WEBVTT
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The figure shows the parabola 𝑥 equals two 𝑦 squared minus 16𝑦 plus 22 with its vertex 𝑣 marked.
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What are the coordinates of 𝑣?
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So notice when we usually see a parabola, it’s opening upwards, not sideways like this.
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The reason why is because this is 𝑥 equals two 𝑦 squared minus 16𝑦 plus 22.
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Usually, it’s 𝑦 equals instead of 𝑥 equals.
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This is what makes it turn sideways.
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So this equation isn’t a normal standard form, and if we wanna know the vertex of this we, need to put it in the vertex form, and we can do that by completing the square.
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Our first step is to group the first two terms together.
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And now let’s take out a GCF, a greatest common factor.
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And when we take a two out from two 𝑦 squared, we get one 𝑦 squared, and when we take a two out of negative 16𝑦, we get negative eight 𝑦.
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Now notice we left a little space.
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We’re gonna be adding something in there.
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We will fill it with 𝑏 divided by two squared.
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So in a normal polynomial, 𝑏 is the coefficient in front of 𝑥 or 𝑦, not the squared term and not the constant.
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So for this, 𝑏 would be negative eight, and negative eight divided by two is negative four.
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And when we square that, we get 16, so we will add a 16 in that spot.
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Now you can’t just add things to an equation; we have to keep things balanced.
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So if we just added in a 16, then technically that 16 is a 16 times two, so really that represents we’ve added in a positive 32.
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So to keep it balanced, we also have to subtract 32 from that same side.
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So adding 32 and subtracting 32 at the same time means we’re really not doing anything; it’s just zero.
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Now the whole point of doing this is this polynomial should be something squared, so what number multiplies to be 16 and adds to be negative eight?
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Well, that’s negative four and negative four, which is 𝑦 minus four squared, and then 22 minus 32 is negative 10.
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So now we’re in vertex form and our vertex- so our vertex is negative 10, four.
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So we go left 10 and up four to get to our vertex.
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Now usually, that negative 10 in our equation represents going up and down, but since it’s 𝑥 equals, it’s left and right, and same thing with the four.
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In our normal equation, it’s 𝑦 minus the number, so 𝑦 minus, we must have plugged in positive four.
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So it’s negative 10, four as our vertex, and that four moved it up and down.
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And again, usually when it’s 𝑦 equals, that’s left and right, but for this, it’s up and down.