Question

The equation `y=0.014x^2+ 0.448x -2.324` models the price of gasoline at a local gas station last March. In the equation, x=1 corresponds to March 1. On which date in March was the gas price the highest? What was the price on that date?

Answer 1

March 31
$25.018

Explanation:

We have an equation where the degree of `y` is 1 and the degree of `x` is 2. Note that the coefficient of the lone term of `y` and the term of `x` with the highest degree are both positive.
The graph of the equation is that of a parabola opening upward.

What does that mean?

We have the parabola's vertex as its lowest point (i.e. price).
The gas price is decreasing from any point (date) before the vertex up to the vertex.
On the other hand, the gas price will increase starting from the vertex and onwards.

To check the trend during March (where `x = 1 =>` March 1), Let's use x = 1 and x = 2.

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`x = 1`
`=> y = 0.014(1^2) + 0.448(1) - 2.324`
`=> y = 0.014 + 0.448 - 2.324`
`=> y = -1.862`

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`x = 2`
`=> y = 0.014(2^2) + 0.448(2) - 2.324`
`=> y = 0.056 + 0.896 - 2.324`
`=> y = -1.372`

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Note that as we increase the value of `x`, the corresponding value of `y` also increases. This means that from `x = 1` onwards, we are already past the lowest point. The price of the gas will keep increasing as the days progress.

This means for the month of March, the gas price will be highest on the last day of the month.

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`x = 31`
`=> y = 0.014(31^2) + 0.448(31) - 2.324`
`=> y = 13.454 + 13.888 - 2.324`
`=> y = 25.018`