Vertex Form of Quadratics Practice 14

Question

Accepted Answer

If we're ever given a quadratic equation in standard form and we need to graph it, we can always rewrite it in vertex form by completing the square, and I'll show you how. Remember, to complete the square, we're going to take our middle term and split it in half. So we'll have negative four x and negative four x. And now we're going to draw our square diagram and split our first term in half as well. So we'll have x times x, which will make x squared, and negative four and negative four. So when we multiply negative four times x, we get negative four x, which is one half of our term. And when we do the other, we also get negative four x. Now completing the square, we're gonna have negative four times negative four, which makes positive 16, which means that x minus four squared is equal to x squared minus eight x plus 16. But we don't want plus 16. We want plus five. So to fix this, we're going to subtract 11 from both sides, which means that X squared minus eight X plus five can also be rewritten as X minus four squared minus 11. And now that we have this in vertex form, by completing the square, we can more easily graph it We know that our vertex is going to be at our h comma k values So our vertex will be at four comma negative 11 Plotting that on the graph, we'll go ahead and label our axes by two. So two, four, six, eight, negative two, negative four, negative six, negative eight, negative 10, and positive two, four, six, eight, 10 as well. So four negative 11 is going to be down here. Next, let's find our x intercepts, which we can do by taking our equation in vertex form. X minus four squared minus 11, setting it equal to zero and solving for x. We'll add 11 to both sides to get X minus four squared equals 11. Use our square root property to square root both sides and get X minus four is equal to positive and negative square root 11. And then adding four to both sides, we get x is equal to four plus or minus the square root of 11. Now here we'll have to do some estimating. The square root of 11 is about 3.32. So adding four to 3.32, we get that one of our x intercepts is at 7.32, which is about here on our graph. And our other x intercept will be at four minus 3.32, which is 0.68, which is about here on our graph. Lastly, we wanna find our y intercept, which we can find by plugging in zero for x in our equation. And it's actually easier to do that with our equation in standard form, because when we plug in zero, we'll get zero squared minus eight times zero plus five is equal to y. And our first two terms easily go to zero, and all we're left with is y equals five for our y intercept. And we can plot that right here. Now, connecting all of our dots, we get this parabola as the graph of x squared minus eight x plus five.

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Vertex Form of Quadratics Practice 14

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