Quadratic Functions: Minimum, Maximum, Domain, and Range

Quadratic Functions

Analyzing the Function $f(x) = 2x^2 - 12x - 9$

Quadratic functions are polynomial functions of degree two and have the general form $f(x) = ax^2 + bx + c$. Their graphs are parabolas, which open upward if $a > 0$ and downward if $a < 0$. The vertex of the parabola represents either the minimum or maximum value of the function.

Minimum or Maximum Value

• Key Point 1: The direction in which the parabola opens is determined by the sign of the leading coefficient $a$.

• Key Point 2: If $a > 0$, the parabola opens upward and the function has a minimum value at its vertex. If $a < 0$, the parabola opens downward and the function has a maximum value at its vertex.

• Example: For $f(x) = 2x^2 - 12x - 9$, $a = 2 > 0$, so the function has a minimum value.

Finding the Minimum Value and Its Location

• Key Point 1: The vertex of a quadratic function $f(x) = ax^2 + bx + c$ occurs at $x = -\frac{b}{2a}$.

• Key Point 2: Substitute $x$ into the function to find the minimum (or maximum) value.

• Calculation:

  • $a = 2$, $b = -12$

  • Vertex: $x = -\frac{-12}{2 \times 2} = \frac{12}{4} = 3$

  • Minimum value: $f(3) = 2(3)^2 - 12(3) - 9 = 2(9) - 36 - 9 = 18 - 36 - 9 = -27$

• Example: The minimum value is $-27$ and it occurs at $x = 3$.

Domain and Range

• Key Point 1: The domain of any quadratic function is all real numbers: $(-\infty, \infty)$.

• Key Point 2: The range depends on whether the function has a minimum or maximum value.

  • If minimum at $y = k$, range is $[k, \infty)$.

  • If maximum at $y = k$, range is $(-\infty, k]$.

• Example: For $f(x) = 2x^2 - 12x - 9$, minimum value is $-27$ at $x = 3$.

  • Domain: $(-\infty, \infty)$

  • Range: $[-27, \infty)$

Summary Table

Property	Value
Minimum/Maximum	Minimum
Location of Minimum	$x = 3$
Minimum Value	$-27$
Domain	$(-\infty, \infty)$
Range	$[-27, \infty)$

Additional info: The vertex formula and the process for finding domain and range are standard for all quadratic functions. These concepts are foundational in Precalculus and are used in graphing and analyzing polynomial functions.