Hampton math - Math 2



Steps for Solving Quadratic Equations by Factoring:1. Get the equation in standard form equal to zero: ax2 + bx + c = 02. Factor the quadratic expression3. Set each factor equal to zero (zero product property)4. Solve each equation5. Check your solutions 1. x2 + 12x +32 = 02. x2 – 6x + 9 = 0 3. x2 - 2x - 7 = 84. (2x -2)(x + 3) = 05. x2 – 36 = 06. 2x2 + 5x -12 = 0Practice: Solve each equation.1. x2 + 7x + 6 = 02. x2 – 36 = 03. x2 – 6x + 9 = 04. x2 – x = 125. x2 + 6x = 406. x2 – 13x + 30 = 07. 2x2 + 7x -15 = 08. 5x2 + 14x = 3 9. 14x2 + 26x – 4 = 0 For each question below, sketch a rough draft of the graph and label the starting height (y-intercept), the maximum, and the second x-intercept/zero (where the object hits the ground). Then answer the questions that follow WITHOUT USING A CALCULATOR! Show all work below.A baseball is thrown straight up from the top of a 128 foot tall building with an initial speed of 32 feet per second. The height of the ball as a function of time can be modeled by the function h(t) = –16t2 + 32t + 128. How long will it take for the baseball to reach its maximum height?What is the baseball’s maximum height?How long will it take for the baseball to hit the ground?How long will it take the baseball to be at the same height as when it was thrown?A bouncy ball is thrown straight up from the top of a 288 foot tall building with an initial speed of 48 feet per second. The height of the ball as a function of time can be modeled by the functionh(t) = –16t2 + 48t + 288. How long will it take for the bouncy ball to reach its maximum height?What is the bouncy ball’s maximum height?When will the bouncy ball reach a height of 320 feet?A rocket is launched straight up from the top of a 24 foot tall building with an initial speed of 92 feet per second. The height of the rocket as a function of time can be modeled by the function h(t) = –16t2 + 92t + 24. How long will it take the rocket to reach its maximum height?What is the rockets maximum height?How long will it take for the rocket to hit the ground?A rock is dropped from a bridge 256 feet above a river. The height of the rock as a function of time can be modeled by the function h(t) = –16t2 + 256. What is the rock’s maximum height and when does it reach that height?When will the rock hit the water?5936615147320000236549A baseball is “popped” straight up by a batter with an initial velocity of 64 ft/sec from an initial height of 3 feet Although the path of the ball is straight up and down, the graph of its height as a function of time is a parabola. The ball goes up fast at first and then more slowly because of gravity before it reaches its peak and falls back to the earth. 00A baseball is “popped” straight up by a batter with an initial velocity of 64 ft/sec from an initial height of 3 feet Although the path of the ball is straight up and down, the graph of its height as a function of time is a parabola. The ball goes up fast at first and then more slowly because of gravity before it reaches its peak and falls back to the earth. Example 1. Verbal Representation (calculator)A. Provide an algebraic representation for this function:Provide an algebraic representation for this function:___________________________________________________________________________-38608053657500B. Consider the graphic representation shown below. Produce a corresponding graph in your calculator.A. What is the maximum height the ball reaches?B. What is the time it takes to reach that maximum?C. When does the ball hit the ground?575373524003000236855264160Suppose a particular star is projected from an aerial firework at a starting height of 520 feet with an initial upward velocity of 72 ft/s. How long will it take for the star to reach maximum height? How far above the ground will it be? 00Suppose a particular star is projected from an aerial firework at a starting height of 520 feet with an initial upward velocity of 72 ft/s. How long will it take for the star to reach maximum height? How far above the ground will it be? Example 2. Verbal Representation:42957754381500Provide an algebraic representation for this function:___________________________________________________Sketch a Graph of this function. Practice: 1. An object is launched directly upward. The path of the object can be modeled by the equation .Sketch the situation graphically:323977012954000Label the following on your graphWhere the object is launched fromMax height of the objectWhere the object hit the groundFind the actual values for A,B, and C.A: ___________________B: ___________________C: ___________________3681095281940002. Nick’s height in meters above the water t seconds after diving from a diving board into a pool can be modeled by . A. What is Nick’s max height above the water and when does this occur?B. When does Nick enter the water?C. At