Finding the Vertex of a Parabola
Find the vertex of any parabola using the vertex formula, understand what it means for the graph, and apply it to maximize revenue, minimize costs, and solve real optimization problems.
The vertex of a parabola is the most valuable single point in a quadratic function. It tells you the maximum or minimum — the peak profit, the lowest cost, the greatest height, the best price. Businesses use it to find the price that maximizes revenue. Engineers use it to find the minimum material that meets a structural constraint. Coaches use it to find the optimal launch angle for a throw. Wherever you need to optimize something that has a "sweet spot" — too much is bad, too little is bad, and there's one best value in the middle — you're working with a parabola and its vertex is the answer.
- Find the vertex \((h, k)\) from standard form \(f(x) = ax^2 + bx + c\) using the vertex formula
- Determine whether the vertex is a maximum or minimum based on the sign of \(a\)
- Identify the axis of symmetry \(x = h\)
- Apply vertex-finding to optimization problems — pricing, profit, projectile height
Given \(f(x) = ax^2 + bx + c\), the x-coordinate of the vertex is:
\[ h = -\frac{b}{2a} \]Substitute \(h\) back into \(f(x)\) to get the y-coordinate:
\[ k = f(h) = f\!\left(-\frac{b}{2a}\right) \]The vertex is \((h, k)\). The axis of symmetry is \(x = h\). The parabola opens up when \(a > 0\) (vertex is a minimum) and down when \(a < 0\) (vertex is a maximum).
Opens Up vs. Opens Down
| Condition | Direction | Vertex Type | Real-World Use |
|---|---|---|---|
| \(a > 0\) | Opens up ∪ | Minimum | Minimize cost, minimize distance |
| \(a < 0\) | Opens down ∩ | Maximum | Maximize profit, maximize height |
The Complete Process
To find the vertex and describe the parabola:
- Confirm the function is in standard form \(ax^2 + bx + c\).
- Read off \(a\), \(b\), \(c\) — note the sign of \(a\).
- Compute \(h = -b/(2a)\).
- Compute \(k = f(h)\) by substituting \(h\) into the original function.
- State the vertex, axis of symmetry, and whether it's a max or min.
For \(f(x) = 2x^2 - 8x + 3\), find the vertex and state whether it is a maximum or minimum.
- Identify: \(a=2, b=-8, c=3\). Since \(a=2>0\), the parabola opens up — vertex is a minimum.
- x-coordinate: \(h = -\dfrac{-8}{2(2)} = \dfrac{8}{4} = 2\)
- y-coordinate: \(k = f(2) = 2(4) - 8(2) + 3 = 8 - 16 + 3 = -5\)
- Vertex: \((2, -5)\). Axis of symmetry: \(x = 2\).
A ball is launched from ground level. Its height is \(h(t) = -16t^2 + 64t\). Find the maximum height and the time it occurs.
- Identify: \(a=-16, b=64, c=0\). Since \(a<0\), vertex is a maximum.
- Time of max: \(t = -\dfrac{64}{2(-16)} = -\dfrac{64}{-32} = 2\) seconds
- Max height: \(h(2) = -16(4) + 64(2) = -64 + 128 = 64\) feet
A store sells widgets. At a price of \(p\) dollars each, weekly revenue is modeled by \(R(p) = -5p^2 + 200p\). At what price is revenue maximized, and what is the maximum revenue?
- Identify: \(a=-5, b=200, c=0\). Since \(a<0\), the vertex gives the maximum revenue.
- Optimal price: \(p = -\dfrac{200}{2(-5)} = -\dfrac{200}{-10} = 20\) dollars
- Maximum revenue: \(R(20) = -5(400) + 200(20) = -2000 + 4000 = \$2{,}000\)
Pricing above or below $20 reduces revenue. At $10: \(R = -500 + 2000 = \$1{,}500\). At $30: \(R = -4500 + 6000 = \$1{,}500\). The symmetry of the parabola means equal price deviations produce equal revenue losses.
For each function, find the vertex and state max or min:
- \(f(x) = x^2 + 6x + 5\)
- \(g(x) = -3x^2 + 12x - 8\)
- A quadratic profit function \(P(x) = -2x^2 + 40x - 150\). Find the quantity that maximizes profit.
▶ Show Answers
- \(h = -6/2 = -3\), \(k = 9 - 18 + 5 = -4\). Vertex: \((-3, -4)\), minimum.
- \(h = -12/(2 \cdot -3) = 2\), \(k = -3(4) + 24 - 8 = 4\). Vertex: \((2, 4)\), maximum.
- \(x = -40/(2 \cdot -2) = 10\). Max profit at 10 units. \(P(10) = -200 + 400 - 150 = 50\).
- Using h as the final answer: \(h = -b/(2a)\) gives only the x-coordinate. You must substitute back to find \(k\) — the y-coordinate of the vertex. Both together form the vertex point \((h, k)\).
- Sign error in the vertex formula: The formula is \(h = -b/(2a)\), not \(b/(2a)\). If \(b = -8\), then \(h = -(-8)/(2a) = 8/(2a)\). The negative sign matters — applying it incorrectly shifts the vertex to the wrong side.
- Confusing opens up/down: Positive \(a\) → opens up → minimum. Negative \(a\) → opens down → maximum. If you mix these up, you'll misidentify the vertex as a max when it's actually a min, leading to wrong conclusions in optimization problems.
- Vertex x-coordinate: \(h = -b/(2a)\) — always compute this first.
- Vertex y-coordinate: \(k = f(h)\) — substitute \(h\) back into the original function.
- Sign of \(a\) determines direction — positive means minimum, negative means maximum.
- Axis of symmetry is always \(x = h\) — the parabola is symmetric about this vertical line.
- In optimization problems, the vertex answer directly gives the optimal input (\(h\)) and optimal output (\(k\)).
Revenue and profit functions are almost always quadratic in introductory business models — price goes up, units sold go down, and revenue traces a parabola. The vertex gives the profit-maximizing price instantly. Operations managers use quadratic cost functions to minimize production costs as a function of batch size. Data scientists fitting parabolas to experimental data use the vertex to locate the optimal parameter value. Any time an analyst says "find the sweet spot," they're finding the vertex of a parabola.
Calculator Connection
The Parabola Vertex Calculator computes the vertex, axis of symmetry, direction, and graphs the parabola — enter any quadratic in standard form to see its vertex highlighted and a full analysis of the curve.
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