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The graph of the function π is produced from that of π of π₯ equals π₯ squared by the following steps: a shift of four to the right, a dilation by a scale factor of one-fourth, and a shift of four up.
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What is the function π?
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When a quadratic is in vertex form, itβs easy to see how it can be transformed or shifted.
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π, β, and π represent different things that we could do to our graph to change it or move it.
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π represents the scale factor.
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It decides how thin or wide the graph will be.
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β will shift our graph left and right.
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A negative β, so plugging in a negative β, will shift the graph left, and if we would plug in a positive β, it would shift the graph right.
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π moves the graph up and down, so if we plug in a positive π, it will shift up, and if we plug in a negative π, it will shift down, so β moves it horizontally and π moves it vertically.
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So first, letβs begin with π.
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A dilation is what makes something bigger or smaller.
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And like we said, our graph could become thinner or wider; itβs the same thing.
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So our π will be one-fourth.
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Next, thereβs a shift of four to the right, so we begin with π₯ minus and then we plug in β.
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So if weβre going four to the right, weβre plugging in a positive four.
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So even though it says π₯ minus four, weβre still going to the right.
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And then lastly, we are shifting four up, so we plug in positive four for π.
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So this is in vertex form.
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Letβs go ahead and put it in standard form.
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So we need to evaluate this, in other words simplify.
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So π₯ minus four squared means we need to multiply it to itself twice.
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So we FOIL.
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π₯ times π₯ is π₯ squared, π₯ times negative four is negative four π₯, and then we have negative four times π₯, so itβs another negative four π₯; thatβs how we get our negative eight π₯.
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And then lastly, we take negative four times negative four, which is 16.
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And now we need to take one-fourth times that polynomial, so we distribute the one-fourth and then we also bring down the four.
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So our last step would be to combine the four and the four by adding.
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So the function π is equal to π₯ squared divided by four minus two π₯ plus eight.