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Mixed Review

Mixed Review. Rewrite in function notation: y= 3x – 2 Which of the following functions equals 14 when n equals 2? a) f( n ) = n + 13 b) f( n ) = 2( n +5) c) f( n ) = 6 n d) f( n ) = Identify the function on the graph 4.Identify the slope of the graphed function

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Mixed Review

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  1. Mixed Review • Rewrite in function notation: y= 3x – 2 • Which of the following functions equals 14 when n equals 2? • a) f(n) = n + 13 • b) f(n) = 2(n+5) • c) f(n) = 6n • d) f(n) = • Identify the function on the graph • 4.Identify the slope of the graphed function • 5. Is the following a function 3 & 4 5

  2. Standard MM1A1.d Investigate and explain the characteristics of a function Maximum and Minimum Values

  3. The third "Characteristic" of functions we need to investigate: maximum and minimum

  4. What do you think would be the maximum? The minimum? minimum maximum

  5. The word "maximum (max)" refers to the highest y-value for the graph. The word "minimum (min)" refers to the lowest y-value for the graph.

  6. Example No max for this graph because it never stops long enough to locate it! The min for this graph is the y-value of the vertex. -8 (-1, -8)

  7. Example (3, 8) The max for this graph is the y-value of the vertex. 8 No min for this graph because it never stops long enough to locate it!

  8. Did you notice, there is not ever a max and a min at the same time for a parabola? Why not? Can you think of a shape that would have both a max and a min? A circle would have a max and a min. But, is a circle a function? Will it pass the vertical line test?

  9. Let's give you a chance to practice finding the max and the min values for yourself.

  10. Name the max or min. There is no max. The min is -9.

  11. Name the max or min. There is no max. The min is +3.

  12. Which of the following have a max, a min, or neither. B C A Max Neither Min E F D Min Neither Neither

  13. Name the max or min. There is no max. The min is +3.

  14. Name the max or min. There is no min. The max is +6.

  15. Name the max or min. There is no min. The max is -4.

  16. Name the max or min. Max is 5. Min is -10 & -4.

  17. Change Gears Using a graph to find the max and min is pretty easy. But what about using an equation to find the max or min?

  18. Keep in mind, we need the vertex to determine the max or min. y = 2(x - 4)2 + 5 Vertex is? (4, 5)

  19. y = 2(x - 4)2 + 5 vertex is (4, 5) We know from this that the 5 is either the max or min but we don't know which. So, we have to know which way the graph is pointing: up or down Max up down Min

  20. y = 2(x - 4)2 + 5 For this, we look at the slope. Slope is 2 positive. So we know the graph opens up. So the 5 is a min because the graph opens up. vertex (4, 5) Rule: If slope is + it opens up and you have a min. If slope is - it opens down and you have a max.

  21. Find the max or min. y = (x + 1)2 + 2 Vertex is ______. Opens _____. Max of 2 Answer is ________.

  22. Find the max or min. y = 4(x - 1)2 - 8 Vertex is ______. Opens _____. Min of -8 Answer is ________.

  23. We eventually run into a problem though. Go figure! Quadratic equations are not always written in this form: y = a(x - h)2 + k y = 2(x - 4)2 + 5 Why is this form called the "vertex" form?

  24. Another form is called standard form. y = ax2 + bx + c y = 2x2 + 8x - 4 We still need the vertex and direction of opening to determine max or min.

  25. First we need a, b, and c. y = ax2 + bx + c y = 2x2 + 8x - 4 a = 2, b = 8, c = -4 Now you try... y = -x2 - 3x + 1 a = ____, b = ____, c = ____ -1 -3 +1

  26. Rule: If a > 0 opens up. If a < 0 opens down. y = 2x2 + 8x - 4 Min a = 2, so it opens up. y = -x2 - 3x + 1 Max a = -1, so it opens down.

  27. Now we need the vertex! y = 2x2 + 8x - 4 Find the x-coordinate of the vertex by this formula b 2a a = ___, b = ___, c = ___ -2 See if you can! Check your answer

  28. y = 2x2 + 8x - 4 a = 2, b = 8, c = -4 b 2a This is also called the axis of symmetry. -2 Next step, plug -2 into original equation, and find the y-coordinate of the vertex. -12 Vertex (-2, -12) check See if you can!

  29. Review what we did! y = 2x2 + 8x - 4 Step 1: a = 2, b = 8, c = -4 Step 2: a > 0; opens up (gives Min) Step 3: Vertex ( -2, -12) Step 4: Plug in x to get y. b 2a Conclusion: we have a Min of -12!

  30. You try! I'll help! y = -2x2 - 8x - 1 Step 1: a = ___, b = ___, c = ___ Step 2: a ? 0; opens ____ Step 3: Vertex ( ___, ___) Step 4: Plug in x to get y. b 2a Conclusion: we have a ___ of ___!

  31. Answer y = -2x2 - 8x - 1 Step 1: a = -2, b = -8, c = -1 Step 2: a < 0; opens down (Max) Step 3: Vertex ( -2 , 7) Step 4: Plug in x to get y. b 2a Conclusion: we have a MAX of 7!

  32. Try one on your own! Find direction of opening, vertex, and max or min. y = x2 + 12x - 2 Opens up Vertex of (-6, -38) Min of -38

  33. Find the max or min of the following equations. 1) y = 2(x + 3)2 - 5 2) y = (x + 1)2 + 2 3) y = (x + 9)2 - 5 4) y = 1.5(x – 1)2 + 1 Min of -5 Max of +2 Max of -5 Min of +1

  34. Drag the equation to sit with its graph. When you think youhaveit, check it! y = x2 + 6x + 5 y = -x2 - 4x - 5 y = 2x2 - 8x + 9 B C A y = 2x2 - 8x + 9 y = x2 + 6x + 5 y = -x2 - 4x - 5

  35. Let's practice finding max and min. Teacher, hand out MM1A1.d Practice Sheet 2, "Finding Max and Min".

  36. MM1A1.d Practice Sheet 2, "Finding Max and Min". See Word Document Attachment

  37. Quiz Time! Quiz on next page.

  38. Quiz: Max & Min 1-3. Find the max and/or min of the functional graphs. 1. 2. 3. 4. Find the max and/or min of the functional equations and state if slope for each Is positive or negative. 4. 5.

  39. Key to Mixed Review 1. f(x) = 3x-2 2. B 14 = 2(n+5) 2(2+5) 2(7)14 3. Absolute Value 4. 2 (up 2 over 1) 5. No (use vertical line test)

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