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The function π¦ is equal to π of π₯ is stretched in the horizontal direction by a scale factor of one-third and in the vertical direction by a scale factor of one-third.
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Write, in terms of π of π₯, the equation of the transformed function.
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In this question, weβre given the curve of a function π of π₯ and weβre told that this is stretched in the horizontal direction by a scale factor of one-third.
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And then itβs also stretched in the vertical direction by a scale factor of, once again, one-third.
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We need to use this information to find the equation of the transformed function in terms of our original function π of π₯.
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We can see that weβre applying two stretches to our function π of π₯, one in the horizontal direction and one in the vertical direction.
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So, to answer this question, weβre first going to need to recall exactly how do we represent stretches of a function.
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And to do this, itβs often easiest to think of an example.
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Letβs start with π¦ is equal to π evaluated at π times π₯.
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Remember, our π₯-values are our inputs, and weβre multiplying all of our inputs by a value of π.
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So weβre multiplying all of our input values of π₯ by a value of π, so this is going to be a stretch in the horizontal direction.
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However, we still need to work out the scale factor.
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And one way of finding the scale factor is to try an example.
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Letβs set our value of π equal to two and letβs let π evaluated at four be equal to 10.
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If π evaluated at four is equal to 10, then π evaluated at two multiplied by two is also π of four, so this is also going to be equal to 10.
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So, in this case, multiplying the π₯ inside of our function by a value of two means we needed to halve our input value of π₯ to reach the same output.
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So weβve halved all of our input values of π₯.
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Remember, the input values of π₯ are on the π₯-axis, the horizontal axis.
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So this is a horizontal stretch by a factor of one-half.
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And, of course, if instead of using π is equal to two, weβd just use the value of π, then weβd be stretching in the horizontal direction by a scale factor of one over π.
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But remember, in this question, we also need to do a vertical stretch.
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And remember, the vertical position on a curve represents the output of the function.
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In other words, the π¦-coordinate represents the output of our function.
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So if we want to scale in the vertical direction, weβre going to want to multiply our outputs of the function.
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So π¦ is equal to π times π of π₯ should be a vertical stretch.
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We do need to work out the scale factor though.
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In this case though, itβs a little easier.
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If weβre just multiplying our output values by a value of π, then weβre just multiplying the π¦-coordinates by π.
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Weβre not changing the input values at all.
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In other words, weβre just stretching our curve by a scale factor of π in the vertical direction.
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Weβre now ready to try and find an equation of our transformed function given to us in the question.
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First, the question wants us to stretch the curve π¦ is equal to π of π₯ in the horizontal direction by a scale factor of one-third.
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And we know exactly how to represent a horizontal stretch by a scale factor of one over π.
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And in the question, we want a scale factor of one-third, so weβre going to need to set our value of π equal to three.
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In other words, all we need to do is multiply the input values of π₯ by three.
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Therefore, weβve shown to stretch the curve π¦ is equal to π of π₯ by a scale factor of one-third in the horizontal direction, we need π¦ is equal to π evaluated at three π₯.
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But thatβs not the only thing this question wants us to do.
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We also need to stretch our curve in the vertical direction by a scale factor of one-third.
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And we know exactly how to do this by using our second rule.
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To vertically stretch a curve by a scale factor of one-third, weβre going to need to set our value of π equal to one-third.
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In other words, all weβre doing is multiplying the outputs of our function by one-third.
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So by applying this to our previously transformed function, we get the equation π¦ is equal to one-third times π evaluated at three π₯.
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And this is our final answer.
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Therefore, we were able to show if the curve π¦ is equal to π of π₯ is stretched in the horizontal direction by a scale factor of one-third and in the vertical direction by a scale factor of one-third, then the equation of the transformed function, written in terms of π of π₯, is given by π¦ is equal to one-third multiplied by π evaluated at three π₯.