Video: Finding the Solution Set of a Quadratic Function When ๐‘“(๐‘ฅ) = 0 by Drawing the Function

By drawing a graph of the function ๐‘“(๐‘ฅ) = 2๐‘ฅยฒ โˆ’ 3๐‘ฅ, find the solution set of ๐‘“(๐‘ฅ) = 0.

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Video Transcript

By drawing a graph of the function ๐‘“ of ๐‘ฅ equals two ๐‘ฅ squared minus three ๐‘ฅ, find the solution set of ๐‘“ of ๐‘ฅ equals zero.

Our equation is ๐‘“ of ๐‘ฅ equals two ๐‘ฅ squared minus three ๐‘ฅ. Before we start drawing our graph, itโ€™s helpful to have a few coordinates so that we can sketch the curve. We can use the tables to do this. If we calculate the ๐‘ฅ-values, negative two, negative one, zero, one, two, that should give us some idea of the shape of this graph. Our first box would be two times negative two squared minus three times negative two, which equals 14. Next, we have two times negative one squared minus three times negative one, which is five. When we plug in zero, the result is zero. When we plug in one, we get negative one. When we plug in two, we get two.

The top values, the ๐‘ฅ-values, represent the domain of the function, what we can plug in for ๐‘ฅ. And the bottom values represent the range, what weโ€™ll need for the ๐‘ฆ-values. In these values, we have a range thatโ€™s lowest point is negative one and highest point is 14. Now, these are just from the points weโ€™ve chosen. That doesnโ€™t mean itโ€™s the range of the full function. But it does give us an idea of what the scale of the ๐‘ฅ- and ๐‘ฆ-axis should be. We can graph a point at negative two, 14; negative one, five; zero, zero; one, negative one; and two, two.

When weโ€™re looking at this, we see that it might be helpful to have an additional point on the ๐‘ฅ-axis, so we could solve for three. When we plug three into the equation, we get nine as the output. And that just gives us one more point to be able to sketch this graph. It wonโ€™t be perfect, but you can try to get a smooth curve between the points. And when we do that, since weโ€™re looking for the solution set of ๐‘“ of ๐‘ฅ equals zero, weโ€™re looking for the ๐‘ฅ-intercepts. Weโ€™re looking for the places where this function crosses the ๐‘ฅ-axis. The first one is very clear. Itโ€™s the point zero, zero. And the second intersection happens halfway between one and two. We have a solution when ๐‘ฅ equals zero and a solution when ๐‘ฅ is one and a half which makes our solution set zero and three-halves or zero and one and a half.

Remember, this is not a coordinate point. These are two different ๐‘ฅ-values which when inputted into the function, the output is zero.

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