Integrated Math 2 / Unit 3: Quadratic Functions
3.1 Parabolas and Key Features
Notes
A quadratic function in standard form is $f(x) = ax^2 + bx + c$ with $a \neq 0$. Its graph is a parabola.
- If $a > 0$ the parabola opens up (minimum). If $a < 0$ it opens down (maximum).
- The axis of symmetry is the vertical line $\displaystyle x = -\frac{b}{2a}$. The vertex lies on it.
- The $y$-intercept is $c$.
- Example: $y = x^2 - 6x + 5$ has axis $x = 3$, vertex $(3, -4)$, and $y$-intercept 5.
Practice
Which function has a graph that opens downward?
A parabola opens down when the coefficient of $x^2$ is negative.
Find the axis of symmetry of $y = x^2 - 4x + 7$.
$\displaystyle x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$.
What is the $y$-intercept of $y = 2x^2 + 5x - 9$?
The $y$-intercept is the value at $x = 0$, which is the constant term $-9$.
Ready for more? Take the Unit 3 test.