Understanding the End Behavior of a Graph or the end behavior of a polynomial function requires good knowledge about graphs and functions. The behavior of a function is usually expressed with a graph that can be easily sketched. So what do you think is the end behavior of a function?

Now, remember that the purpose of most graphs in mathematics is to successfully plot a function. The y axis generally represents the function and independent variable, while the x axis represents the variable that determines the function. The Y variable is generally dependent on the x variable, which means that its values are determined by the value of x.

Now consider the function y = 5x + 2. We can never determine the value of y unless we know what x is. Just as the x variable determines the y variable during calculations, it also determines how the y variable will behave when plotted in a graph. In essence, the end behavior of a function simply means how it is bound to behave onto infinity based on the values of x.

This piece will provide a deeper explanation of what the end behavior of a function means, and what you can expect anytime it comes up mathematically.

**What Is End Behavior?**

**The end behavior of a polynomial function f(x) explains how the function will behave in a graph as x approaches positive or negative infinity.**

Y = 5x^{2 }+ 3 is a function.

Now in the function above, x is the independent variable because its value is never dependent on any other variable. On the other hand, y is the dependent variable because the value of the independent variable x defines it. Y is also often replaced by f(x) where f(x) means a function dependent on x.

Like this function, every function in mathematics has an end behavior. This means that every function can be plotted in a graph up to infinity. Of course, infinity means that the graph of the function can continue unendingly.

However, as we know, it is impossible for us to continue plotting a graph function without stopping. It is time exhausting, and it will look as if we are being redundant, so we usually plot a few values and use them to predict how the function will look like anytime x is negative or positive. Predicting how the graph will look is the informal way of saying that we are studying the end behavior of the function or the end behavior of the graph representing the function.

Now we will consider this graph below.

Clearly, from this graph, we can see that the curve kept going up on the positive x axis. It is obvious that as long x positive values continue to increase, there will be no stopping the curve from going up. So based on the graph, we can actually say that as x approaches positive infinity, the function represented in the graph will also approach positive infinity.

If we take a look at the third quadrant, we can also see that the curve tends to slide downward as the values of negative x continue to grow bigger. This move means that as x approaches negative infinity, the function will move towards the negative infinity.

So based on the overall information of the above graph, we can express the function in the format

The arrows at each curve’s end state that the curves can continue in the upward and downward direction as bigger values from each end are added to them.

Now another important information to know is how the leading coefficient of a function determines how the graphed function will behave.

Now we will systematically provide several explanations to explore our End Behavior Definition. This will be put in several points so you can easily understand how the End Behavior of Function work

**First Point:** You need to know which coefficient is considered the leading coefficient if you want to compute the end Behavior of Polynomials successfully. The leading coefficient is the first coefficient in an equation.

For the function y = x + 3, it is clear that the first coefficient is 1 as X comes before 3.

For the function F(x) = x^{4} – x^{2} + 5x, the leading coefficient is also 1 because it is the coefficient for x^{4,} which is the first variable. The second coefficient is -1 because X^{2} follows next.

It is important to know the leading coefficient of a polynomial if you want to know is end behavior.

Second Point: The leading variable order also plays a major role. If the order is even or odd, it will influence the behavior of the graph.

For example, in this equation x^{4} – x^{2} + 5x,

The leading order is 4. Clearly 4 is even.

**How to Find End Behavior?**

Generally, finding the end behavior of the function can be quite easy, and you may not even need to compute the function to know. You mostly only need to play to the gallery. This section will highlight how you can determine the end behavior of a polynomial function without necessarily calculating the function.

**Step 1: Understand the role of the leading coefficient and its contribution to the end Behavior Graph**

If the leading coefficient is positive and the variable degree is even i.e. 2, 4, 6, … then the function is infinitely positive as x approaches positive infinity. Also, the function will remain positively infinite as x approaches negative infinity.

i.e

If the leading coefficient is negative and its degree is even i.e 2, 4, 6, … then the function is infinitely negative as x approaches positive infinity. Also, the function will remain negatively infinite as x approaches negative infinity.

i.e

If the leading coefficient is positive and its degree is odd i.e 1, 3, 5, … then the function is infinitely positive as x approaches positive infinity. Also, the function will become negatively infinite as x approaches negative infinity.

i.e

If the leading coefficient is negative and its degree is odd i.e 1, 3, 5, … then the function is infinitely negative as x approaches positive infinity. Also, the function will become positively infinite as x approaches negative infinity.

i.e

**Step 2:** Calculate the function values and graph them accordingly. Ensure that your graph End Behavior falls in line with the first step.

The next section will consider examples of how to calculate the end Behavior of a Function with your graph.

**How To Find End Behavior Of A Function?**

So we will now consider some examples that explain the end behavior of graphs without necessary plotting it.

Example

Y = x^{2}

Now it is obvious that the leading coefficient 1 is positive, and the degree 2 is also even. If we consider our step 1, we will see that the first property is the most suitable.

So it is expected that for the above function.

So we already know the end behavior of the graph. However, let us test the function with values to see if it will move according to our predicted direction as outlined.

For the positive x we will use the following numbers, 2, 3, 5, and 6

For the negative x we will use the same values but in their negative forms – 2, – 3, – 5 and – 6

Now we will first substitute the respective values with the function.

For x = 2,

Y = 2^{2 }= 9

For x = 3,

Y = 3^{2}

Y = 3^{2 }= 9

For x = 5,

Y = 5^{2 }= 25

For x = 6,

Y = 6^{2 }= 36

Now for the negative x’s we will have the same trend just like the positive values.

For x = – 2,

Y = – 2^{2 }= 4

For x = – 3,

Y = – 3^{2 }= 9

For x = – 5,

Y = – 5^{2 }= 25

For x = – 6,

Y = – 6^{2 }= 36

Based on the above calculations for all positive x’s, y = 4, 9, 25 and 36

Also, for all negatives x’s, y = 4, 9, 25 and 36 So, we will have a graph of the form

The graph indeed ends up in our predicted direction, which is given as

So it is clear that our earlier assumption about this function end behavior.

Since we have been able to show that this is true. We can attempt to predict the end behavior of the subsequent functions listed below

**Conclusion**

This piece has considered what the end behavior of a function means. As you can see, you only need to play to the gallery to get your answer. The solution is very straightforward, and what you have to dos remember step 1 so you can easily tell the behavior of a function every time. You can continue to try different exercises until you master them properly. If you are dealing with a theoretical problem, you may be required to plot the graph, and that is why we showed how you could do that effortlessly.