5.2 Quadratic Functions: Vertex, Axis, Intercepts & Concavity

5.2 Quadratic Functions: Vertex, Axis, Intercepts & Concavity

Quadratic functions are non-linear polynomial functions of degree $2$ whose graphs form parabolas with vertical axes of symmetry. On the FTCE Mathematics 6–12 examination, candidates are expected to demonstrate structural mastery across three canonical algebraic representations, convert fluidly between forms, derive key geometric landmarks (vertex, axis of symmetry, intercepts), and apply quadratic models to solve optimization problems in science, business, and geometry.

---

1. The Three Canonical Forms of Quadratic Functions

Every quadratic function $f: \mathbb{R} \to \mathbb{R}$ can be expressed in three distinct mathematical forms, each tailored to specific analytical tasks:

---

2. Graphical Anatomy: Vertex, Axis of Symmetry, Intercepts & Concavity

Derivation of the Vertex Coordinates

To locate the vertex of $f(x) = ax^2 + bx + c$ analytically without calculus, we complete the square:

$$f(x) = a\left(x^2 + \frac{b}{a}x\right) + c$$

Adding and subtracting $\left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2}$ inside the parentheses:

$$f(x) = a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}\right) + c - a\left(\frac{b^2}{4a^2}\right)$$ $$f(x) = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)$$

Comparing this directly with the vertex template $f(x) = a(x - h)^2 + k$ reveals:

$$h = -\frac{b}{2a}, \quad k = f(h) = c - \frac{b^2}{4a}$$

Axis of Symmetry

Because the squared quantity $\left(x + \frac{b}{2a}\right)^2$ is symmetric about $x = -\frac{b}{2a}$, the parabola possesses bilateral reflection symmetry across the vertical line:

$$x = -\frac{b}{2a}$$

Educators must remind students that an axis of symmetry is a geometric line in the coordinate plane and must always be written as a linear equation $x = h$, never as a solitary scalar number.

Concavity & Extrema

The second derivative of $f(x) = ax^2 + bx + c$ is $f''(x) = 2a$, which is a non-zero constant:

• Concave Upward ($a > 0$): As $x \to \pm\infty$, $f(x) \to +\infty$. The function decreases on $(-\infty, h)$ and increases on $(h, \infty)$. The vertex $(h, k)$ represents the absolute global minimum. The minimum value is $k$, attained at $x = h$.
• Concave Downward ($a < 0$): As $x \to \pm\infty$, $f(x) \to -\infty$. The function increases on $(-\infty, h)$ and decreases on $(h, \infty)$. The vertex $(h, k)$ represents the absolute global maximum. The maximum value is $k$, attained at $x = h$.

          Concave Upward (a > 0)             Concave Downward (a < 0)
               y                                  y
               |       |                          |         (h, k) Max
               |   *   |   *                      |       *   -----   *
               |    *  |  *                       |     *       |       *
               |     * | *                        |    *        |        *
               +-------*------- x                 +-------------+---------- x
                     (h, k) Min                                 |
                       x = h                                  x = h

Finite Differences in Tabular Data

When analyzing discrete data tables with constant step size $\Delta x = 1$:

• Linear functions exhibit constant first differences $\Delta y$.
• Quadratic functions exhibit linear first differences $\Delta y$ and constant second differences $\Delta^2 y = 2a$.

For example, if $y$ values at $x = 0, 1, 2, 3, 4$ are $3, 8, 17, 30, 47$:

• First differences: $8-3=5$, $17-8=9$, $30-17=13$, $47-30=17$.
• Second differences: $9-5=4$, $13-9=4$, $17-13=4$.
• Constant second difference is $4 = 2a \implies a = 2$.

---

3. Real-World Optimization Models

Secondary mathematics curricula emphasize modeling real-world extrema via quadratic vertex determination.

Context A: Geometric Optimization (Enclosing Area)

A farmer wishes to construct a rectangular corral bordering a straight river bank, requiring fencing on only three sides. The farmer has $1200$ linear meters of fencing.

1. Define variables: Let $x$ represent the width perpendicular to the river (two sides). The length parallel to the river is $L = 1200 - 2x$.
2. Formulate objective function: Area $A(x) = x \cdot L = x(1200 - 2x) = -2x^2 + 1200x$.
3. Identify concavity: $a = -2 < 0$, ensuring a global maximum at the vertex.
4. Locate vertex coordinates: $$x = -\frac{b}{2a} = -\frac{1200}{2(-2)} = \frac{1200}{4} = 300\text{ meters}$$
5. Evaluate optimal dimensions and maximum area: $$L = 1200 - 2(300) = 600\text{ meters}$$ $$A_{\max} = 300 \cdot 600 = 180{,}000\text{ square meters}$$

Context B: Revenue Maximization in Business

A performing arts center currently charges $\text{\textdollar}20$ per ticket and sells an average of $300$ tickets per performance. Market analysis indicates that for each $\text{\textdollar}2$ increase in ticket price, attendance decreases by $15$ patrons.

1. Express ticket price and demand in terms of price increases $n$: $$p(n) = 20 + 2n, \quad q(n) = 300 - 15n$$
2. Formulate revenue as a quadratic function of $n$: $$R(n) = p(n) \cdot q(n) = (20 + 2n)(300 - 15n) = 6000 - 300n + 600n - 30n^2 = -30n^2 + 300n + 6000$$
3. Find optimal number of price increases: $$n_{\text{opt}} = -\frac{b}{2a} = -\frac{300}{2(-30)} = \frac{300}{60} = 5$$
4. Compute optimal price, attendance, and maximum revenue: $$\text{Optimal Price } p = 20 + 2(5) = \$30$$ $$\text{Optimal Attendance } q = 300 - 15(5) = 225\text{ patrons}$$ $$R_{\max} = 30 \times 225 = \$6{,}750$$

Context C: Projectile Motion & Kinematics

An object propelled vertically under Earth's gravity follows the position equation:

$$h(t) = -16t^2 + v_0 t + h_0 \quad (\text{feet}), \quad \text{or} \quad h(t) = -4.9t^2 + v_0 t + h_0 \quad (\text{meters})$$

where $v_0$ is initial vertical velocity and $h_0$ is launch height.

• Time to apex: $t_{\text{apex}} = -\frac{v_0}{2(-16)} = \frac{v_0}{32}$.
• Maximum height attained: $h_{\max} = h(t_{\text{apex}})$.
• Total flight time: Solve the quadratic equation $h(t) = 0$ for positive $t$.

---

4. Common Student Misconceptions & Pedagogical Strategies