7.3 Quadratic Equations, Vertex Form & Parabolic Functions

Quadratic Equations, Vertex Form & Parabolic Functions

Core Competency: Quadratic equations and parabolic models are foundational to intermediate and advanced college mathematics. The ACCUPLACER AAF assessment rigorously evaluates your skill in factoring multi-term polynomials, manipulating the three canonical forms of quadratic functions, completing the square, interpreting the discriminant, and calculating maximum or minimum vertex values in real-world applied scenarios.

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The Three Canonical Quadratic Forms

A quadratic function is a second-degree polynomial function. Depending on the analytical goal, quadratics are represented in one of three algebraic forms:

+-----------------------------------------------------------------------------------------+
|                        THE THREE CANONICAL FORMS OF QUADRATICS                          |
+-----------------------+---------------------------+-------------------------------------+
| Quadratic Form        | Algebraic Structure       | Primary Analytical Advantage        |
+-----------------------+---------------------------+-------------------------------------+
| **Standard Form**     | y = ax² + bx + c          | Direct y-intercept at (0, c);       |
|                       |                           | Axis of symmetry x = -b / (2a)      |
+-----------------------+---------------------------+-------------------------------------+
| **Vertex Form**       | y = a(x - h)² + k         | Immediate vertex coordinates (h, k);|
|                       |                           | Max/Min value identification        |
+-----------------------+---------------------------+-------------------------------------+
| **Factored Form**     | y = a(x - r₁)(x - r₂)     | Immediate x-intercepts at (r₁, 0)   |
|                       |                           | and (r₂, 0); Root identification   |
+-----------------------+---------------------------+-------------------------------------+

Parameter Behaviors Across All Forms

• Leading Coefficient ($a$):
  • If $a > 0$, the parabola opens upward (concave up, $\bigcup$), possessing a global minimum value at its vertex.
  • If $a < 0$, the parabola opens downward (concave down, $\bigcap$), possessing a global maximum value at its vertex.
  • The magnitude $|a|$ dictates vertical stretch ($|a| > 1$, narrow) or vertical compression ($0 < |a| < 1$, wide).
• Horizontal & Vertical Shifts in Vertex Form: In $y = a(x - h)^2 + k$, the vertex is located at $(h, k)$. Note the subtraction sign inside the parenthetical expression: $y = 3(x - 4)^2 + 5$ has its vertex at $(+4, 5)$, whereas $y = 3(x + 4)^2 + 5$ has its vertex at $(-4, 5)$.

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Factoring Methodologies for Quadratic Expressions

Solving quadratic equations of the form $ax^2 + bx + c = 0$ by factoring relies on the Zero-Product Property: if $A \cdot B = 0$, then $A = 0$ or $B = 0$.

Factoring Decision Matrix

1. Greatest Common Factor (GCF): Always extract the highest common monomial factor first: $$6x^3 - 18x^2 = 6x^2(x - 3)$$
2. Difference of Two Squares ($a^2 - b^2$): Factors into conjugate binomials: $$a^2 - b^2 = (a - b)(a + b) \implies 16x^2 - 49 = (4x - 7)(4x + 7)$$
3. Perfect Square Trinomials ($a^2 \pm 2ab + b^2$): $$a^2 + 2ab + b^2 = (a + b)^2 \quad \text{and} \quad a^2 - 2ab + b^2 = (a - b)^2$$ $$9x^2 - 24x + 16 = (3x - 4)^2$$
4. Trinomial Factoring with $a = 1$ ($x^2 + bx + c$): Find two integers $p$ and $q$ such that $p \cdot q = c$ and $p + q = b$: $$x^2 - 7x + 12 = (x - 3)(x - 4) \implies x = 3, \, x = 4$$
5. The $ac$-Method (Grouping) for $a \neq 1$ ($ax^2 + bx + c$):
   • Step 1: Multiply $a \cdot c$.
   • Step 2: Find two factors of the product $ac$ that add up to $b$.
   • Step 3: Split the middle linear term $bx$ into these two components.
   • Step 4: Factor by grouping the first two terms and the last two terms.

Worked Example: The $ac$-Method

Factor and solve: $6x^2 + 11x - 10 = 0$.

• Step 1: Compute $a \cdot c = 6 \times (-10) = -60$.
• Step 2: Find factors of $-60$ that sum to $+11$. The factors are $+15$ and $-4$ (since $15 \times (-4) = -60$ and $15 + (-4) = 11$).
• Step 3: Split $11x$ into $+15x - 4x$: $$6x^2 + 15x - 4x - 10 = 0$$
• Step 4: Group terms in pairs: $$3x(2x + 5) - 2(2x + 5) = 0$$
• Step 5: Factor out the common binomial $(2x + 5)$: $$(3x - 2)(2x + 5) = 0$$
• Step 6 (Zero-Product Property): $3x - 2 = 0 \implies x = \frac{2}{3}$; $2x + 5 = 0 \implies x = -\frac{5}{2}$.

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Solving Quadratics by Completing the Square

Completing the square converts a quadratic expression from standard form into vertex form and serves as the algebraic proof for the quadratic formula.

