Section 4.4 Quadratic Functions in Standard Form

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1 Section 4.4 Quadratic Functions in Standard Form A quadratic function written in the form y ax bx c or f x ax bx c is written in standard form. It s not right to write a quadratic function in either vertex form or standard form. Some ideas, like using a to decide whether the parabola opens up or down remain the same. Some ideas, like finding the vertex, become more difficult in standard and some, like finding the y-intercept, become easier. Whether written in standard form or vertex form the quadratic function itself is the same Reviewing Vocabulary for Quadratic Functions Let s use some functions written in standard form to help review vocabulary. Practice Reviewing Vocabulary for Quadratic Functions a) For each function decide whether the parabola will open up or down and find the y- intercept. Then finish the data table and, assuming the shape will follow the pattern, graph the function. Using the graph estimate any x-intercepts, the vertex, and the axis of symmetry. Since a is 1 the function opens up. Substituting 0 for x, and simplifying, shows the y-intercept is,. There are two x-intercepts at about 1, 0 and 3, 0 and the vertex is close to The axis of symmetry would be the line x 1. b),. The function opens down and has a y-intercept of 0, 5. There are no x-intercepts and the vertex is close to, 3. The axis of symmetry is the line x. Homework 4.4 For each function decide whether the parabola will open up or down and find the y-intercept. Then finish the data table and, assuming the shape will follow the pattern, graph the function. Using the graph estimate any x-intercepts, the vertex, and the axis of symmetry. 1) ) 3) 4) Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 147

2 4.4. Finding the Vertex and the Axis of Symmetry Algebraically The vertex is the lowest point when a parabola opens up or the highest point when a parabola opens down. When the function has the form f x ax bx c the x-coordinate of the vertex is the value b. The y-coordinate of the vertex is f which tells us to substitute the value for the x-coordinate into a b a the function and simplify. Here s some practice. Practice 4.4. Finding the Vertex and the Axis of Symmetry Algebraically f x x 4x 7 a) For each quadratic function find the vertex and the axis of symmetry The x-coordinate of the vertex is. Used b 4 a 1 The y-coordinate of the vertex is 3 b) The vertex is 3,. The axis of symmetry is x f x 5. 5x 40x 60 The x coordinate of the vertex is approximately 38.. Substituted for x and solved for y to find the y-coordinate of the vertex. f x x 4x 7 f 4 7 f 3 The axis of symmetry is the line parallel to the y-axis which runs through the x axis at. Used b and rounded. a 5. 5 The y coordinate is 16.. The vertex is 3. 8, 16.. The axis of symmetry is x 38.. Substituted 38. for x and solved for y using a calculator to find the y coordinate of the vertex. f f.. Rounding to another place value would have been fine also. Homework 4.4 Find the vertex and the axis of symmetry for each quadratic function. f x x 8x 3 6) 5) f x x x ) 1 y x 9 7) 4 y x 87x Finding the x-intercepts and Uses of the Discriminant To find the x-intercepts of a quadratic function written in standard form set the value of f x to 0 and solve the resulting equation 0 ax bx c for x. Recall from our earlier work that the discriminant helps predict the number and type of solutions to expect. If the discriminant is greater than zero there will be two real x-intercepts. If the discriminant is zero the vertex is on the x-axis, and there will be one real x- intercept. If the discriminant is less than zero there will be no real x-intercepts. This occurs if the vertex is Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 148

3 above the x-axis and the function opens up or if the vertex is below the x-axis and the function opens down. Here s some practice with finding x-intercepts. Practice Finding the x-intercepts and Uses of the Discriminant a) f x x x For each quadratic function use the discriminant to decide on the number of x- intercepts. If there are x-intercepts substitute 0 for f x and solve the resulting 4 7 equation to find them. 4 7 Wrote in the form f x x x f x ax bx c b) 7 There are no x-intercepts f x x x There are two x-intercepts. Since b 4ac which is less than 0 there are no x-intercepts which is positive Since b ac there are two x-intercepts. Substituted 0 for f x and solved for x. The x-intercepts are 00, and 1 0 7,. 0 7x x b b 4ac x a x x or x f x x x 1 c) There is one x-intercept there will be one real Since b ac x-intercept. Substituted 0 for f x and solved for x. The x-intercept is 10,. 0 x x1 0 x 1 1 Homework 4.4 For each quadratic function use the discriminant to decide on the number of x- intercepts. If there are x-intercepts substitute 0 for y and solve the resulting equation to find them. 9) f x x x 5 10) 1) f x x 87x 11) y 5x 3 f x 7. 8x 1. 3x ) y 5. 5x 40x ) y x x 3 Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 149

