2. Start with the quadratic function: f(x) = ax² +bx+c and follow the steps below. (a) Step 1: Show that f(x) a (x + 2)² - 10² + å 2a Assume that a 0. Hint: We did something like this when solving for the quadratic roots formula. = (b) Step 2: That part of f(ª) that depends on ä is given by (x + 2)². We want to 2a a study exactly how (x + 2)² depends upon x. Show that: 2a i. (x + 2)² ≥ 0 for all values of ii. (x + 2)² = 0 only when x = to b 2a -2/a so (x + 2)² = 0 has 2 identical roots equal 2a b +h. 2a b 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 = 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. h. Using a 2a b 2a -
2. Start with the quadratic function: f(x) = ax² +bx+c and follow the steps below. (a) Step 1: Show that f(x) a (x + 2)² - 10² + å 2a Assume that a 0. Hint: We did something like this when solving for the quadratic roots formula. = (b) Step 2: That part of f(ª) that depends on ä is given by (x + 2)². We want to 2a a study exactly how (x + 2)² depends upon x. Show that: 2a i. (x + 2)² ≥ 0 for all values of ii. (x + 2)² = 0 only when x = to b 2a -2/a so (x + 2)² = 0 has 2 identical roots equal 2a b +h. 2a b 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 = 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. h. Using a 2a b 2a -
ChapterP: Prerequisites
SectionP.4: Factoring Polynomials
Problem 84E: The rate of change of an autocatalytic chemical reaction is kQxkx2 where Q is the amount of the...
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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(x)
a
(x + 2)² - 10² + å
2a
Assume that a 0. Hint: We did something like this when solving for the
quadratic roots formula.
=
(b) Step 2: That part of f(ª) that depends on ä is given by (x + 2)². We want to
2a
a
study exactly how (x + 2)² depends upon x. Show that:
2a
i. (x + 2)² ≥ 0 for all values of
ii. (x + 2)² = 0 only when x =
to
b
2a
so (x + 2)² = 0 has 2 identical roots equal
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.
b
2a
b
2a
b
2a
iv. Now, can be positive, negative or zero. So, we can conclude that if
za
= 0 then (x + 2)² behaves exactly like x². When - 20, then (x+ 2)²
b
2a
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.
v. So, we know that (x + 2)² opens upward exactly like x². Moreover, (x+)²
2a
has a minimum (smallest numerical value) of 0 when x =
b
2a
Illustrate
b
2a
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
x = -2 is the average of the 2 roots of the quadratic equation.
2a
Transcribed Image Text:(c) Step 3: Go back to Step 1, and look at that part of f(x) that does not depend upon
a
6²
a
x. Call it K = - 12+. Show that K can be positive or negative depending upon
the values of the parameters (a, b, c). Show that the effect of K is to shift (x+2)²
up or down by a constant amount. Conclude that (2)
6²
-
= (x + 1) ². + is
2
simply the function r2 shifted horizontally so it sits at x =
and then it is shifted up or down by the amount K: upwards when K > 0 and
downwards when K<0. See hint in Step 4.
- on the x-axis
(d) Step 4: Provide an intuitive explanation why f(x) ax²+bx+c = a
a f(x)
a(x+2)²-ab²+a will behave exactly like the function ax² shifted horizontally
so it sits at x = -za on the x-axis and then shifted up or down by the amount
ak. Hint: Start with the function g(x) = ax². Next, figure out what the function
g(x+2) looks like. Discover that adding a constant to x in a function like g(x)
shifts the function horizontaly to the left or the right depending upon the sign of
the constant () that is added.
=
=
2a
(e) Step 5: Show that if a < 0 then f(x) has a maximum x = - and when a > 0,
f(x) has a minimum x = - Show this with the special cases f(x) = x²+x+1
and f(x) = −x² + x + 1. When a < 0, f(x) opens downward and when a > 0,
f(x) opens upward.
(f) Step 6: Show that the sum of two quadratic functions will usually be a quadratic
function but it may be a linear function or a constant function.
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