what height is Nick after 2 seconds?3. Leo throws a rock upward from the top of a 50-foot cliff. The rock lands at the base of the cliff. The rock’s height above the base of the cliff after t seconds is represented by .453326521971000A. What is the rocks max height above the ground and when does this occur?B. When does the rock hit the ground?C. When is the rock at the same height as the top of the cliff?4142740330200004. Suppose that a bottle rocket is launched upward so that its path can be modeled by the equation . A. What is the max height that the rocket reaches andwhen does it occur?B. When does the rocket hit the ground?C. How high was the rocket after 10 seconds?5. At a festival, pumpkins are launched with large catapults and air cannons. On one launch, the height of a pumpkin in feet above the ground after t seconds is modeled by. 430212515938500A. Sketch the path of a pumpkin on the axes.B. Label the important parts of the path.(Launch point, max point, landing)C. Find the maximum height of the pumpkin.D. After how many seconds did the pumpkin reach its max height?E. When did the pumpkin reach the ground?Real World Connections Practice1. Manufacturing An electronics company from 1995 developed a line of portable radios with CD players. Their research suggested that the daily sales s for the product could be modeled by , where p was the price of each unit.Find the maximum daily sales.What price will result in that maximum?Do you think this model would still hold true today?4686300291465002. Architecture The shape of the Gateway Arch in St. Louis, Missouri, is a catenary curve, which loosely resembles a parabola. The function models the shape of the arch, where y is the height in feet and x is the horizontal distance from the base of the left side of the arch in feet.According to the model, what is the maximum height of the arch?What is the width of the arch at the base?3. Field Hockey Suppose a player makes a scoop that releases the ball from the ground with an upward velocity of 34 ft/s. a. Write a function rule to model the flight of the ball.b. Will the ball ever reach a height of 20 ft? Explain.4. Business The weekly revenue, R, for a company is , where p is the price of the company’s product. When will the weekly revenue reach $1500? Explain.-113665528320005. Woodland Jumping Mouse The woodland jumping mouse can hop surprisingly long distances given its small size. A relatively long hop can be modeled by where x and y are measured in feet.How far can a woodland jumping mouse hop?Can a woodland jumping mouse jump a tree stump that is 2.5 ft high?Using the Quadratic FormulaSolve each equation with the quadratic formula.Helpful hints to solving word problems:1.2.3.4.1. If each of the sides of a square is lengthened by 6 cm, the area becomes 144 cm2. Find the length of one side of the original square.2. Bob is fencing in a pen along an outside wall of his house for his dog. Bob has 60 meters of fencing and wants to give his dog the greatest possible area. Find the dimensions of the pen.3. Steve is planning a garden that is 25 m longer than it is wide. The garden will have an area of 7500 m2. What will its dimensions be?4. A flower bed is to be 3 m longer than it is wide. The flower bed will have an area of 108 m2. What will its dimensions be?5. The length of a pool at a local YMCA is 10 feet more than its width. A walkway 4 feet wide surrounds the outside of the pool. If the total area of the walkway and pool is 999 square feet, find the dimensionsof the pool. 6. A rectangular pen is to be built along a wall from 72 yards of fencing. Find the maximum area that can be enclosed and the dimensions of the pen.7. A rectangular piece of ground is to be enclosed on three sides by 160 ft of fencing. The fourth side is the barn. Find the dimensions of the enclosure and the maximum area that can be enclosed.8. The length of a tropical garden at a local conservatory is 5 feet more that its width. A walkway 2 feet wide surrounds the outside of the garden. If the total area of the walkway and garden is 594 square feet, find the dimensions of the garden.9. Helen has a rectangular garden 25 feet by 50 feet. She wants to increase the garden on all sides by an equal amount. If the area of the garden will be increased by 400 square feet, by how much will each dimension be increased?10. A rectangle is 5 centimeters longer than it is wide. Find the possible dimensions if the area of the rectangle is 104 square centimeters.1. From 4 feet above a swimming pool, Susan throws a ball upward with a velocity of 32 feet per second. The height of the ball t seconds after Susan throws it is given by . a. Find the maximum height reached by the ball and the time this height is reached.b. When was the ball at the same height as when it was thrown?2. Marta throws a baseball with an initial upward