4 4.4.4 Graphing a Quadratic Function Written in Standard Form You now have a set of ideas which are helpful when graphing a quadratic function in standard form. Procedure Graphing a Quadratic Function Standard Form 1) Write the function in the form f x ax bx c ) Decide if the graph opens up a 0 or opens down 0 3) Substitute 0 for x and simplify to find the y-intercept. a. b 4) Find the vertex. The x-coordinate is. The y-coordinate is f a b a. 5) Find the axis of symmetry x b a 6) If the discriminant shows x-intercepts exist, find them. Practice Graphing a Quadratic Function Use the procedure to graph the quadratic function f x x 6x 5 a) The graph opens up a 1 which is greater than 0. The y-intercept is 05, Substituted 0 for x and solved to find y 5. The vertex is 34, The axis of symmetry is x 3 The x-coordinate of the vertex is b a y then 6 1 There are two x-intercepts. b 4ac which is greater than zero. The x-intercepts are 50, and 10, x x 5 or x When x 3 Homework 4.4 Use the procedure to graph the quadratic function 15) y x x 8 16) 18) y 5. 5x 40x ) y x x 1 f x x 4 17) y x x 4 f x 0. 5x x 4 0) Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 150

5 4.4.5 Applying a Quadratic Function Written in Standard Form We can use quadratic functions to ask and answer interesting questions. Practice Applying a Quadratic Function Written in Standard Form Between 198 and 1997 the function C t 0.9t 7.5t models the number of cases of chicken pox (in thousands) in the United States. t stands for the year since 198. a) Discuss why you suspect this function will have a maximum value. Since (so the parabola opens down) and the vertex is maximum value between 198 (year 0) and 1997 (year 15).., there will be a b) Find the year the number of cases reached a maximum and find the number of cases that year. The vertex 8., tells us that the number of cases will reach a maximum around 1990 when there will be about 169,000 cases. c) How would you use functional notation to ask the question, How many cases of chickenpox were there in the United States in 1994? C 1 d) Answer the question that Ct 150 asks. Solving t 7.5t for t shows that in year 199 (year 10) there were 150,000 cases. You could also argue that in year 1980 (year ) there may also have been around 150,000 cases. Homework 4.4 1) The number of farms in the United States between 1975 and 009 (in thousands) can be modeled using the function F t 0.6t 3t 544. a) Find the year the number of farms reached a minimum and find the number of farms that year. b) Find the year there were,000 more farms than in c) Find and discuss the meaning of F 0. d) Translate, What year do you expect there to be,800,000 farms? into functional notation. ) If an object is thrown into the air with an initial speed of 88 feet per second the function H t 16t 88t describes its height as a function of time. (Disregard the height of the source that is throwing the object.) a) Find the maximum height the object will reach. b) Find when the object will have a height that s 100 feet less than it s maximum height. c) Ask the question, How high will the object after one second? using functional notation. d) How many seconds will it take for the object to hit the ground? Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 151

6 3) A person who makes vintage bicycles finds if they charge $p per bicycle then the function P p 0. p 4p 4, 000 estimates their profit (or loss) for the year. a) Find the price that will maximize the profit and find this maximum profit. b) Find P 0 and suggest a reason for the answer. c) Find p if P p 30, 600 and speculate on the meaning of each answer. d) Find the change in profit/loss if prices are raised from $10 to $310. Find the change in profit if prices are raised from $500 to $600. Discuss this difference in profit for the same $100 increase in price. 4) The function R t 6. 7x 158x 137 predicts the cases of Rocky Mountain Spotted Fever (in thousands) in the United States using the year since a) Predict the year the number of cases reached a minimum and find this minimum number of cases. b) Translate, Find the year(s) there were 1,500,000 cases of Rocky Mountain Spotted Fever. into functional notation. c) Find R 14. Use the axis of symmetry to predict the other year this number of cases occurred. Verify your answer using the function. Homework 4.4 1) ) 3) The function opens down, has a y- intercept of 0, x-intercepts at about 4. 5, 0 and 0. 5, 0 vertex at 6,. The axis of and a symmetry is the line x and the maximum y value is 6. The function opens up, has a y- intercept of 06, an x-intercept of 0, and a vertex at 0,. The axis of symmetry is the line x and the minimum y value is 0. The function opens down, has a y- intercept of 0,8 no x-intercepts and a vertex at 4, 4. The axis of symmetry is the line x 4 and the maximum y value is 4. Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 15

