Intermediate Algebra Activity
Finding a Quadratic Model
 A quadratic model can be generated by hand using the vertex form of a quadratic
function: y  a  x  h   k where (h, k) are the coordinates of the vertex.
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 Define the variables.
 Find the vertex (h, k) and at least one other point.
 Substitute the vertex in for (h, k) in y  a  x  h   k
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 Pick another point (x,y) , and substitute it into y  a  x  h   k for x and y and
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solve for a.
 Write the equation of the model. Plug in a, h and k. Leave x and y in the equation.
 Check the model by graphing it on the rectangular coordinate system.
Practice Problems:
1. Find the equation of a quadratic function with a vertex of  8, 40  passing through the
point  2,55 .
2. Find the equation of a quadratic function with a vertex of  2.5, 4  passing through the
point  3.5, 7  .
3. Find the equation of a quadratic function with a vertex of  9, 5.2  passing through the
point  3, 4.1 .
4. Find the equation of a quadratic function with a vertex of 10 days,$567  passing
through the point 17days,$948 .
5. Mark threw a rock off a 243 foot cliff at the Grand Canyon. At zero seconds the rock
was at a height of 243 feet. The rock reached its highest point of 271 feet in 1.5 seconds.
Let y represent the height of the rock after x number of seconds. Find the equation of the
quadratic function that models this. Draw a graph of your quadratic model on the
rectangular coordinate system. Use your function to give the height of the rock after 2.5
seconds.
6. Let’s look at Delta College Solar Energy. The following data is found at
http://www3.delta.edu/solar/solarArchive.html . It gives the amount of solar energy
obtained in kWh (kilowatt hours) during each month of the year 2009. January is month
1 and December is month 12. The maximum solar energy of 1483.6 kWh happened in
month 5 (May). By December (month 12) the solar energy had fallen to 537.2 kWh. Let
the month be the x coordinate and the solar energy by the coordinate and the solar energy
by the y coordinate. Find the equation of the quadratic function that models this. Draw
a graph of your quadratic model on the rectangular coordinate system. Use your function
to give the amount of solar energy in March (month 3).
7. A company that manufactures transmissions for Ford cars and trucks found that they had
a minimum cost of $24500 if the number of work hours per employee was kept at 40
hours per week. When employees had to work 50 hours per week, their overall costs
increased to $29600. Assuming that costs can be modeled with a quadratic function, find
the equation of the quadratic function that fits this situation. Let x represent the average
number of hours employees work per week, and let y represent the cost in dollars. Draw a
graph of your quadratic model on the rectangular coordinate system. Use your function
to give the costs if the employees are working 45 hours per week.
8. In business, it is important to manufacture the right amount of product. Looking again at
the Transmission company in #3, we found that if the company manufactures 137
transmissions per week, they had a maximum profit of $521,800. If they make less than
that, they do not meet their supply and they lose profit. If they make too many
transmissions, then they can’t sell them all and they also lose profit. One week the
company made 108 transmissions due to some equipment problems. The profits that
week dropped to $411,300. Assuming that costs can be modeled with a quadratic
function, find the equation of the quadratic function that fits this situation. Let x
represent the number of transmission built per week, and let y represent the profit in
dollars. Draw a graph of your quadratic model on the rectangular coordinate system. Use
your function to give the profit if the company makes 142 transmissions.