Using the equation representing the height of the firework (h = -16t2 + v0t + h0), algebraically determine the extreme value of f(t) by completing the square and finding the vertex. interpret what the value represents in this situation.

The idea is to write the equation of the parabola in vertex form, i.e. in the form that immediately tells you where the vertex of the parabola. The extreme value (minimum or maximum) of the parabola occurs at the -coordinate of the vertex. Whether this point is a minimum or a maximum is determined by the sign of the leading coefficient; in this case, that coefficient is -16, which means the parabola is "opening" downward so that the vertex is the maximal point.

The general form would be

where the vertex is located at . There are two useful ways we can do this.

(1) We can work backwards and solve for by expanding the expected vertex form.

This tells us we need to have

(2) We could also work forwards, starting with the given equation for and completing the square:

which is the same expression we ended up with before. Using either method, we find that the vertex is , giving an extreme value of .

The expression tells you the height of the firework at a specific time . From the vertex form we've derived above, we've shown that the greatest height that the firework can achieve is determined exactly by its launch velocity and how high the firework was when it was launched .

step-by-step explanation:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

step-by-step explanation:

given that megan owns a clothing boutique.

the profits of 2010, 11, are recorded and shown as follows:

years since     2010 0     1     2     3       4       5

profits made in 1000's   39       84     111     120   111     84

we find that profit gradually increases form 2010 to 2013 upto 120,   then starts decreasing.

the profits for 2012 and 2014 are the same while that of 2011 and 2015 are also the same.

this shows that there is symmetry about the line profit = 120000

hence answer is:

a.   since the profit made in 2012 is equal to the profit made in 2014, the profit is           symmetric about the line y = 120,000.

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answer: 230 miles ( vote ! )

step-by-step explanation:

180.39 minus 17.99 equals 162.4

then divide 162.4 by .80 to get 230