Question Video: Finding the Solution Set of a Quadratic Function from Its Graph for Different 𝑓(π‘₯) Values Mathematics

The graph shows the function 𝑓(π‘₯) = 2π‘₯Β² βˆ’ 4π‘₯ βˆ’ 6. What is the solution set of 𝑓(π‘₯) = 0? What is the solution set of 𝑓(π‘₯) = βˆ’6?

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Video Transcript

The graph shows the function 𝑓 of π‘₯ equals two π‘₯ squared minus four π‘₯ minus six. What is the solution set of 𝑓 of π‘₯ equals zero? What is the solution set of 𝑓 of π‘₯ equals negative six?

The solution set for 𝑓 of π‘₯ equals zero will be the places where two π‘₯ squared minus four π‘₯ minus six equals zero. And on the graph, that’s the place where they cross the π‘₯-axis. The 𝑦 equals zero across the π‘₯-axis. In this case, we have a solution at negative one and at positive three. So for 𝑓 of π‘₯ equal to zero, the solution set is negative one, three. Remember, this is not a coordinate. These are the π‘₯-values for which 𝑓 of π‘₯ equals zero. And we’ve given them in set notation.

The π‘₯-axis, as we’ve said, is the place where 𝑦 equals zero. The line 𝑓 of π‘₯ equals zero is the π‘₯-axis. But what would we do if we were using a graph to look at 𝑓 of π‘₯ equals negative six? This would be the place or the places where our quadratic cross the line 𝑦 equals negative six. And in this graph, we see that’s happening twice. It happens when π‘₯ equals zero and when π‘₯ equals two. In set notation, we’re saying the two π‘₯-values for which 𝑓 of π‘₯ equals negative six are zero and two.

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