written by Sunny Yoon
Make up linear functions f(x) and g(x). Explore, with different pairs of f(x) and g(x) the graphs for
h(x) = f(x) + g(x)
h(x) = f(x)*g(x)
h(x) = f(x)/g(x)
h(x) = f(g(x))
Summarize and illustrate.
Assumption:
When you add two linear functions f(x) and g(x), the sum of two functions h(x) is going to be another linear function.
When you multiply two linear functions, the product h(x) is going to be a quadratic function.
When you divide two linear functions, the quotient h(x) is going to be a rational function.
When you compose two linear functions, the composition h(x) is going to be another linear function.
Adding two linear functions
f(x) = ax + b
g(x) = cx + d
h(x) = (ax + b) + (cx + d)
When a > 0 and c > 0, then the slope of h(x) is going to be steeper than f(x) and g(x) since the slope of h(x) is going to be (a + c) > 0.
f(x) = 3x + 1(Green); g(x) = 5x + 7(Red)
h(x) = (3x + 1) + (5x + 7); (Blue)
Similar situation arises when a < 0 and c < 0.
f(x) = x  2(Green); g(x) = 4x + 5(Red)
h(x) = (x  2) + (4x + 5); (Blue)
It's easier to notice the steepness of each function as x goes to positive infinity. Since the yintercept of g(x) is greater than h(x), it's harder to determine the steepness on the left side of the graph.
What if the sign of a and c are different? Then, the sign of the bigger absolute value slope will determine the slope of h(x).
Here are two examples to illustrate that point.
f(x) = x  2(Green); g(x) = 3x + 1(Red)
h(x) = (x  2) + (3x + 1); (Blue)
The slope of h(x) is in between f(x) and g(x) and it it positive just like g(x) since the c > a.
f(x) = 4x + 5(Green); g(x) = 3x + 1(Red)
h(x) = (4x + 5) + (3x + 1); (Blue)
In order to determine the steepness of the functions, please look at the right side of the graph. Once again, the slope of h(x) is in between of f(x) and g(x) and the sign is negative just like f(x). Since a > c, the slope of h(x) is likely to follow the behavior of f(x).
What if a and c have different signs, but same absolute value? For example, f(x) = 2x  2 (Green), g(x) = 2x + 4 (Red) and h(x) = (2x  2) + (2x + 4) (Blue).
h(x) will always become a constant function since the slope of f(x) and g(x) will cancel each other out when you add them together.
Mutiplying two linear functions
f(x) = ax + b
g(x) = cx + d
h(x) = (ax + b) * (cx + d) = (ac)x^2 + (ad + bc)x + (bd)
We can find out that h(x) will always be a parabola since it is a quadratic equation.
Explanation  Graphical Representation  
Same sign (a,c are both positive or negative) 
If a and c are same signs, that the parabola will be a Ushape. It doesn't matter whether a > c or a < c because the new slope of h(x) is going to be a*c. Multiplying two positives or two negatives always result in a positive answer.
h(x) is represented by the blue parabola. 

Different Sign (a and c have a different sign) 
If a and c have different signs, then the parabola will be a upside down Ushape because a*c will result in a negative number.
h(x) is represented by the blue parabola. 
Dividing two linear functions
f(x) = ax + b
g(x) = cx + d
h(x) = (ax + b)/(cx + d)
h(x) = (ax +b)/(cx + d)  Example Used for the animation (n is between 0 and 10) 
Graphical Representation 
When a and c have same signs 
y = (nx +1)/(x  3)  
When a and c have different signs

Y = (nx + 1)/(x3) 
Composition of two linear functions
f(x) = ax + b
g(x) = cx + d
h(x) = a(cx + d) + b = acx + (ad + b)
When composing two linear functions, it becomes another linear function. However, the slope of the composed linear function changes. The new slope is going to be a product of two previous linear function.
h(x) = a(cx + d) + b  Example used for the graphical representation (n is between 0 and 10) 
Graph 
When a and c have same sign  y = 2(nx  1) + 3 The slope will be positive since two positives or two negatives equal to positive. 

When a and c have different signs  y = 2(nx  1) + 3 The slope will be a negative value since the product of two values with different signs equal to negative. 
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