4.4 Forms of Quadratic Functions & Feature Recognition

4.4 Forms of Quadratic Functions & Feature Recognition

Quick Summary: Every parabola in the $xy$-plane can be represented by three equivalent algebraic equations: Standard Form, Vertex Form, and Factored Form. The Digital SAT specifically tests your ability to recognize which algebraic form reveals a particular geometric property (such as the vertex, maximum/minimum value, $y$-intercept, or $x$-intercepts) directly as constants or coefficients within the equation.

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The Three Canonical Quadratic Forms

                               +-----------------------------------+
                               |   Forms of Quadratic Functions    |
                               +-----------------+-----------------+
                                                 |
         +---------------------------------------+---------------------------------------+
         |                                       |                                       |
         v                                       v                                       v
+------------------------------+ +------------------------------+ +------------------------------+
|      STANDARD FORM           | |        VERTEX FORM           | |       FACTORED FORM          |
|    f(x) = ax^2 + bx + c      | |     f(x) = a(x - h)^2 + k    | |    f(x) = a(x - r_1)(x - r_2)|
|                              | |                              | |                              |
| - Displays:                  | | - Displays:                  | | - Displays:                  |
|   - y-intercept (0, c)       | |   - Vertex (h, k)            | |   - x-intercepts (r_1, 0)    |
|   - Opening direction (a)    | |   - Max/Min value (k)        | |     and (r_2, 0)             |
|                              | |   - Axis of symmetry x = h   | |   - Line of symmetry midpoint|
+------------------------------+ +------------------------------+ +------------------------------+

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Deep Dive into Each Form

1. Standard Form: $f(x) = ax^2 + bx + c$

• Directly Revealed Feature: The $y$-intercept is $(0, c)$ because $f(0) = a(0)^2 + b(0) + c = c$.
• Opening Direction: If $a > 0$, the parabola opens upward (cup/valley); if $a < 0$, the parabola opens downward (cap/peak).
• Axis of Symmetry Formula: $x = -\frac{b}{2a}$.
• Vertex $y$-coordinate: $k = f\left(-\frac{b}{2a}\right)$.

2. Vertex Form: $f(x) = a(x - h)^2 + k$

• Directly Revealed Feature: The vertex is $(h, k)$.
• Extremum (Maximum/Minimum):
  • If $a > 0$: The function has a minimum value of $k$, occurring at $x = h$.
  • If $a < 0$: The function has a maximum value of $k$, occurring at $x = h$.
• Axis of Symmetry: The vertical line $x = h$.
• Inner Sign Caution: Notice the negative sign inside: $(x - 5)^2 + 3$ has vertex $(5, 3)$, whereas $(x + 5)^2 + 3 = (x - (-5))^2 + 3$ has vertex $(-5, 3)$.

3. Factored (Intercept) Form: $f(x) = a(x - r_1)(x - r_2)$

• Directly Revealed Feature: The $x$-intercepts (zeros/roots) are $(r_1, 0)$ and $(r_2, 0)$.
• Axis of Symmetry: By symmetry, the vertex $x$-coordinate is the exact arithmetic mean of the zeros: $$h = \frac{r_1 + r_2}{2}$$
• Inner Sign Caution: The function $f(x) = 2(x - 3)(x + 7)$ has $x$-intercepts at $x = 3$ and $x = -7$.

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Feature-Recognition Comparison Matrix

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Converting Between Quadratic Forms

                  +-----------------------------------------+
                  |             STANDARD FORM               |
                  |           f(x) = ax^2 + bx + c          |
                  +--------------+-------------------+------+
                                 |                   ^
            Complete the Square  |                   |  Expand & Combine
            or find h = -b/(2a)  |                   |
                                 v                   |
                  +---------------------------+      |
                  |        VERTEX FORM        +------+
                  |    f(x) = a(x - h)^2 + k  |
                  +---------------------------+

Worked Example 1: Converting Standard Form to Vertex Form

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

• Method A: Completing the Square

  1. Factor the leading coefficient $a = 2$ from variable terms: $f(x) = 2(x^2 - 6x) + 23$.
  2. Add and subtract $\left(\frac{-6}{2}\right)^2 = 9$ inside the parentheses: $$f(x) = 2(x^2 - 6x + 9 - 9) + 23$$ $$f(x) = 2(x^2 - 6x + 9) - 2(9) + 23$$ $$f(x) = 2(x - 3)^2 - 18 + 23 = 2(x - 3)^2 + 5$$

• Method B: Vertex Formula ($h = -b / 2a$)

  1. $h = -\frac{-12}{2(2)} = \frac{12}{4} = 3$.
  2. $k = f(3) = 2(3)^2 - 12(3) + 23 = 18 - 36 + 23 = 5$.
  3. Plug into $a(x - h)^2 + k$: $2(x - 3)^2 + 5$.

Worked Example 2: Identifying Intercepts from Factored Form

Given $g(x) = -3(x + 4)(x - 2)$:

• Setting $g(x) = 0$ yields $x$-intercepts at $(-4, 0)$ and $(2, 0)$.
• The axis of symmetry is the midpoint: $h = \frac{-4 + 2}{2} = -1$.
• The peak output value is $g(-1) = -3(-1 + 4)(-1 - 2) = -3(3)(-3) = 27$.
• In vertex form, this function is written as $g(x) = -3(x + 1)^2 + 27$.

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The Classic SAT "Which Equation Displays..." Question Archetype

On the Digital SAT, all 4 answer options will frequently be algebraically equivalent equations of the exact same parabola. You do not need to do any algebraic manipulation; you only need to match the requested feature to its canonical form!

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Digital SAT Desmos Strategies for Quadratic Forms

1. Verify Equivalence: When given four answer choices, graph the original equation in Line 1 and each answer choice in Lines 2-5. The correct equivalent equation will trace over the original graph with identical curvature and intercepts.
2. Slider Exploration: Type y = a(x - h)^2 + k to observe how $h$ controls horizontal translation, $k$ controls vertical translation, and $a$ controls vertical dilation/reflection.
3. Reading Feature Constants: Hover over the vertex or intercepts on the Desmos canvas. Note the exact numerical coordinates $(h, k)$ and $(r_1, 0), (r_2, 0)$, then look for those identical numbers appearing directly in the equation options.

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Common Pitfalls & Exam Traps

• Sign error inside vertex form: $(x + 4)^2 - 3$ has vertex $(-4, -3)$, not $(+4, -3)$.
• Picking standard form when asked for vertex: $y = x^2 - 6x + 8$ has vertex $(3, -1)$, but it does not display the vertex as constants. Vertex form $y = (x - 3)^2 - 1$ must be chosen.
• Forgetting the leading coefficient $a$: When completing the square on $3x^2 + 12x + 5$, factoring out $3$ yields $3(x^2 + 4x) + 5 = 3(x + 2)^2 - 12 + 5 = 3(x + 2)^2 - 7$.