- Expand the function \(h(x) = 2(x-3)^2\) to identify the leading term as \(2x^2\).
- Recognize that the degree 2 is even and the leading coefficient 2 is positive, indicating that the graph opens upward.
- Determine that the function approaches positive infinity as \(x\) approaches both positive and negative infinity.
- Final Answer: As \(x \to -\infty\), \(h(x) \to \infty\); as \(x \to \infty\), \(h(x) \to \infty\).
Explanation
- Analyzing the Function and Identifying the Leading Term
We are given the quadratic function \(h(x) = 2(x - 3)^2\). To analyze its end behavior, we first expand the expression into its standard polynomial form. Using the algebraic identity \((x - a)^2 = x^2 - 2ax + a^2\), we have:
\[h(x) = 2(x^2 - 6x + 9) = 2x^2 - 12x + 18.\]
The end behavior of any polynomial is governed by its leading term, which is the term with the highest power of \(x\). In this case, the leading term is \(2x^2\).
- Determining End Behavior based on Polynomial Properties
For a polynomial, the end behavior as \(x \to \pm \infty\) is determined by the degree of the polynomial and the sign of its leading coefficient.
- The degree of \(h(x) = 2x^2 - 12x + 18\) is 2, which is an even number.
- The leading coefficient is 2, which is a positive number.
Because the degree is even and the leading coefficient is positive, the function will increase without bound as \(x\) moves toward both positive and negative infinity. This is because the squaring of large positive or large negative numbers results in a large positive value, and multiplying by 2 maintains that positive direction.
- Evaluating the Limits and Concluding the End Behavior
Now we evaluate the limits of the function:
- As \(x \to -\infty\), the term \(x^2\) becomes arbitrarily large and positive, so \(2x^2 \to \infty\), and thus \(h(x) \to \infty\).
- As \(x \to \infty\), the term \(x^2\) becomes arbitrarily large and positive, so \(2x^2 \to \infty\), and thus \(h(x) \to \infty\).
Therefore, both limits result in positive infinity. The final answer for the end behavior is that \(h(x)\) approaches positive infinity in both cases.
Examples
Understanding end behavior is essential in fields like economics, where you might model the total cost of production. If a cost function behaves like a positive even-degree polynomial, it tells you that costs will escalate rapidly regardless of whether you produce very little (negative scenario conceptually) or a massive amount of product. Predicting this growth helps businesses plan for long-term scalability and financial sustainability.