AdvAlg6.4GraphingQuadratics.notebook. March 07, Newton s Formula h(t) = 1 gt 2 + v o t + h o 2. time. initial upward velocity
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1 Notes Lesson 6 4 Applications of Quadratic Functions Newton s Formula h(t) = 1 gt 2 + v o t + h o 2 Height of object time Constant (accel. due to gravity) *32 ft/sec 2 *9.8 m/sec 2 **MEMORIZE THESE** initial upward velocity initial height Velocity: Rate of change of position with respect to time Acceleration: Measure of how fast velocity changes Jul 30 5:13 PM 1
2 1. What are the two forms for the equation of a parabola? (A quadratic equation) Vertex Form y k = a(x h) 2 Standard Form y = ax 2 + bx + c 2. Study example 1 3. Now try the following problem in your notes: a. Are the following equations equivalent? y = 3(x + 1) 2 4 and y = 3x 2 + 6x 1 Try to change one to look like the other. y = 3(x + 1) 2 4 3(x + 1)(x + 1) 4 3(x 2 + 2x + 1) 4 3x 2 + 6x x 2 + 6x 1 they are equivalent Dec 2 8:29 PM 2
3 4. When comparing the two forms below: 5. Theorem dealing with parabola congruency 6. Given the equation of a parabola f(x) = ax 2 + bx + c What is the y intercept? What is the domain? f(0) = a(0)+ + b(0) + c = c x can be any real number What is the range? If a > 0 If a < 0 Dec 2 8:36 PM 3
4 7. Given the equation of a parabola y k = a(x h) 2 What is the vertex? The vertex is (h, k) What is the axis of symmetry? What is the domain? What is the range? The line of symmetry is x = h x can be any real number If a > 0, f(x) >= k If a < 0, f(x) <= k Dec 2 8:42 PM 4
5 8. In the equation a. What does h represent? b. What does g represent? The height of an object after t seconds The acceleration due to gravity 9.8 m/sec 2 or 32 ft/sec 2 c. What does v o represent? d. What does h o represent? The initial velocity of the object The initial height of the object e. What does t represent? time typically measured in seconds Dec 2 8:54 PM 5
6 9. Study example Study example Now try the following problem in your notes: a. Suppose a toy rocket is launched so that its height h in meters after t seconds is given by How high is the rocket after 1 second? g = 9.8 How high is the rocket when launched? How high is the rocket after 12 seconds? When is the rocket 13 meters in the air? When will the rocket hit the ground? Dec 2 9:00 PM 6
7 Graph y = x 2 + 2x 1 for values of x between 3 and 3. use the graph to write the vertex form of the graph. Sep 13 11:10 AM 7
8 1. Describe the differences and similarities between the graph of y 3 = 2 ( x + 9 ) 2 and y = x 5 2. Using a preimage of y = 7x 2 write the equation for the image under a translation of 8 units up and 5 units right. What is the vertex and the equation for the axis of symmetry? 3. Expand and simplify (4x 3) 2 ( 5x 1) 2 Dec 17 6:48 AM 8
9 Object tossed or thrown upward: H = 1gt 2 + v o t + h o 2 1. Suppose a ball is thrown upward from a height of 5 feet with an initial velocity of 30ft/sec. A) Write an equation relating the time t in seconds and the height H of the ball in feet. B) Find the height of the ball after 1.5 seconds. Jul 30 5:58 PM 9
10 H = 1gt 2 + v o t + h o 2 Object is dropped 1. Suppose a ball is dropped from the top of a 79 foot tall tree. A) Write an equation that describes the relationship between H, the height in feet of the ball above the ground, and time t in seconds. h t Jul 30 6:01 PM 10
11 H = 1gt 2 + v o t + h o 2 2. Johanna threw a water balloon upward at a speed of 10m/sec while standing on the roof of a building 12 meters high. A) What was the height of the balloon after 2 seconds? B) Assume the balloon did not land on the roof, and estimate how long it took the balloon to reach the ground. Jul 30 6:06 PM 11
12 Vertex Form: Standard Form: y = 2(x+5) 2 7 y + 5 = 2( x 6) 2 Jul 30 6:07 PM 12
13 Suppose a toy rocket is launched so that its height h meters after t seconds is given by the equation h = 4.9 t t How high is the rocket when launched? How high is the rocket after 12 sec? When does the rocket hit the ground? Sep 13 11:06 AM 13
14 Write in standard form y = ax 2 + bx + c y 1 = 2 ( x 4 ) 2 y = 3 ( x + 5 ) y 2 = ( x 9 ) Dec 1 1:34 PM 14
15 Jethro chucked a watermelon up from a height of 5'. The initial velocity was 35 ft/ sec. How high is the watermelon after 2 sec.? Is the watermelon still in the air after 3 sec? Dec 1 1:37 PM 15
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26 The graph of y=(x 2) 2 is the graph of y=x 2 shifted to the right two units. Dec 2 9:17 PM 26
27 The graph of y+4= x 5 is the graph of y= x shifted to the right five units and down 4 units. Dec 2 9:17 PM 27
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30 a. b. c. d. e. f. g. h. Dec 2 9:18 PM 30
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43 Attachments Chapter 6 Lesson Masters B.pdf
There are two types of solutions
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2 If ax + bx + c = 0, then x = b) What are the x-intercepts of the graph or the real roots of f(x)? Round to 4 decimal places.
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