Mathematics Education

EMAT 6680, Professor Wilson

Exploration 1, Combining Functions by Ursula Kirk

Make up a linear function f(x) and g(x). Explore with different pairs of f(x) and g(x) the graph for:

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·

·

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Summarize, explain and illustrate

Graph Number 1

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we add two linear functions, the new function is also a linear function. Our new function is a straight line with a positive slope.

Graph Number 2

For and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we multiply two linear functions, the result is a quadratic function. Our new function is a concave up parabola.

Graph Number 3

For and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we divide two linear functions the result is a rational function. Our new function has an asymptote at

Graph Number 4

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we find the composite of two linear functions, the result is another linear function. Our new function is a straight line with a positive slope.

Exploration Number 2

Make up a linear function f(x) and g(x). Explore with different pairs of f(x) and g(x) the graph for:

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·

·

·

Summarize, explain and illustrate

Graph Number 1

Next, we will repeat the exploration with a new set of functions

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we add two linear functions, the new function is also a linear function. Our new function is a straight line with a negative slope.

Graph Number 2

For and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we multiply two linear functions, the result is a quadratic function. Our new function is a concave down parabola.

Graph Number 3

For and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we divide two linear functions the result is a rational function. Our new function has an asymptote at .

Graph Number 4

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we find the composite of two linear functions, the result is another linear function. Our new function is a straight line with a negative slope.

Exploration Number 3

Make up a linear function f(x) and g(x). Explore with different pairs of f(x) and g(x) the graph for:

·

·

·

·

Summarize, explain and illustrate

Graph Number 1

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we add two linear functions, the new function is also a linear function. Our new   function is a straight line with a slope of zero.

Graph Number 2

For and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we multiply two linear functions, the result is a quadratic function. Our new function is a concave down parabola.

Graph Number 3

and, we will calculate

);

Now, we can graph our new function

Here we can observe that when we divide two linear functions the result is a rational function. Our new function has an asymptote at .

Graph Number 4

For and, we will calculate

;

Now, we can graph our new function

Here we can observe that when we find the composite of two linear functions, the result is another linear function. Our new function is a straight line with a negative slope.

Conclusion

After completing the three explorations, we can conclude that:

·         When we add two linear functions, the result is another linear function.

·         When we multiply two linear functions, the result is a quadratic function.

·         When we divide two linear functions, the result is a rational function.

·         When we find the composite of two linear functions, the result is a linear function.

1.      When we add two linear functions, the result is another linear function.

Given two functions f(x) = mx + b and g(x) = nx + d, we can add them together so:

h(x) = f(x) + g(x)= (mx + b) + (nx + d)= (mx + nx) + (b + d)= (m + n)x + (b + d)

As we can see, the addition results in a new linear function, having a slope of m + n and a y-intercept of b + d.

2. When we multiply two linear functions, the result is a quadratic function.

Given two functions f(x) = mx + b and g(x) = nx + d, we can multiply them together so:

f(x)g(x) = (mx + b)(nx + d)= mxnx + mxd + bnx + bd= mnx2 + mdx + bd

The coefficients are: a = mn, b = md and c = bd

3. When we divide two linear functions, the result is a rational function.

Given two functions f(x) = mx + b and g(x) = nx + d, we can divide them together so:

h(x) = f(x)/g(x) = (mx + b)/(nx + d)

Vertical Asymptote

nx + d = 0

x = -d/n

Horizontal Asymptote

yn - m = 0

y = m/n

4. When we find the composite of two linear functions, the result is a linear function.

Given two functions f(x) = mx + b and g(x) = nx + d, we can find their composition so:

h(x) = f(g(x))= m(nx + d) + b= mnx + md + b= (mn)x + (md + b)

k(x) = g(f(x))= n(mx + b) + d= nmx + nb + d= (nm)x + (nb + d)