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Quadratics

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Reverse Foil Works with TRInomials (most of the time they start with an “x².”) Always look for a GCF first!!! x² + 5x + 6 Using what you already know about FOIL, try to come up with a basic idea of what the factored form would look like.

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**Factor Each Expression**

x² + 7x + 6 x² - 6x + 8 x² + 11x + 24 x² - 7x – 18 x² + 3x – 10 x² - 4x – 12 2x² + 3x + 1

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**Special cases in factoring…**

Factor x² - 9 How could we re-write x² - 9 with an “x” term in the middle? Now factor. x² - 100 4x² - 49

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**The Basics… Vertex Axis of symmetry (x = -b/2a ) X-intercepts**

Y-intercepts Maximum / minimum Opens Domain / Range

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**y = 4x² - 16x - 30 Vertex Axis of symmetry X-intercepts Y-intercepts**

Maximum / minimum Opens Domain / Range

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**y = - 4x² - 16x - 30 Vertex Axis of symmetry X-intercepts Y-intercepts**

Maximum / minimum Opens Domain / Range

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**Quadratic Applications**

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1 Miranda throws a set of keys up to her brother, who is standing on a third-story balcony with his hands 38 feet above the ground. If Miranda throws the key with an initial velocity of 40 feet per second, the equation h = -16t² + 40t + 5 gives the height h of the keys after t seconds.

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1 continued… a) How long does it take the keys to reach their highest point? b) How high do the keys reach? c) Will her brother be able to catch the keys, why or why not?

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2 A stone was thrown from the top of a cliff 60 meters above sea level. The height H meters, of the stone above sea level t seconds after it was released is given by H(t) = -5t² - 20t + 60.

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2 continued… a) Find the time taken for the stone to reach its maximum height. ___________ b) What is the maximum height above sea level reached by the stone? __________ c) How long is it before the stone strikes the water? _________

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3 The height H metres of a cannonball t seconds after it is fired into the air is given by

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3 continued… a) Find the time taken for the cannonball to reach its maximum height. _________ b) What is the maximum height reached by the cannonball? __________ c) How long does it take for the cannonball to fall back to earth? _____________

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6 A vegetable gardener has 40m of fencing to enclose a rectangular garden plot where one side is an existing brick wall. If two equal sides are x m long… Show that the area is given by

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6 continued… Find the dimensions of the vegetable garden of maximum area.

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7 A farmer wants to build two rectangular pens of the same size next to a river so they are separated by one fence. If she has 240 meters of fencing and does not fence the side next to the river:

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Draw a picture What equation will give us the area? What dimensions will maximize area?

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For a typical basketball shot, the ball’s height (in feet) will be a function of time in flight (in seconds), modeled by an equation such as h = -16t2 +40 t +6. a) What is the maximum height of the ball? b) When will the shot reach the height of the basket? (10 feet) c) When will the ball hit the floor, if it missed the basket entirely?

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**a) What is the maximum height of the ball?**

Put it in your calculator! Use your zooms and change your window until you see the maximum. Find the maximum! Answer: The maximum height of the ball is 31 feet!

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**b) When will the shot reach the height of the basket? (10 feet)**

Key words to highlight: When (so we are looking for our x) Height of the basket (10 feet) Put 10 in for y2 and find the… INTERSECTION! Answer: 2.4 seconds!

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**c) When will the ball hit the floor, if it missed the basket entirely?**

What do we put in for y2? y2 = 0 Now find the intersection! Answer: The ball will hit the floor after 2.64 seconds!

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YOU DO: The height, H metres, of a rocket t seconds after it is fired vertically upwards is given by How long does it take for the rocket to reach its maximum height? What is the maximum height reached by the rocket? How long does it take for the rocket to fall back to earth?

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1.8 QUADRATIC FUNCTIONS A function f defined by a quadratic equation of the form y = ax 2 + bx + c or f(x) = ax 2 + bx + c where c 0, is a quadratic.

1.8 QUADRATIC FUNCTIONS A function f defined by a quadratic equation of the form y = ax 2 + bx + c or f(x) = ax 2 + bx + c where c 0, is a quadratic.

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