Video: Sketching Curves from their Roots

Solve π‘₯Β² βˆ’ π‘₯ βˆ’ 6 = 0 by factoring and sketch the graph.

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Video Transcript

Solve π‘₯ squared minus π‘₯ minus six equals zero by factoring and sketch the graph.

So in order to factor this quadratic, we need to come up with numbers that multiply to be negative six and add to be negative one.

So we could multiply one and six and either one would be negative or six would be negative, or we can multiply two and three where either two would be negative or three would be negative. However, which pair would add to be negative one. It would be two and negative three.

Therefore, π‘₯ plus two and π‘₯ minus three are factors, and we should set each of those equal to zero. And we get π‘₯ equals negative two and π‘₯ equals positive three. So our graph would cross the π‘₯-axis at negative two and three. We can find where crosses the 𝑦-axis by plugging in zero into our function.

And we get negative six. Since the leading coefficient is one, we know that will open upward. So it’ll be shaped like this. And then the last thing that we could do just to be a little more specific is to find the vertex, which is β„Ž, π‘˜. β„Ž is equal to negative 𝑏 over two π‘Ž, and π‘˜ is equal 𝑓 of β„Ž.

So for our quadratic, π‘Ž is one 𝑏 is negative one and 𝑐 is negative six. So β„Ž would be equal to one-half. And to find π‘˜, we will plug in one-half into our function. So π‘˜ equals negative six and one-fourth.

So our Vertex is one-half and negative six fourth, which would be here. So here would be the sketch of our graph. And solving by factoring, we got π‘₯ equals negative two and π‘₯ equals positive three.

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