End Behavior of Quadratic Functions
Grade 9 students in California Reveal Math Algebra 1 learn to describe the end behavior of quadratic functions based solely on the sign of the leading coefficient a. When a>0 the parabola opens upward and y approaches positive infinity as x goes to either positive or negative infinity — the vertex is a minimum. When a<0 the parabola opens downward and y approaches negative infinity on both sides — the vertex is a maximum. Unlike linear functions, a quadratic's end behavior is the same in both directions because x² is always non-negative, making only the sign of a decisive.
Key Concepts
For a quadratic function $f(x) = ax^2 + bx + c$ with $a \neq 0$, the end behavior describes what happens to $y$ as $x \to +\infty$ or $x \to \infty$.
$$\text{If } a 0: \quad y \to +\infty \text{ as } x \to +\infty \text{ and } y \to +\infty \text{ as } x \to \infty$$.
Common Questions
What determines the end behavior of a quadratic function?
The sign of the leading coefficient a in f(x)=ax²+bx+c determines end behavior. If a>0, both ends go to positive infinity. If a<0, both ends go to negative infinity.
Why is a quadratic's end behavior the same on both sides?
Because the x² term dominates and is always non-negative, meaning the outputs ultimately rise or fall based only on whether a is positive or negative — the same in both directions.
What is the end behavior of f(x)=2x²-4x+1?
Since a=2>0, the parabola opens upward. As x goes to positive infinity or negative infinity, y goes to positive infinity. The vertex is a minimum point.
What is the end behavior of f(x)=-3x²+6x-2?
Since a=-3<0, the parabola opens downward. As x goes to either infinity, y goes to negative infinity. The vertex is a maximum point.
How does end behavior relate to projectile motion?
A ball's height h(t)=-16t²+64t has a=-16<0, so both ends fall to negative infinity. This confirms the ball eventually returns to the ground and the vertex represents maximum height.
Which unit covers end behavior of quadratic functions?
This skill is from Unit 10: Quadratic Functions in California Reveal Math Algebra 1, Grade 9.