Quadratic Function in General Form: y = ax^2 + bx + c

A quadratic function is a function whose graph is a parabola. The general form (also called the standard form) of a quadratic function is written as:

\( y = ax^2 + bx + c \)

Here, \(a\), \(b\), and \(c\) are real number coefficients, and \(a \ne 0\) (if \(a = 0\), the \(x^2\) term disappears and the function becomes linear instead of quadratic). This is the form you will see most often when a quadratic equation in standard form or a quadratic expression in standard form is first given to you, before any factoring or completing the square has been done.

Reading off the coefficients is just a matter of matching terms. For example, in \( y = 3x^2 - 5x + 2 \):

• \(a = 3\) — the coefficient of \(x^2\)
• \(b = -5\) — the coefficient of \(x\)
• \(c = 2\) — the constant term

Watch for missing terms. In \( y = x^2 - 9 \), there is no \(x\) term, so \(b = 0\) and \(c = -9\). In \( y = -2x^2 + x \), there is no constant term, so \(c = 0\).

Each coefficient controls a specific feature of the graph, which is covered in more depth on the characteristics of quadratic functions page. As a quick reference:

• \(a\) controls the direction of opening and the width. If \(a > 0\), the parabola opens upward; if \(a < 0\), it opens downward. A larger \(|a|\) makes the parabola narrower.
• \(c\) is always the \(y\)-intercept, since substituting \(x = 0\) gives \(y = c\).
• \(b\), together with \(a\), determines the horizontal position of the vertex and the axis of symmetry.

Even though vertex form is built to show the vertex directly, you can still find it from general form using the formula:

\( x = -\dfrac{b}{2a} \)

This gives the \(x\)-coordinate of the vertex (and the equation of the axis of symmetry). Substitute that value back into \( y = ax^2 + bx + c \) to find the \(y\)-coordinate. If you would rather rewrite the whole function in vertex form, see converting general form to vertex form by completing the square.

Consider \( y = x^2 - 2x - 3 \), so \(a = 1\), \(b = -2\), \(c = -3\).

• y-intercept: set \(x = 0\), so \(y = c = -3\), giving the point \((0, -3)\).
• Vertex x-coordinate: \( x = -\dfrac{b}{2a} = -\dfrac{-2}{2(1)} = 1 \).
• Vertex y-coordinate: \( y = (1)^2 - 2(1) - 3 = -4 \), so the vertex is \((1, -4)\).
• x-intercepts: factoring gives \((x - 3)(x + 1) = 0\), so \(x = 3\) or \(x = -1\).

Since \(a = 1 > 0\), the parabola opens upward, matching the graph shown above.

To sketch a parabola from general form, find the \(y\)-intercept, the vertex, and (if possible) the \(x\)-intercepts, then plot a few extra points using a table of values on either side of the axis of symmetry. The graph is always symmetric about the vertical line through the vertex, so once you have points on one side you can mirror them on the other. If instead you already have a graph and need to work backward to build the equation, that process is covered in finding the quadratic function for a given parabola.

• General form: \( y = ax^2 + bx + c \), with \(a \ne 0\).
• \(c\) gives the y-intercept directly.
• The sign of \(a\) tells you whether the parabola opens up or down.
• The vertex x-coordinate is \( x = -\dfrac{b}{2a} \); substitute to get the y-coordinate.

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