Quadratic Functions and Models

Standard Form and Properties of Quadratic Functions

• Quadratic function forms:

  • General form: $f(x) = ax^2 + bx + c$, where $a \neq 0$

  • Standard (vertex) form: $f(x) = a(x - h)^2 + k$, where $a \neq 0$

• Key geometric properties from standard form:

  • Vertex: $(h, k)$

  • Axis of symmetry: $x = h$

  • Parabola opening direction:

  • Opens upward if $a > 0$ (vertex is a minimum)

  • Opens downward if $a < 0$ (vertex is a maximum)

• Constructing standard form from a vertex and point:

  • Given vertex $(4, -1)$ and point $(3, 5)$, substitute into standard form: $5 = a(3 - 4)^2 - 1$

  • Solving yields $a = 6$, resulting in $f(x) = 6(x - 4)^2 - 1$

Converting to Standard Form by Completing the Square

• Function $f(x) = x^2 - 6x + 7$:

  • Standard form: $f(x) = (x^2 - 6x + 9) + 7 - 9 = (x - 3)^2 - 2$

  • Vertex: $(3, -2)$

  • Axis of symmetry: $x = 3$

  • $x$-intercepts: $(3 + \sqrt{2}, 0)$ and $(3 - \sqrt{2}, 0)$

• Function $f(x) = -2x^2 - 4x + 3$:

  • Standard form: $f(x) = -2(x^2 + 2x + 1) + 3 - (-2)(1) = -2(x + 1)^2 + 5$

  • Vertex: $(-1, 5)$

  • Axis of symmetry: $x = -1$

  • $x$-intercepts: set $f(x) = 0$ to obtain $x = -1 \pm \sqrt{\frac{5}{2}}$

Maximum and Minimum Values of Quadratic Functions

• Vertex formula for general form $f(x) = ax^2 + bx + c$:

  • Vertex location: $\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$

  • If $a < 0$, $f$ attains a maximum at $h = -\frac{b}{2a}$ with maximum value $f\left(-\frac{b}{2a}\right)$

  • If $a > 0$, $f$ attains a minimum at $h = -\frac{b}{2a}$ with minimum value $f\left(-\frac{b}{2a}\right)$

• Revenue optimization model $R(p) = -30p^2 + 1500p$ ($R$ in thousands of dollars, $p$ is price per unit):

  • Revenue at $p = \$20$: $R(20) = -30(20)^2 + 1500(20) = 18{,}000$ thousand dollars ($\$18{,}000{,}000$)

  • Price yielding maximum revenue: $p = -\frac{b}{2a} = -\frac{1500}{2(-30)} = \$25$

  • Maximum revenue: $R(25) = -30(25)^2 + 1500(25) = 18{,}750$ thousand dollars ($\$18{,}750{,}000$)