelled in the $$7$$ hours after leaving.​