Defining End Behavior
End behavior describes what happens to f(x) as x approaches positive infinity or negative infinity, revealing the long-range trend of a function graph. In Grade 11 enVision Algebra 1 (Chapter 10: Working With Functions), students describe end behavior using notation like 'as x → ∞, f(x) → ∞' or 'as x → −∞, f(x) → 3.' The output may approach a specific value (horizontal asymptote) or increase/decrease without bound. Determining end behavior is essential for classifying function families.
Key Concepts
Property End behavior describes the values of a function, $f(x)$, as the input, $x$, approaches positive infinity ($x \to \infty$) and negative infinity ($x \to \infty$). We write this as: As $x \to \infty$, $f(x) \to ?$ As $x \to \infty$, $f(x) \to ?$.
Examples For the quadratic function $f(x) = x^2$, as $x \to \infty$, $f(x) \to \infty$. As $x \to \infty$, $f(x) \to \infty$. For the linear function $g(x) = 2x + 3$, as $x \to \infty$, $g(x) \to \infty$. As $x \to \infty$, $g(x) \to \infty$. For the exponential function $h(x) = 2^x$, as $x \to \infty$, $h(x) \to \infty$. As $x \to \infty$, $h(x) \to 0$.
Explanation End behavior describes the long term trend of a function''s graph on the far left and far right sides. To determine this, we look at what happens to the y values as the x values get extremely large in the positive or negative direction. The y values can approach a specific number (like a horizontal asymptote) or they can increase or decrease without bound, approaching positive or negative infinity.
Common Questions
What is end behavior of a function?
End behavior describes the values that f(x) approaches as x gets extremely large or extremely small, showing the function's trend at the far left and right of its graph.
How do you write end behavior notation?
Use arrow notation: 'As x → ∞, f(x) → ?' and 'As x → −∞, f(x) → ?' Fill in whether f(x) approaches infinity, negative infinity, or a specific number.
What does it mean if the end behavior shows f(x) approaching a specific number?
The function has a horizontal asymptote — its output levels off and gets increasingly close to that value without ever reaching it.
What is the end behavior of the parent quadratic function f(x) = x²?
As x → ∞, f(x) → ∞, and as x → −∞, f(x) → ∞. Both ends of the parabola rise upward without bound.
What is the end behavior of a basic exponential growth function?
As x → ∞, f(x) → ∞ (grows without bound), and as x → −∞, f(x) → 0 (approaches the horizontal asymptote).
How does end behavior differ between linear and exponential functions?
A linear function has opposite end behaviors on its two ends. An exponential growth function rises steeply on one end and flattens toward zero on the other.