Answered

The function f(x) = -3x^3 + 2x^2 - x + 5 is graphed on a coordinate grid. Which statements accurately describe the end behavior of the graph of the function?


AAs x approaches negative infinity, y approaches positive infinity. As x approaches positive infinity, y approaches positive infinity.
BAs x approaches negative infinity, y approaches negative infinity. As x approaches positive infinity, y approaches positive infinity.
CAs x approaches negative infinity, y approaches negative infinity. As x approaches positive infinity, y approaches negative infinity.
DAs x approaches negative infinity, y approaches positive infinity. As x approaches positive infinity, y approaches negative infinity.

Verified Answer :

  • Simplify the polynomial by combining constant terms to get \(f(x) = 2x^2 - x + 2\).
  • Identify that the function is an even-degree polynomial (\(n=2\)) with a positive leading coefficient (\(a_n=2\)).
  • Conclude that both ends of the graph extend towards positive infinity.
  • Select the matching option, which is \(\boxed{A}\).

Explanation

  1. Simplify and Identify the Function Characteristics

The function given is \(f(x) = -3 + 2x^2 - x + 5\). To analyze its end behavior, we first simplify the expression by combining the constant terms, \(-3\) and \(5\):
\[f(x) = 2x^2 - x + (-3 + 5) = 2x^2 - x + 2.\]
The function is a quadratic polynomial, which is a polynomial of degree \(n = 2\). The leading term is \(2x^2\), and the leading coefficient is \(a_n = 2\).

  1. Analyze End Behavior

The end behavior of a polynomial function \(f(x) = a_n x^n + \dots + a_0\) depends primarily on its degree \(n\) and the sign of its leading coefficient \(a_n\).

  1. The degree \(n = 2\) is an even number.
  2. The leading coefficient \(a_n = 2\) is positive.

For any polynomial with an even degree and a positive leading coefficient, the graph will open upwards. This means that as the input value \(x\) grows larger in either the positive or negative direction, the output value \(y\) will increase towards positive infinity.

  • As \(x \to \infty\), \(f(x) \to \infty\).
  • As \(x \to -\infty\), \(f(x) \to \infty\).
  1. Conclusion

Comparing our findings with the given options:

AAs \(x \to -\infty\), \(y \to \infty\). As \(x \to \infty\), \(y \to \infty\).
BAs \(x \to -\infty\), \(y \to -\infty\). As \(x \to \infty\), \(y \to \infty\).
CAs \(x \to -\infty\), \(y \to -\infty\). As \(x \to \infty\), \(y \to -\infty\).
DAs \(x \to -\infty\), \(y \to \infty\). As \(x \to \infty\), \(y \to -\infty\).

The analysis matches option A. Therefore, the end behavior is correctly described by: As \(x \to -\infty\), \(y \to \infty\), and as \(x \to \infty\), \(y \to \infty\). The correct choice is \(\boxed{A}\).

Examples

In many real-world scenarios, such as modeling the trajectory of a thrown ball or predicting the profit of a business based on production volume, quadratic functions are commonly used. Understanding the end behavior of these functions helps in forecasting long-term trends—for instance, knowing that as production increases significantly beyond the break-even point, costs will eventually rise rapidly, or confirming that a projectile will always return to the ground. This specific skill is essential for interpreting graphs in economics, physics, and engineering.

Related Questions