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Flip to reveal answersWhat graph shape does a quadratic model produce?
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All 16 Flashcards — Quadratic models
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Question
What graph shape does a quadratic model produce?
Answer
A parabola — a symmetric U-shape. Opens upward (∪) if a > 0, downward (∩) if a < 0.
Question
Give a real-world example of a quadratic model.
Answer
A ball thrown upward: h(t) = −5t² + 20t + 3. Height rises, reaches a maximum, then falls — the parabolic path of projectile motion.
Question
How does a quadratic model differ from a linear model?
Answer
Linear: constant rate of change, straight line. Quadratic: changing rate of change, has a maximum or minimum turning point (vertex), curved graph.
Question
R(p) = −2p² + 80p gives revenue R at price p. What does the downward parabola tell you?
Answer
Revenue increases, reaches a maximum at the vertex (optimal price), then decreases. There is one best price for maximum revenue.
Question
Formula: x-coordinate of the vertex of y = ax² + bx + c.
Answer
x = −b/(2a). The y-coordinate is found by substituting this x back into the equation.
Question
Find the vertex of y = 2x² − 8x + 3.
Answer
x = −(−8)/(2·2) = 2. y = 2(4) − 8(2) + 3 = 8 − 16 + 3 = −5. Vertex at (2, −5).
Question
IB asks "Find the minimum value of f(x) = x² − 6x + 11."
Answer
x = −(−6)/(2·1) = 3. f(3) = 9 − 18 + 11 = 2. Minimum value is 2 (at x = 3).
Question
How do you know whether the vertex is a maximum or a minimum?
Answer
If a > 0 (parabola opens up), the vertex is a minimum. If a < 0 (parabola opens down), the vertex is a maximum.
Question
IB asks for the "maximum value" of f(x) = −x² + 6x − 5. Student writes x = 3. What is wrong?
Answer
x = 3 is the x-coordinate of the vertex, not the maximum value. The maximum value is f(3) = −9 + 18 − 5 = 4.
Question
Student uses x = b/(2a) for the vertex (forgot the negative). What goes wrong?
Answer
The formula is x = −b/(2a). Forgetting the negative gives the wrong x-value and hence the wrong vertex.
Question
Can a quadratic with a > 0 have a maximum? Explain.
Answer
No — if a > 0 the parabola opens upward and only has a minimum. Only quadratics with a < 0 have a maximum.
Question
A context says "the ball is on the ground." What equation does this give for h(t) = −5t² + 20t?
Answer
h(t) = 0. Set −5t² + 20t = 0 → −5t(t − 4) = 0 → t = 0 or t = 4. Ball is on the ground at t = 0 and t = 4.
Question
h(t) = −5t² + 20t + 1. Find the maximum height.
Answer
t = −20/(2·−5) = 2. h(2) = −5(4) + 40 + 1 = 21. Maximum height = 21.
Question
P(n) = −n² + 10n − 16. Find the production level for maximum profit.
Answer
n = −10/(2·−1) = 5. Maximum profit at n = 5 units.
Question
IB gives a quadratic and asks "for what values of n is P positive?" How do you answer?
Answer
Find x-intercepts (set P = 0, solve). P is positive between the roots if a < 0, or outside them if a > 0.
Question
R = −3p² + 120p. What do the x-intercepts represent in the revenue context?
Answer
R = 0 at p = 0 and p = 40. These are the prices at which revenue is zero: free (no payment) or so expensive no one buys.
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