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Determine the quadratic function 𝑓 with the following properties.
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Its graph has a vertex at three, negative 17.
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And 𝑓 of four equals five.
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We’ve been given a vertex and a solution point, which means we should think about the vertex form of quadratic functions.
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Which is 𝑓 of 𝑥 equals 𝑎 times 𝑥 minus ℎ squared plus 𝑘, where ℎ, 𝑘 is the vertex, and 𝑎 is some constant value.
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Using this form 𝑓 of 𝑥 equals 𝑎 times 𝑥 minus ℎ squared plus 𝑘, we can plug in three for ℎ and negative 17 for 𝑘.
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Which gives us 𝑓 of 𝑥 equals 𝑎 times 𝑥 minus three squared plus negative 17.
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Plus negative 17 can be simplified to, say, minus 17.
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But in order to find 𝑎, we’ll need to use the point we were given.
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If 𝑓 of four equals five, then we plug in four for 𝑥 and five for 𝑓 of 𝑥.
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Four minus three is one.
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One squared is one.
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𝑎 times one equals 𝑎.
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So, we have five equals 𝑎 minus 17.
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And if we add 17 to both sides, we see that 𝑎 equals 22.
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And so, we can go back to our original equation and plug in 22 for 𝑎.
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And we get the equation 𝑓 of 𝑥 equals 22 times 𝑥 minus three squared minus 17.