Algorithmic Procedure

To solve $ax^2 + bx + c = 0$:

1. Isolate Variable Terms: Subtract the constant $c$ to the right-hand side: $ax^2 + bx = -c$.
2. Normalize the Leading Coefficient: If $a \neq 1$, divide all terms by $a$: $x^2 + \frac{b}{a}x = -\frac{c}{a}$.
3. Add the Square of Half the Linear Coefficient: Compute $\left(\frac{b}{2a}\right)^2$ and add this exact value to both sides of the equation: $$x^2 + \frac{b}{a}x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2$$
4. Factor the Left Side as a Binomial Square: $$\left(x + \frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2}$$
5. Apply the Square Root Property: Take the square root of both sides, including the $\pm$ operator: $$x + \frac{b}{2a} = \pm \frac{\sqrt{b^2 - 4ac}}{2a} \implies x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Worked Example: Converting to Vertex Form by Completing the Square

Convert $f(x) = 2x^2 - 12x + 23$ to vertex form and state its vertex.

• Step 1: Factor out leading coefficient $2$ from the variable terms only: $$f(x) = 2(x^2 - 6x) + 23$$
• Step 2: Complete the square inside parentheses. Half of $-6$ is $-3$, and $(-3)^2 = 9$.
• Step 3: Adding $9$ inside the parentheses actually adds $2 \times 9 = 18$ to the function. Maintain equality by subtracting $18$ outside: $$f(x) = 2(x^2 - 6x + 9) + 23 - 18$$
• Step 4: Factor the trinomial: $$f(x) = 2(x - 3)^2 + 5$$
• Result: The vertex is at $(3, 5)$, and since $a = 2 > 0$, the minimum value is $y = 5$.

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The Quadratic Formula & Discriminant Analysis

For any quadratic equation $ax^2 + bx + c = 0$ ($a \neq 0$), the exact solutions are determined by the Quadratic Formula:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

The Discriminant ($\Delta = b^2 - 4ac$)

The radicand expression under the square root, $\Delta = b^2 - 4ac$, is the discriminant. It dictates the number and mathematical nature of the roots without requiring full computation.

+-----------------------------------------------------------------------------------------+
|                         DISCRIMINANT DIAGNOSTIC MATRIX                                  |
+-----------------------+---------------------------------+-------------------------------+
| Discriminant Value (Δ)| Nature and Number of Roots      | Graphical x-Intercept Profile |
+-----------------------+---------------------------------+-------------------------------+
| Δ > 0 (Perfect Square)| 2 Distinct Rational Real Roots  | Parabola crosses x-axis twice |
|                       | (e.g., Δ = 25, 49, 144)         | at rational coordinates       |
+-----------------------+---------------------------------+-------------------------------+
| Δ > 0 (Non-Square)    | 2 Distinct Irrational Real Roots| Parabola crosses x-axis twice |
|                       | (e.g., Δ = 13, 28; conjugate)   | at radical coordinates        |
+-----------------------+---------------------------------+-------------------------------+
| Δ = 0                 | 1 Repeated Rational Real Root   | Parabola is tangent to x-axis;|
|                       | (Multiplicity 2)                | Vertex touches at (-b/(2a), 0)|
+-----------------------+---------------------------------+-------------------------------+
| Δ < 0                 | 2 Complex Conjugate Roots       | Parabola NEVER touches x-axis;|
|                       | (Non-real roots: u ± vi)        | Entirely above or below axis  |
+-----------------------+---------------------------------+-------------------------------+

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Parabola Anatomy & Applied Vertex Optimization

Every parabola modeled by $y = ax^2 + bx + c$ exhibits symmetric geometry defined by key landmarks.

Geometric Features of Parabolas

• Axis of Symmetry: The vertical line $x = -\frac{b}{2a}$. It passes directly through the vertex and bisects the parabola into two mirror images.
• Vertex Coordinates: $\left(-\frac{b}{2a}, \, f\left(-\frac{b}{2a}\right)\right)$.
• $y$-Intercept: $(0, c)$, obtained by setting $x = 0$.
• $x$-Intercepts (Zeroes): Points where $f(x) = 0$, found via factoring or the quadratic formula.

Applied Optimization Word Problems

On the ACCUPLACER, quadratic vertex problems typically appear in two major applied contexts:

1. Projectile Motion Modeling

The vertical height $h(t)$ of an object projected into the air at time $t$ seconds is modeled by:

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

• Time to Maximum Height: $t_{\text{max}} = -\frac{v_0}{2(-16)} = \frac{v_0}{32}$.
• Maximum Height Attained: Evaluate $h(t_{\text{max}})$.
• Total Flight Time (Landing): Solve $h(t) = 0$ for $t > 0$.

Worked Example: Projectile Motion

A flare is launched from a platform $80\text{ feet}$ high with an initial upward velocity of $64\text{ ft/s}$. Its height function is $h(t) = -16t^2 + 64t + 80$.

1. Time to peak height: $t = -\frac{64}{2(-16)} = \frac{64}{32} = 2\text{ seconds}$.
2. Maximum peak height: $h(2) = -16(2)^2 + 64(2) + 80 = -64 + 128 + 80 = 144\text{ feet}$.
3. Time when flare strikes ground ($h(t) = 0$): $$-16t^2 + 64t + 80 = 0 \implies -16(t^2 - 4t - 5) = 0 \implies -16(t - 5)(t + 1) = 0$$ Since time cannot be negative, the flare hits the ground at $t = 5\text{ seconds}$.

2. Revenue Optimization Modeling

Revenue equals unit price times quantity demanded: $R(p) = p \cdot q(p)$. When demand decreases linearly with price ($q(p) = q_0 - mp$), revenue becomes a downward-opening quadratic function whose vertex maximizes revenue.