WEBVTT
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What is the rate of change shown by this graph of a function?
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Well for straight line functions, the rate of change is defined as, by how much does the 𝑦-coordinate change when I increase the 𝑥-coordinate by one?
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Now at first glance, that’s actually quite tricky to work out on this particular graph.
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So for e𝑥ample, if I go from here where my 𝑦-coordinate is this, to increase my 𝑥-coordinate by one, so up to here where my 𝑦-coordinate is this, this difference here is the rate of change of my line.
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Now obviously, I’m not able to be very accurate about that.
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But if we look closely at the graph, they’ve helpfully put a couple of little green dots on here.
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So these are where the line passes through e𝑥act coordinate pairs.
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So the left-hand point there has an 𝑥-coordinate of negative five and a 𝑦-coordinate of negative three.
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And the second point has an 𝑥-coordinate of zero and a 𝑦-coordinate of negative one.
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Now in moving from the first point to the second point, we can see that the 𝑥-coordinate here has increased by five, and the 𝑦-coordinate here has increased by two.
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So let’s write that down.
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When 𝑥 increases by five, then 𝑦 increases by two.
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Now the definition of rate of change is, by what does 𝑦 increase when 𝑥 increases by one.
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So in our e𝑥ample, we’ve increased the 𝑥-coordinate five times too much.
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So if we divide that by five, so five divided by five is equal to one, so we’re going a fifth as far in the 𝑥-direction.
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And because it’s a straight line, because it’s a linear relationship, because it’s proportional, this means that we’re going o- to only go a fifth as far in the 𝑦-direction as well.
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So the change in 𝑦-coordinate is the original two and we’re going to divide that by five.
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So rather than calculate that as a decimal, we just literally write it as two divided by five, two fifths.
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Now this rate of change is sometimes called the slope of the line, or the gradient of the line.
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But in this question, we called it rate of change.
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So the answer is that the rate of change is two over five, or two fifths.