The Investigation of the Linear Function

By

Ronnachai Panapoi

In this write-up assignment, I will discuss on the graph of function which is formed by the different types of two linear functions.

Let f(x) and g(x) be linear functions given by

f(x) = ax + b

g(x) = cx + d

Firstly, I will explore the graph of h(x) defined as

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

In this case, we surely know that h(x) is still the linear function since h(x) is in the form of (a + c)x + (b+d).

After that, I investigate how the h(x) changes when each constant varies. By adjusting the constant with the four sliders, we have seen the changes of the graph, h(x).

Click HERE to investigate the characteristics of graph of h(x)

In the case of h(x) = f(x) + g(x), we know that slope of the graph of h(x) is (a + c). So, as the a, or c changes, the slope of h(x) also changes at the same time.

In addition, the y-intercept of h(x) changes when the y-intercepts of f(x) and g(x) vary. We can investigate again to make sure about this notion by clicking HERE.

Secondly, we will study another type of h(x) defined as

h(x) = f(x)g(x) for different types of the linear functions, f(x) and g(x).

In this case, we know that the graph of h(x) is a parabola as a result of the product of two linear functions, f(x) and g(x), which is a quadratic equation.

I will investigate how each constant affects to the characteristics of h(x) by replacing different constant into the equation. Click HERE to see the change of h(x) after the constants are adjusted.

While exploring, I have seen that the parabola (h(x)) has the lowest point when both values of a, c are positive and negative at the same time. By contrast, the h(x) has the highest point when a, c are not positive or negative at the same time. Click HERE to see whether or not what I have investigated is true.

Further, if only the value of b changes, I have discovered that h(x) moves up and down. Click HERE to see the picture.

The graph of h(x) moves up and down around the x-intercept point of g(x). Particularly, the path of the vertex of h(x) looks like the parabola.

Next, I will talk about what I have found as investigating h(x) in the form of h(x) = f(x)/g(x)

I have seen that the graph of h(x) in this case is the disconnected curve.

Click HERE to see the graph of h(x) with various constants in f(x) and g(x).

Noticeably, the graph of h(x) intersects the x-axis and

y-axis at the points of which the graph of f(x) intersects x-axis and y-axis.

Looking at the graph of h(x), one of the curves goes from the left-hand side close to the line: y = a and moves down near the line: x = -d/c.

The other curve goes from the right-hand side closely-above the line: y = a and then move up near the line: x = -d/c.

The criteria to make a decision when the graph of h(x) will move up at the left or the right depends on the intersection point of f(x) and g(x). The graph of h(x) will move up at the left if the intersection point of f(x) and g(x) is on the left side of the line: x = -d/c. Similarly, the graph of h(x) will move up at the right if the intersection point of f(x) and g(x) is on the right side of the line: x = -d/c.

Click HERE to see the picture about this idea.

Finally, we will explore and summarize what we have seen of the graph of h(x) defined by h(x) = f(g(x)).

Let investigate the graph of h(x) first by clicking HERE.

Of course, we know that h(x) is still the linear function.

This is because

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

= f(cx + d) = a (cx + d) + d = (ac)x + d

Although, the graph of h(x) is the linear which we know very well about its y-intercept point and its slope by looking at d and ac, respectively, there remains some interesting point in this case.

As exploring, it is noticeable that sometimes the graph of h(x) is parallel to f(x) or g(x).

Click HERE to look for when do these events occur?

What have you seen as you explore the graph of h(x)?

Some facts I have got from exploring the graph are as follows:

1. When the product of a and c is prime, the graph of h(x) is parallel to one of the two graphs, f(x) and g(x).

This suggests that one of these graphs has slope equal to1 and the other is the prime.

I, therefore, obtain the first fact that the graph of h(x) is parallel to f(x) or g(x) if it has the slope as a prime.

2. Apparently, these three graphs will be parallel to each other if their slopes are equal to 1.