velocity of 70 feet per second. This equation models the situation. a. Ignoring Marta’s height, how long after she releases the ball, will it hit the ground?b. What is the maximum height of the baseball?3. A volcanic eruption blasts a boulder upward with an initial velocity of 240 feet per second. This is modeled by the equation .a. How long will it take the boulder to hit the ground? b. How high was the boulder after 5 seconds?4. A baseball player hits a high pop-up with an initial upward velocity of 30 meters per second, 1.4 meters above the ground. The height of the ball (in meters) t seconds after being hit is modeled by . a. How long will it take for the baseball to hit the ground? b. What time will the ball be 15 meters high?5. A rectangular lot is 50 feet wide and 60 feet long. If both the width and the length are increased by the same amount, the area is increased by 1200 square feet. Find the amount by which both the width and the length are increased.6. A rectangular lawn is 60 feet by 80 feet. How wide of a uniform strip must be cut around the edge when mowing the grass in order for half of the lawn to be cut?7. A rectangular lawn has dimensions of 24 feet by 32 feet. A sidewalk will be constructed along the inside edges of all four sides. The remaining lawn will have an area of 425 square feet. How wide is the walk?8. A picture frame measures 12 cm by 20 cm (uniform width). The picture without the frame measures 84 . Find the width of the frame. (challenge)9. A rectangular picture is 12 inches by 16 inches. If a frame of uniform width contains an area of 165 square inches (frame only), what is the width of the frame?10. A farmer has 400 feet of fencing. He wants to fence off a rectangular field that borders a straight river (no fence along the river). What are the dimensions that will give him the largest area?Consecutive Integer Word Problems.Definition of consecutive: ______________________________________________________________Examples of consecutive integers: _______________________________________________________1st: 2nd:3rd:Examples of consecutive odd integers: ____________________________________________________1st: 2nd:3rd:Examples of consecutive even integers: ____________________________________________________1st: 2nd:3rd:COMPARING FUNCTIONS 1Name of FunctionGeneral Shape of GraphSketchLinearQuadraticExponentialComplete the following tables and answer the questions to the right. (a) Xy = 2x86550540640This function is. ??linear ? quadratic? exponentialWhat methods can you use to verify the type of function selected?00This function is. ??linear ? quadratic? exponentialWhat methods can you use to verify the type of function selected?1st Diff -3 -2 -1 0 1 2 3 (b) x y = x21st Diff8375652540This function is. ? linear ? quadratic ? exponentialHow do you know?What do the 1st and 2nd differences represent in this function? 00This function is. ? linear ? quadratic ? exponentialHow do you know?What do the 1st and 2nd differences represent in this function? 2nd Diff -3 -2 -1 0 1 2 3(c) x y = 2x1st Diff90043020955What do you notice about the differences in this function? ______________________________________________By what number is the first difference multiplied by to get the next term in the sequence of y-values? ______________________________________________How does this value connect to the function?______________________________________________This function is. ??linear ? quadratic ? exponentialWhat methods can you use to verify the type of function selected?This is called an _________________ function00What do you notice about the differences in this function? ______________________________________________By what number is the first difference multiplied by to get the next term in the sequence of y-values? ______________________________________________How does this value connect to the function?______________________________________________This function is. ??linear ? quadratic ? exponentialWhat methods can you use to verify the type of function selected?This is called an _________________ function2nd Diff -3 -2 -1 0 1 2 32. Use differences to identify the type of function represented by the table of values. Then label which type of function each table of values models.xyxY xyxy-45-532-2-2.750.50.9-38-4160-20.751.1-213-382111.3-120-244131.251.5029-126611.51.71400182531.751.9Identify the following equations as linear, quadratic or exponential.1. 2. 3. 4. 5. 6. All _______________ functions have _______________________.All _______________ functions have a _____________________________.All _______________ functions must have a _______________ in the _______________. Identify the following equations as linear, quadratic or exponential.1. y = 4x + 6 2. 3. y = x2-5x+64. y = -2(4)x5. y = 3x +36. fx=x-22+7 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download