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2.6 (2) Quadratic Functions

2.6 (2) Quadratic Functions. Applications and Graphs, Using the Vertex. POD. What are the two ways to find the vertex of a parabola?. Why do we care what the vertex is?. The vertex represents the maximum (a < 0) or minimum (a > 0) point on a parabola.

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2.6 (2) Quadratic Functions

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  1. 2.6 (2) Quadratic Functions Applications and Graphs, Using the Vertex

  2. POD What are the two ways to find the vertex of a parabola?

  3. Why do we care what the vertex is? The vertex represents the maximum (a < 0) or minimum (a > 0) point on a parabola. Look at example 9 on page 181 in your books. The equation: h(t) = -16t2 + 803t +600 Keep in mind here that the equation models height as a function of time-- it is not necessarily the path the projectile makes. Graph it on calculators to find maximum height-- you find the right window. How does this compare to an algebraic result? What information does this point give us?

  4. Why do we care what the vertex is? The vertex represents the maximum (a < 0) or minimum (a > 0) point on a parabola. h(t) = -16t2 + 803t +600 The vertex is (25.1, 10675). This means that at approximately 25.1 seconds, the projectile will reach its max height of 10,675 feet.

  5. Why do we care what the vertex is? h(t) = -16t2 + 803t +600 When is it first 5000 feet above the ground? When is it more than 5000 feet above the ground? (What are two ways to do this?) How long is it in flight?

  6. Why do we care what the vertex is? h(t) = -16t2 + 803t +600 When is it first 5000 feet above the ground? You can plug 5000 in for y, or graph y = 5000 and find the intersections. The time is about 6.3 seconds. When is it more than 5000 feet above the ground? (What are two ways to do this?) We can find only the first intersection and use the symmetric nature of the arc to find the other point. Between 6.3 and 43.9 seconds. How long is it in flight? This is when y = 0. For 50.9 seconds.

  7. Why do we care what the vertex is? Let’s use the vertex to solve other types of problems. Look at p. 185, #36. We have a line and a parabola. Where is the maximum difference between the two graphs, and what is it? The difference: d(x) = (-x + 3) - (2x2 + 8x + 4). See how that works?

  8. Why do we care what the vertex is? The difference: d(x) = (-x + 3) - (2x2 + 8x + 4). Simplify this difference, and see another quadratic to represent the distance between the curve and the line. d(x) = - 2x2 -9x -1 Find the vertex for this distance parabola.

  9. Why do we care what the vertex is? d(x) = - 2x2 -9x -1 The vertex for this difference/ distance parabola is (-2.25, 9.125). The max distance is 9.125, and it occurs at -2.25.

  10. Why do we care what the vertex is? Let’s use the vertex to solve other types of problems. Look at p. 185, #38. At what altitude is ozone density above Edmonton, Canada, the greatest in spring? D(h) = -0.078h2 + 3.811h - 32.433

  11. Why do we care what the vertex is? At what altitude is ozone density above Edmonton, Canada, the greatest in spring? D(h) = -0.078h2 + 3.811h - 32.433 The first graph has a domain from 0 to 35. The second has a domain of 20 to 35, which is actually the domain for the model. What point do we want?

  12. Why do we care what the vertex is? At what altitude is ozone density above Edmonton, Canada, the greatest in spring? D(h) = -0.078h2 + 3.811h - 32.433 The vertex of the graph is (24.43, 14.12). The ozone density is greatest at 24.42 km. At that altitude, the density is 14.11 x 10-3 cm/km.

  13. Why do we care what the vertex is? Let’s use the vertex to solve other types of problems. Look at p. 185, #44. A classic for the SAT? Two numbers whose difference is 40 and whose product is a minimum.

  14. Why do we care what the vertex is? Two numbers whose difference is 40 and whose product is a minimum. Once we have the parabolic equation for the product, find the vertex.

  15. Why do we care what the vertex is? Two numbers whose difference is 40 and whose product is a minimum. The vertex is at x = -20, halfway between the zeros of 0 and -40. The other number, y, is 20. The product is -400.

  16. Why do we care what the vertex is? Let’s use the vertex to solve other types of problems. Look at p. 185, #46. Also applicable to construction. A rectangular field divided is into three smaller rectangles, and 1000 yards of fencing. Maximize the area using only that fencing. What would be a decent equation?

  17. Why do we care what the vertex is? A rectangular field divided is into three smaller rectangles, and 1000 yards of fencing. Maximize the area using only that fencing.

  18. Why do we care what the vertex is? Graph it with the following window: -100 < x < 300, -5000 < y < 35000 What’s the vertex? How does that relate to the answer?

  19. Why do we care what the vertex is? Vertex at (125, 31250). One side of the field is 125 yd, the other is 250 yd. The area bounded by that fencing is 31250 yd2.

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