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Video: Finding the Maximum or Minimum Value of a Quadratic Function

Kathryn Kingham

Find the maximum or minimum value of the function 𝑓(𝑥) = 1 + 3𝑥², given 𝑥 ∈ [−3, 3].


Video Transcript

Find the maximum or minimum value of the function 𝑓 of 𝑥 equals one plus three 𝑥 squared, given that 𝑥 is contained in negative three, three. Before we try to answer this question, let’s think about what we know about quadratic functions. Quadratic functions are functions in the format 𝑓 of 𝑥 equals 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐. Our function here can be written as 3𝑥 squared plus one and fits into that format. The 𝑏 in our equation would be zero.

The graphs of quadratic functions are called parabolas, and they open upward for positive functions and downward for negative functions. Our instructions were to look at 𝑥 from negative three to three, so let’s see if we can sketch this graph. In a quadratic formula, the 𝑐-value, in our case one, is always the 𝑦-intercept. So our parabola will start at zero, one. Now I’m going to plug in some values here: 𝑓 of one equals one plus three times one squared. This means that 𝑓 of one equals four. We can also plug in 𝑓 of negative one, plugging negative one for 𝑥. One plus three times negative one squared also equals four.

Now that we have these three points, we can sketch a graph. Our parabola would look like this. Once you have this graph, it’s really easy to spot that this function has a minimum value. The minimum value of the smallest value of this function is at zero, one. By graphing our function, we were able to visually see that the minimum value of this function is one.