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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?  LetŐs explore using the following two functions.

 

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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.

 

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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.  LetŐs take a look.

 

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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?

 

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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.  LetŐs check to see that this is the case.

 

formula1.png

 

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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.