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Calculus/Translation

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Expert: - 2/2/2009


Question
Find a formula for a translation of f(x)=x^3 that passes through the two points on the graph.  Explain how you found the solution.  The two points are (2,-2) (4,4)

Answer

Translations
Questioner:   Christy
Category:  Calculus
Private:  No
 
Subject:  Translation
Question:  Find a formula for a translation of f(x)=x^3 that passes through the two points on the graph.  Explain how you found the solution.  The two points are
(2,-2) (4,4)
.................................

This ain't going to be pretty.  If you have two points on y = x^3 whose y-values differ by 6, then they won't have integer coordinates:

f(-2) = -8
f(-1) = -1
f(0) = 0
f(1) = 1
f(2) = 8

No two of those differ by 6.  Bad news.
.....................
OK, then:

A translation has the form:

g(x) = (x - a)^3 + b.

Now observe that:

g(2) = -2, which gives you an equation in (a,b).
 
g(4) =  4, which gives you an equation in (a,b).

You have two equations in a and b, which you can solve.

(2 - a)^3 + b = -2

(4 - a)^3 + b =  4

(4 - a)^3 - (2 - a)^3 = 6

64 - 48a + 12a^2 - a^3 - (8 - 12a + 6a^3 - a^3) = 6

64 - 48a + 12a^2 - a^3 - 8 + 12a - 6a^3 + a^3 = 6

50 - 48a + 12a^2  + 12a - 6a^2  = 0

6a^2  - 36a  + 50  = 0

3a^2  - 18a  + 25  = 0


D = 324 - 300 = 24, not a perfect square (we knew this wouldn't be good)

   18 +- sqrt(24)
a = ---------------
        6


   18 +- 2 sqrt(6)
a = ---------------
        6

   9 +- sqrt(6)
a = ------------
        3

a = 3.816497
a = 2.183503

Now (4 - a)^3 + b =  4

b = 3.993821
b = -1.993821

g(x) = (x - 3.816497)^3 + 3.993821

g(x) = (x - 2.183503)^3 - 1.993821

This pretty much does it.

See attached.

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

Education/Credentials
(See above.)

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