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Sample Test: Quadratics II

Module by: Kenny M. Felder. E-mail the author

Summary: A second sample test on quadratics.

Exercise 1

x = 3y 2 + 5 x = 3y 2 + 5 size 12{x= - 3y rSup { size 8{2} } +5} {}

  • a. Opens (up / down / left / right)
  • b. Vertex: ___________
  • c. Sketch a quick graph on the graph paper

Exercise 2

y + 9 = 2x 2 + 8x + 8 y + 9 = 2x 2 + 8x + 8 size 12{y+9=2x rSup { size 8{2} } +8x+8} {}

  • a. Put into the standard form of a parabola.
  • b. Opens (up / down / left / right)
  • c. Vertex: ___________
  • d. Sketch a quick graph on the graph paper

Exercise 3

For what xx size 12{x} {}-values is this inequality true? 2x2+8x+8<92x2+8x+8<9 size 12{2x rSup { size 8{2} } +8x+8<9} {}

Exercise 4

For what xx size 12{x} {}-values is this inequality true? x210x28x210x28 size 12{ - x rSup { size 8{2} } - "10"x <= "28"} {}

Exercise 5

A rock is thrown up from an initial height of 4' with an initial velocity of 32 ft/sec. As I’m sure you recall, h(t)=ho+vot16t2h(t)=ho+vot16t2 size 12{h \( t \) =h rSub { size 8{o} } +v rSub { size 8{o} } t - "16"t rSup { size 8{2} } } {}.

  • a. Write the equation of motion for the rock.
  • b. At what time (how many seconds after it is thrown) does the rock reach its peak? How high is that peak? (Don’t forget to answer both questions…)
  • c. During what time period is the rock above ground?

Exercise 6

A hot dog maker sells hot dogs for $2 each. (So if he sells xx size 12{x} {} hot dogs, his revenue is 2x2x size 12{2x} {}.) His cost for manufacturing xx size 12{x} {} hot dogs is 100+12x+11000x2100+12x+11000x2 size 12{"100"+ { {1} over {2} } x+ { {1} over {"1000"} } x rSup { size 8{2} } } {}. (Which I just made up.)

  • a. Profit is revenue minus cost. Write a function P(x)P(x) size 12{P \( x \) } {} that gives the profit he will make, as a function of the number of hot dogs he makes and sells.
  • b. How many hot dogs should he make in order to maximize his profits? What is the maximum profit?
  • c. How many hot dogs does he need to make, in order to make any profit at all? (The answer will be in the form “as long as he makes more than this and less than that or, in other words, between this and that, he will make a profit.”)

Extra Credit:

  • a: Find the vertex of the parabola y=ax2+bx+cy=ax2+bx+c size 12{y= ital "ax" rSup { size 8{2} } + ital "bx"+c} {}. This will, of course, give you a generic formula for finding the vertex of any vertical parabola.
  • b: Use that formula to find the vertex of the parabola y=3x2+5x+6y=3x2+5x+6 size 12{y= - 3x rSup { size 8{2} } +5x+6} {}

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