Key takeaway: A parabola is a smooth, mirror-symmetric U-shaped curve traced by a quadratic equation, with exactly one turning point, the vertex.
Grade 9 – Algebra 1F-IF.C.7aFunctions and Graphs
Definition of Parabola
A parabola appears whenever you plot \(y = ax^{2} + bx + c\). The curve is smooth and turns around only once. It is also perfectly balanced: fold the picture along its vertical line of symmetry and the two halves match exactly.
The sign of \(a\) decides the opening. For \(a > 0\) the arms go up and the vertex is the minimum; for \(a < 0\) they go down and the vertex is the maximum. A larger \(|a|\) squeezes the curve narrower, and a smaller \(|a|\) spreads it wider; with \(a = 1\) you get the basic model \(y = x^{2}\), and with \(a = 4\) the same shape becomes four times as steep at each input. The arc of a basketball shot, the main cables of a suspension bridge and the cross-section of a satellite dish all have this shape, which makes this curve useful far outside the math classroom, in sports, engineering and astronomy alike.
Formula and notation
Two common equations for an upright parabola:
\[y = ax^{2} + bx + c \qquad y = a(x – h)^{2} + k\]
In the second form the vertex is \((h, k)\). In either form, the number \(a\) controls the direction and the width, and a curve with a vertical axis never fails the vertical line test.
Key facts
- The vertex is the lowest point on an upward curve, or the highest on a downward one.
- A vertical mirror line passes through it.
- The arms keep spreading apart forever; they never curve back together.
- Zero, one or two crossings of the horizontal axis are possible.
Parabola example
The graph of \(y = -2x^{2} + 8x – 3\) is a parabola. Say how it opens and find its vertex.
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Since \(a = -2 < 0\), it opens downward and the vertex is a maximum.
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The vertex has \(x = -\frac{b}{2a} = -\frac{8}{2(-2)} = 2\).
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Substitute: \(y = -2(4) + 8(2) – 3 = -8 + 16 – 3 = 5\).
The curve opens downward and its vertex is \((2, 5)\). The largest value this function ever reaches is therefore 5, at \(x = 2\), and every other height is smaller.
Common mistake
Watch out
Ravi says the shape is just two straight rays that meet at the vertex, like a V.
Why it is wrong
A parabola is curved. Its steepness changes smoothly: near the vertex it is flat, and farther away it gets steeper. A V shape belongs to the absolute value function. Plot a few extra points before deciding what a parabola looks like: for \(y = x^{2}\), the values 0, 1, 4, 9 grow faster than the equal steps of a V.
Related terms and lessons
- Quadratic Function
- Vertex of a Parabola
- Axis of Symmetry
- Vertex Form
- Absolute Value Function
- Zeros of a Function
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