4.5 Parabolas: Vertex, Axis of Symmetry & Extrema in Context

4.5 Parabolas: Vertex, Axis of Symmetry & Extrema in Context

Quick Summary: A parabola is the graph of a quadratic function. Its defining geometric landmarks are the vertex (the peak or trough), the axis of symmetry (the vertical line splitting the parabola into mirror halves), and its direction of opening. Understanding how these mathematical components translate into real-world contexts—such as projectile height and economic optimization—is a major focus of the Digital SAT.

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The Anatomy of a Parabola

                  PARABOLA: a < 0 (Opens Down)               PARABOLA: a > 0 (Opens Up)
                       Axis of Symmetry: x = h                    Axis of Symmetry: x = h
                                 |                                          |
                          (h,k)  v                                          |
                            * Vertex (MAXIMUM)                              |
                           /|\                                        \     |     /
                          / | \                                        \    |    /
                         /  |  \                                        \   |   /
                        /   |   \                                        \  |  /
     (x_1, y_0) *-----------+-----------* (x_2, y_0)                      \ | /
                      /     |     \                                        \|/
                     /      |      \                                        * Vertex (MINIMUM)
                    /       |       \                                     (h,k)

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Core Formulas for Vertex and Symmetry

For any quadratic function in standard form $f(x) = ax^2 + bx + c$:

1. The $x$-coordinate of the Vertex ($h$)

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

2. The Axis of Symmetry

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

3. The $y$-coordinate of the Vertex ($k$ / Extreme Value)

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

4. Domain and Range

• Domain: All real numbers $(-\infty, \infty)$.
• Range if $a > 0$ (Opens Up): $[k, \infty)$ — the minimum value is $k$.
• Range if $a < 0$ (Opens Down): $(-\infty, k]$ — the maximum value is $k$.

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The Symmetry Property of Parabolas

The axis of symmetry is the perpendicular bisector of every horizontal line segment connecting two points on the parabola that share the same $y$-value.

$$\text{If } f(x_1) = f(x_2), \quad \text{then } h = \frac{x_1 + x_2}{2}$$

High-Yield SAT Problem Type: Midpoint Symmetry

Problem: The graph of $f(x) = ax^2 + bx + c$ contains the points $(3, 14)$ and $(11, 14)$. What is the $x$-coordinate of the vertex of $f(x)$?

• Because both points share the same $y$-value ($y = 14$), the vertex $x$-coordinate must be their midpoint: $$h = \frac{3 + 11}{2} = \frac{14}{2} = 7$$
• The axis of symmetry is the vertical line $x = 7$.

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Contextual Modeling: Word Problem Interpretations

SAT questions frequently embed quadratic functions in applied scenarios. Translating test questions into mathematical components is critical:

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Worked Examples in Context

Worked Example 1: Projectile Trajectory

A flare is launched vertically from an elevated platform. Its height $h(t)$, in meters, after $t$ seconds is modeled by:

$$h(t) = -4.9t^2 + 39.2t + 14.7$$

1. Find the time at which the flare reaches its peak height:

   $$t = -\frac{b}{2a} = -\frac{39.2}{2(-4.9)} = \frac{39.2}{9.8} = 4.0 \text{ seconds}$$

2. Calculate the peak height:

   $$h(4) = -4.9(4)^2 + 39.2(4) + 14.7 = -4.9(16) + 156.8 + 14.7 = -78.4 + 156.8 + 14.7 = 93.1 \text{ meters}$$

3. Determine the initial launch height:

   $$h(0) = -4.9(0)^2 + 39.2(0) + 14.7 = 14.7 \text{ meters}$$

Worked Example 2: Revenue Optimization

A digital subscription service models daily revenue $R(x)$, in dollars, as a function of subscription price $x$ via $R(x) = -20(x - 35)^2 + 24500$.

• What subscription price $x$ maximizes revenue? The function is in vertex form $a(x - h)^2 + k$ with $h = 35$ and $k = 24500$. Since $a = -20 < 0$, the maximum occurs at price $x = 35$ dollars.
• What is the maximum daily revenue? The maximum value is the constant $k = 24,500$ dollars.

Worked Example 3: Finding Unknown Coefficients from Vertex

Problem: The quadratic function $f(x) = x^2 + bx + c$ has its vertex at $(4, -7)$. What is the value of $c$?

• Using vertex form: $f(x) = 1(x - 4)^2 - 7$.
• Expand to standard form: $f(x) = (x^2 - 8x + 16) - 7 = x^2 - 8x + 9$.
• Matching coefficients gives $b = -8$ and $c = 9$.

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Digital SAT Desmos Workflow for Parabola Extrema & Vertices

1. Graph the Function: Enter the equation directly (e.g., y = -16x^2 + 64x + 80). Use $x$ in place of $t$ for Desmos graphing.
2. Locate the Extrema Point: Desmos automatically computes the peak or trough and marks it with a distinct gray dot.
3. Click the Coordinate Dot: Click the gray dot to lock the readout coordinates $(2, 144)$. The $x$-value ($2$) represents the time to peak, and the $y$-value ($144$) represents the maximum height.
4. Window Adjustment for Applied Scenarios: Real-world modeling numbers often exceed standard $[-10, 10]$ grid bounds. Open the wrench tool (Settings) in the top-right corner to set appropriate domain and range intervals (e.g., $x \in [0, 10]$, $y \in [0, 200]$).

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SAT Exam Traps & Best Practices

• $x$ vs. $y$ Confusion: Read carefully whether the question asks for the input that optimizes the function ($x = -b/2a$, such as time or price) or the optimal output value itself ($y = f(-b/2a)$, such as max height or max profit).
• Sign of $a$: Remember that negative $a$ creates a mountain/peak (maximum), while positive $a$ creates a valley/trough (minimum).
• Misinterpreting the Ground Impact Time: Hitting the ground means height $h(t) = 0$ ($x$-intercept), not the vertex.
• Eyeballing Graph Readouts: Always use algebraic formulas or Desmos coordinate clicks rather than visually estimating vertex coordinates on exam diagrams.