Megan Langford

We know how linear functions operate. The slope indicates the steepness of the line, and the additive term indicates the y intercept. But what happens when we combine two linear functions via some operation? Let’s explore using the following two functions.

We know that the pink function increases by two units vertically for every one unit increase horizontally, and we can see this behavior in the graph. Additionally, we know that the y intercept is 1, since this is our additive term. The same is true for the red function, since the line appears to increase by 3 vertical units for every 1 unit horizontal increase. In fact, the y intercept does appear to be -2, which is our additive term in this case.

Adding the Functions

So what happens if we add these two functions? Mathematically, we have . So the graph should increase by 5 vertical units for every horizontal unit increase, and it should cross the y axis at y=-1.

Indeed, our predictions have proven to be true.

Multiplying the Functions

So now, what would we get if we were to multiply the two functions? Mathematically, we would arrive at . This is a quadratic equation. Let’s take a look.

Indeed because this is a quadratic equation, our resulting graph is in the shape of a parabola. Because our leading coefficient is a rather large value of 6, the parabola takes a narrow shape. Additionally, because this value is positive, the parabola is oriented up rather than down. The -2 final term also indicates the y value at the y intercept.

Further research and analysis would tell us that the x value of the vertex is located at . By plugging this value in to the equation, we can see that this means the y value is , so our vertex lies at .

Dividing the Functions

So then what would be the effect of dividing one function by the other? Mathematically, we would get . How about the graph? Well, we know that the function would be undefined at , since this would result in a 0 value for the denominator. What other behavior could we observe?

We can see there does appear to be a vertical asymptote where . Additionally, as the absolute value of x gets larger and larger, the y value gets closer and closer to 0, since the numerator will be smaller than the denominator. This is why the y value approaches 0 on both the right and left tails of the graph as x gets further away from the center of the graph.

Function Composition

So what happens when we make one of the functions a composition of the other? Well, we end up with either , which simplifies to , or , which simplifies to . Because these functions share the same slope, we know that they will be parallel to each other. Also, the first function will cross over the y axis at y=-3, and the second function will cross over the y axis at y=1. Let’s check to see that this is the case.

Indeed our assumptions have proven to be accurate, and we now can see how much different the operation can make on the outcome of a graph.