When we examine the behavior of graphs of functions, we often look at how the graph behaves as x approaches positive infinity and negative infinity. This is known as end behavior, and it can tell us a great deal about the function itself. In this article, we will look at which graph shows the same end behavior as the graph of f(x) = 2×6 – 2×2 – 5.
To begin, let us first review what end behavior is in the context of graphs of functions. End behavior is the behavior of a graph as x approaches positive and negative infinity, regardless of any local extrema or intercepts that may be present. This behavior is often determined by the highest degree of the polynomial and the sign of its leading coefficient. For example, if a polynomial has an even degree and a positive leading coefficient, it will approach positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity. The same is true for polynomials with an odd degree and a negative leading coefficient.
Now that we have reviewed the basics of end behavior, let us return to the question of which graph shows the same end behavior as the graph of f(x) = 2×6 – 2×2 – 5. In this case, the graph of f(x) = 2×6 – 2×2 – 5 has an even degree (6) and a positive leading coefficient (2). This means that the graph will approach positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.
Therefore, any graph that shows the same end behavior as the graph of f(x) = 2×6 – 2×2 – 5 must also have an even degree and a positive leading coefficient. Examples of graphs that would show the same end behavior include f(x) = 2×8 – 2×4 – 5, f(x) = 2×10 – 2×6 – 5, and f(x) = 2×12 – 2×8 – 5. All of these graphs would approach positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity.
In conclusion, the graph of f(x) = 2×6 – 2×2 – 5 has an even degree (6) and a positive leading coefficient (2). This means that the graph will approach positive infinity as x approaches positive infinity and negative infinity as x approaches negative infinity. Therefore, any graph that shows the same end behavior as the graph of f(x) = 2×6 – 2×2 – 5 must also have an even degree and a positive leading coefficient. Examples of graphs with the same end behavior as the graph of f(x) = 2×6 – 2×2 – 5 include f(x) = 2×8 – 2×4 – 5, f(x) = 2×10 – 2×6 – 5, and f(x) = 2×12 – 2×8 – 5.
