Assignment 1: Graphs

June 18, 2003

Write-up: #3. Find two linear functions f(x) and g(x)

Key Points:

Click the Graphing Calculator File  to run any of these problems.

Problem Statement

 Find two linear functions f(x) and g(x) such that their product h(x) = f(x).g(x) is tangent to each of f(x) and g(x) at two distinct points. Discuss and illustrate the method and the results.

Solution

Easiest way seems to be to assume lines are reflection across y axis, so slopes equal and negative of one another for the tangent lines, although there would be cases of this (seems) for the rotation of above figure.

 To help visualize parabolic of this shape, the following is one of simplest formulas y = (-x+2)(x+2). Now if lines are tangent to the parabola then slopes = and the (x,y) are equal at that tangent point. So we have y=mx+b and lets see if we can find matching lines for the simple parabola above.  Taking derivatives for the line y'(-x+2) = -1 and parabola  y'(-x^2+4) = -2x, so slopes are = at -1 = -2x ;  so at x= ½ and -½.    We just solve for b on lines by solving equations.  We can't assume this event will actually be at the y=-x+2 line we modeled above, so we solve two equations knowing (x,y) are equal at this tangent point ( ½ , y) for both line and parabola. o        Line is  y = -x+b = - ½ +b.  o        for parabola y = - ½ ^2 + b^2 so setting equal and solving for b.  b^2 –b = - ½ + ¼ = - ¼ .  or (b- ½ b)^2 = 0, so b = ½ So tangent lines are really y = -x+ ½ and y = x + ½ Tryed on GSP below

 Seems to check out.

Notes on Other Problems

1. Examine:

.

Call 4 the a variable

Call 1 the b variable

What happens if the a is replaced by other numbers (not necessarily integers)? Try 5, 3, 2, 1, 1.1, 0.9, -3. Any unusual event?

Changing a adjusts the x intercepts out to +/- sqrt(number).  Curves expand up (out) as well.

Changing b adjusts the y intercepts out to +/- sqrt(number).

At a=b the graph is a line and an ellipse.  (factor equations) when a-b>0 – ellipse expands for n>0.

Interpret. What equation would give the following graph:

What happens if a constant is added to one side of the equation? Try several graphs in some systematic way. Click here for one set of graphs.

Try graphing

2. Make up linear functions f(x) and g(x).

Explore, with different pairs of f(x) and g(x) the graphs for

i. h(x) = f(x) + g(x)

ii. h(x) = f(x).g(x)

iii. h(x) = f(x)/g(x)

iv. h(x) = f(g(x))

Summarize and illustrate.

 Lines of course follow expectations and addition of them gives average intercepts & slope. Multiplication of linear eq is more interesting.  X & y intercepts predictable by solving for 0's. For division (x+3)/(3x-2) -> think about later

3. Find two linear functions f(x) and g(x) such that their product

h(x) = f(x).g(x)

is tangent to each of f(x) and g(x) at two distinct points. Discuss and illustrate the method and the results.

See above

4. Repeat Problem # 3 above where f(x) and g(x) are quadratic functions

and each function, f(x) and g(x) is tangent to h(x) in two different points. That is, h(x) is a fourth degree equation and each of the second degree equations, f(x) and g(x) is tangent to h(x) in two points.

 Use same basic logic as for #3.  Just explore to visualize a bit Assume no variable on main quadratic again so F(x) or y = x^2 + ax + c: Do two simple equations first: y=x^2 + 1x+0, and y=x^2+

5. Examine graphs of

y = a sin(bx + c)

for different values of a, b, and c.

6. Graph

What do you expect for the graph of

or

7. Let f(x) = a sin(bx + c) and g(x) = a cos(bx + c).

For selected values of a, b, and c, graph and explore:

i. h(x) = f(x) + g(x)

ii. h(x) = f(x).g(x)

iii. h(x) = f(x)/g(x)

iv. h(x) = f(g(x))

8. Explore the following equation for different values of a.

Some examples:

| a = 0 | | a = 1 | | a = 3 | | a = 5 | | a = 10 |

| a = -10 | | a = -3 | | a = -5 | | a = .10 |

| a = .50 | | a = -.50 | | a = -.10 |

| Multiple Graphs |

9. Consider two points (3,4) and (-5,-2). For any point (x,y) we can write the distance equations for these as

Explore graphs with these two distance equations. For example,

a. Consider when each is set to a non-zero constant. Circles are graphed.

b. Consider the sum

for various values of C.

c. Consider the product

for various values of C.

d. Other?