7 4) The function opens up, has a y- intercept of 05, x-intercepts at 1, 0 and 5, 0 and a vertex at 3 4,. The axis of symmetry is the line x 3 and the minimum y value is 4. 5) The x coordinate of the vertex is 8 8. The y coordinate is b a 4 The vertex is, 11 and the axis of symmetry is x. b 0 0 6) The x coordinate is a The y coordinate is The vertex is 0, 9 4 and the axis of symmetry is x 0, which is the y axis. 7) The x coordinate is b The y coordinate is a , The vertex is x ) The x coordinate is b The y coordinate is a The vertex is x ) The discriminant is b ac intercepts..,. and the axis of symmetry is.,. and the axis of symmetry is which is less than 0. There are no real x- b 4ac , 517 which is greater than 0 so there are 10) The discriminant is real x-intercepts. The x-intercepts can be found using. The x-intercepts are , 0 and 35., 0. 11) The discriminant is b ac which is greater than 0 so there are real x- intercepts. The x-intercepts can be found using 0. 77, 0 and 0. 77, 0. 1) The discriminant is b ac... intercept can be found using The x-intercepts are so there is one real x-intercept. The x ) The discriminant is real x-intercepts. 14) The discriminant is intercepts.. The x-intercept is 30,. b 4ac , 60 which is less than 0, so there are no 1 1 b 4ac 1 4 which is less than 0 so there are no real x- 3 3 Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 153

8 15) Opens up, y-intercept 0, 8, vertex 1 9,, axis of symmetry x 1, the radicand is 36 so there are two real x-intercepts, 0 and 4, 0. 16) Opens down, y-intercept 0, 4, vertex 0 4,, axis of symmetry x 0, the radicand is 8 so there are no x- intercepts. 17) Opens up, y-intercept 00,, vertex symmetry 1 1,, axis of x, The radicand is 1 and the real x- 4 1 intercepts are 0, 0 and, 0 18) Opens up, y-intercept 0, 660, vertex ,., axis of symmetry x 38., The radicand is 1, 60 so there are no x-intercepts. 19) Opens down, y-intercept 0,, vertex 5 1,, axis of symmetry x 1, The radicand is 5 so there are two real x-intercepts. The x-intercepts are 3. 4, 0 and 1. 4, 0. Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 154

9 0) Opens up, y-intercept 04,, vertex 40,, axis of symmetry x 4, The radicand is 0 so there is one real x- intercept. The x-intercepts is 40,. 1) a) The vertex 6. 7, tells us that in 00 the number of farms reached a minimum at around,117,300 farms. b) In 1980 there were about F 5 399,399,000 farms. Solving t 3t 544 will give the year(s) there were 100,000 more farms. Since t 1 and t 5 there will be 100,000 more farms around 07 if the function continues to be accurate. c) Since F there will be around,144,000 farms in d) Ft 800 ) a) The vertex. 75, 11 tells us that after.75 seconds the object will reach its maximum height of 11 feet. b) Solve Ht 1 shows that after a quarter of a second and after five and quarter seconds the object will be 1 feet off the ground. c) H 1. d) 0 F t when t 0 and t 5. 5 so after five and a half seconds the ball will hit the ground. 3) a) The vertex 605, 3105 tells us that setting the price at $605 per bicycle will maximize profit at $31,05 per year. b) P 0 4, 000 which implies setting the price at $0 will lead to a $4,000 loss. Probably for initial expenses such as parts. c) Solving for p gives the answers $550 and $660. This implies if we set the price at $55 less than the maximum price (of $605) we will lose profit due to charging too little. If we set the price at $55 more than the maximum price we will again lose profit probably due to selling fewer bicycles at the higher price. d) The profit at $10 is $0, while the profit at $310 is $1,600 for a gain of $1,600. The profit at $500 is $9,000, while the profit at $600 is $31,00 for a gain of $,00. In both cases profit is increasing but the increase is getting smaller for the same amount of change in price. 4) a) The vertex 11. 8, tells us that around 199 the number of cases reached a minimum at 440,508 cases. b) R t 1, 500. c) R The axis of symmetry is. Since 14 is about. units to the right of the axis of symmetry moving two units to the left, or x = 9.6 should yield a y value of close to R which is close as was expected. Copyright 013 Scott Storla 4.4 Quadratic Functions in Standard Form 155