Question

A video game designer decides to include a character in a new game that only moves along a parabolic path. The designer uses the function x²+50+10 to model this character’s path, where y is the height of the character in pixels and x is the time in seconds that it takes the character to move across one parabolic path.

a. Is the vertex of the function a maximum or minimum point and how can you tell?




b. Find the x-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.



   c. Find the y-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.




d. Write the vertex as an ordered pair (x, y). What does the vertex represent for this situation? Write 1-2 sentences to explain your answer.


2. A satellite dish has the shape of a parabola, the U-shaped graph of a quadratic function. Suppose an engineer has determined that the shape of one of the satellite dishes offered by the company can be modeled by the quadratic function y=2/27x²-4/3x , where y is the vertical depth of the satellite dish in inches and x is the horizontal width in inches.

a. Is the vertex of the function a maximum or minimum point and how can you tell?




b. Find the x-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.



   c. Find the y-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.

d. Write the vertex as an ordered pair (x, y). What does the vertex represent for this situation? Write 1-2 sentences to explain your answer.




3. Suppose an investor builds a stock portfolio with a variety of shares in various high tech companies. The value of the stock portfolio is modeled by the function 2x²-20x+100 , where y is the value of the portfolio in hundreds of dollars and x is the time in months.

a. Is the vertex of the function a maximum or minimum point and how can you tell?




b. Find the x-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.



   c. Find the y-coordinate of the vertex. Show all work leading to your answer and write the answer in simplest form.




d. Write the vertex as an ordered pair (x, y). What does the vertex represent for this
situation? Write 1-2 sentences to explain your answer.

Answer 1

1.The given function is f(x) = x²+50x+10 = (x²+2*25x +625)-615 = (x+25)²-615.

1. Since the coefficient of the leading term of the given function is positive, the function represents a parabola whichopens upwards. Hence, the vertex of the function is a minimum point.
2. The vertex form of the parabola is f(x) = (x+25)²-615. Hence the x-coordinate of the vertex is -25.
3. The y-coordinate of the vertex is -615.
4. The vertex as an ordered pair is (-25,-615). The vertex represents a point in the 3^rd quadrant as both the x-coordinate and the y-coordinate are negative. Further, since the coefficient of the leading term of the given function is positive, the function represents a parabola which opens upwards. The vertex is the lowest point of the parabola.

2. The given function is f(x) = (2/27)x²-(4/3)x =2/27(x²-18x) = (2/27)(x²-2*9x+81) -6 =(2/27)(x-9)²-6.

1. a. Since the coefficient of the leading term of f(x) = (2/27)x²-(4/3)x is positive, the function represents a parabola whichopens upwards. Hence, the vertex of the function is a minimum point.
2. The vertex form of the parabola is f(x) = (2/27)(x-9)²-6. Hence the x-coordinate of the vertex is 9.
3. The y-coordinate of the vertex is -6.
4. The vertex as an ordered pair is (9,-6). The vertex represents a point in the 4^th quadrant as the x-coordinate is positive and the y-coordinate is negative. Further, since the coefficient of the leading term of the given function is positive, the function represents a parabola which opens upwards. The vertex is the lowest point of the parabola.

Note:

If the vertex form of a parabola is a(x-h)² +k, then the vertex of the parabola is the point (h,k). Further, if a > 0, the parabola opens upwards and if a < 0, the parabola opens downwards.