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2. Start with the quadratic function: f(x) = ax² +bx+c and follow the steps below. 62 (a) Step 1: Show that f(x) (x + 2)² − b 2 + c - a 2a Assume that a 0. Hint: We did something like this when solving for the quadratic roots formula. = (b) Step 2: That part of (a) that depends on x is given by (x + 2)². We want to f(x) a study exactly how (x + 2)² depends upon x. Show that: i. (x+)² ≥ 0 for all values of ii. (x + 2)² = 0 only when x = - so (x + 2)² = 0 has 2 identical roots equal 2a 2a to b 2a b +h. 2a iii. This is an interesting result. Suppose that h is a positive number, it can be large of small. Show that (x + 2)² takes the value h² when x = Next show that (x + 2)² takes the value h² when x = h. Using a diagram show that this means that (x + 2)² is symmetric about x = - It behaves just like the function x² and we know x² is symmetric about x = 0. 2a

2. Start with the quadratic function: f(x) = ax² +bx+c and follow the steps below. 62 (a) Step 1: Show that f(x) (x + 2)² − b 2 + c - a 2a Assume that a 0. Hint: We did something like this when solving for the quadratic roots formula. = (b) Step 2: That part of (a) that depends on x is given by (x + 2)². We want to f(x) a study exactly how (x + 2)² depends upon x. Show that: i. (x+)² ≥ 0 for all values of ii. (x + 2)² = 0 only when x = - so (x + 2)² = 0 has 2 identical roots equal 2a 2a to b 2a b +h. 2a iii. This is an interesting result. Suppose that h is a positive number, it can be large of small. Show that (x + 2)² takes the value h² when x = Next show that (x + 2)² takes the value h² when x = h. Using a diagram show that this means that (x + 2)² is symmetric about x = - It behaves just like the function x² and we know x² is symmetric about x = 0. 2a

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2. Start with the quadratic function: f(x) = ax² +bx+c and follow the steps below. (a) Step 1: Show that f(¹) a Assume that a 0. Hint: We did something like this when solving for the quadratic roots formula. - (b) Step 2: That part of f(x) that depends on x is given by (x + b)². We want to a (x + 2)² − b ² + € a study exactly how (x + 2)² depends upon x. Show that: i. (x + 2)² ≥ 0 for all values of 2a b 2a ii. (x + 2)² = 0 only when x = 2a to iii. This is an interesting result. Suppose that h is a positive number, it can be large of small. Show that (x + 2)² takes the value h² when x = h. Using a b +h. 2a 2a b Next show that (x + 2)² takes the value h² when x = diagram show that this means that (x + 2)² is symmetric about x = - It behaves just like the function x² and we know x² is symmetric about x = 0. b 2a iv. Now, - X=- b so (x + 2)² = 0 has 2 identical roots equal 2a b can be positive, negative or zero. So, we can conclude that if 2a 2a 2a = 0 then (x + 2)² behaves exactly like x². When -20, then (x+21) ² behaves exactly like x² if the x² function had been moved horizontally so that it rested on the x-axis at x = Horizontal movements like this are sometimes called translations. b 2a* 2a v. So, we know that (x+ 2)² opens upward exactly like x². Moreover, (x + 2)² has a minimum (smallest numerical value) of 0 when x = - Illustrate this with a diagram. The line x = - is called the axis of symmetry of the quadratic function: the function is symmetric about this line. Note also that is the average of the 2 roots of the quadratic equation. b 2a 2a
Transcribed Image Text:2. Start with the quadratic function: f(x) = ax² +bx+c and follow the steps below. (a) Step 1: Show that f(¹) a Assume that a 0. Hint: We did something like this when solving for the quadratic roots formula. - (b) Step 2: That part of f(x) that depends on x is given by (x + b)². We want to a (x + 2)² − b ² + € a study exactly how (x + 2)² depends upon x. Show that: i. (x + 2)² ≥ 0 for all values of 2a b 2a ii. (x + 2)² = 0 only when x = 2a to iii. This is an interesting result. Suppose that h is a positive number, it can be large of small. Show that (x + 2)² takes the value h² when x = h. Using a b +h. 2a 2a b Next show that (x + 2)² takes the value h² when x = diagram show that this means that (x + 2)² is symmetric about x = - It behaves just like the function x² and we know x² is symmetric about x = 0. b 2a iv. Now, - X=- b so (x + 2)² = 0 has 2 identical roots equal 2a b can be positive, negative or zero. So, we can conclude that if 2a 2a 2a = 0 then (x + 2)² behaves exactly like x². When -20, then (x+21) ² behaves exactly like x² if the x² function had been moved horizontally so that it rested on the x-axis at x = Horizontal movements like this are sometimes called translations. b 2a* 2a v. So, we know that (x+ 2)² opens upward exactly like x². Moreover, (x + 2)² has a minimum (smallest numerical value) of 0 when x = - Illustrate this with a diagram. The line x = - is called the axis of symmetry of the quadratic function: the function is symmetric about this line. Note also that is the average of the 2 roots of the quadratic equation. b 2a 2a
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