Key takeaway: The standard form of a quadratic is y = ax² + bx + c with a not equal to 0, where a sets the opening, c is the y-intercept and −b/(2a) gives the axis of symmetry.
Grade 9 – Algebra 1F-IF.C.8aFunctions and Graphs
Definition of Standard Form of a Quadratic
The standard form of a quadratic lists the terms from the highest power to the lowest: an \(x^{2}\) term, an \(x\) term and a constant. Each coefficient carries information about the parabola.
The coefficient \(a\) decides whether the graph opens up or down and how wide it is. The constant \(c\) is where the graph meets the y-axis. The coefficient \(b\) works together with \(a\) to place the axis of symmetry at \(x = -\frac{b}{2a}\). This arrangement is the form needed by the quadratic formula and by factoring, while vertex form is better for reading the turning point directly.
Formula and notation
Written out with its parts named:
\[y = ax^{2} + bx + c\]
Here \(a\) is the leading coefficient, \(b\) is the linear coefficient and \(c\) is the constant term. To use the standard form of a quadratic, make sure that one side of the equation is zero or isolated, with terms in descending order.
Key facts
- The y-intercept is the point (0, c), read straight from the constant term.
- The axis of symmetry is x = −b/(2a); the vertex lies on it.
- A positive a opens up, a negative a opens down.
- Setting y = 0 gives the equation that the quadratic formula solves.
Standard Form of a Quadratic example
For \(y = -2x^{2} + 12x – 10\), identify \(a\), \(b\) and \(c\), then find the \(y\)-intercept, the axis of symmetry and the vertex.
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Reading off: \(a = -2\), \(b = 12\), \(c = -10\). Since \(a < 0\), the parabola opens downward.
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The \(y\)-intercept is \((0, -10)\). The axis is \(x = -\frac{12}{2(-2)} = 3\).
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Vertex height: \(y = -2(9) + 12(3) – 10 = -18 + 36 – 10 = 8\).
The \(y\)-intercept is \(-10\), the axis is \(x = 3\) and the vertex is \((3, 8)\).
Common mistake
Watch out
Seeing \(c = -10\), Mara says the vertex of the parabola is at height \(-10\).
Why it is wrong
The constant \(c\) is the height where the graph crosses the y-axis, not the turning point, unless the vertex happens to lie on the y-axis. Here the vertex is at \(x = 3\), and its height is 8. Find the axis first, then substitute.
Related terms and lessons
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