Quadratic Function in Vertex Form

A quadratic function is any function that can be written with a squared term as its highest power. When it's written as \( y = a(x-p)^2 + q \), we say it is in vertex form. This form is especially useful because it tells you the vertex of the parabola, that is the point \( (p, q) \), without any extra work.

Compare that to the quadratic general form, \( y = ax^2 + bx + c \), where the vertex is hidden inside the coefficients. Vertex form puts the most important feature of the graph, the vertex, right in front of you.

Each letter in \( y = a(x-p)^2 + q \) controls a different feature of the parabola:

• \( a \) controls whether the parabola opens upward (\( a > 0 \)) or downward (\( a < 0 \)), and how narrow or wide it is. When \( |a| > 1 \) the graph is stretched vertically (narrower); when \( 0 < |a| < 1 \) the graph is compressed vertically (wider).
• \( p \) is the x-coordinate of the vertex, and \( x = p \) is the equation of the axis of symmetry.
• \( q \) is the y-coordinate of the vertex, the maximum or minimum value of the function.

Notice the sign inside the parentheses is a subtraction: if you see \( y = 3(x+4)^2 - 5 \), rewrite it as \( y = 3(x-(-4))^2 + (-5) \) so you can read off \( p = -4 \) and \( q = -5 \). The vertex is \( (-4, -5) \).

Because the vertex sits directly in the equation, you never need to complete the square or use a formula to locate it, you just read \( (p, q) \) straight from \( y = a(x-p)^2 + q \). For example, in \( y = (x-2)^2 + 1 \) the vertex is \( (2, 1) \) and the axis of symmetry is \( x = 2 \). Since \( a = 1 > 0 \), the parabola opens upward and \( (2, 1) \) is a minimum point.

To finish the graph, choose a couple of x-values on either side of \( p \), find the matching y-values, and use the symmetry of the parabola to plot points on both sides of the axis of symmetry.

Graph \( y = -\frac{1}{2}(x+1)^2 + 3 \).

1. Rewrite the bracket as a subtraction: \( x + 1 = x - (-1) \), so \( p = -1 \).
2. Read \( q = 3 \) directly from the equation.
3. The vertex is \( (-1, 3) \) and the axis of symmetry is \( x = -1 \).
4. Since \( a = -\frac{1}{2} \) is negative, the parabola opens downward, so \( (-1, 3) \) is a maximum point.
5. Since \( |a| < 1 \), the graph is a little wider than \( y = x^2 \).

Once you're comfortable reading the vertex and direction from the equation, you're ready to look more closely at other characteristics of quadratic functions, such as intercepts, domain, and range.

You can always expand vertex form into general form by multiplying out the bracket. For instance:

\( y = 2(x-3)^2 + 4 \)

\( y = 2(x^2 - 6x + 9) + 4 \)

\( y = 2x^2 - 12x + 18 + 4 \)

\( y = 2x^2 - 12x + 22 \)

Going the other direction, from general form back to vertex form, takes a technique called completing the square. That process is covered in detail in standard form to vertex form, which walks through turning \( y = ax^2 + bx + c \) into \( y = a(x-p)^2 + q \) step by step.

Vertex form is the fastest way to sketch a parabola, state its maximum or minimum value, and describe transformations of the basic function \( y = x^2 \): a shift of \( p \) units horizontally, a shift of \( q \) units vertically, a vertical stretch or compression by a factor of \( |a| \), and a reflection over the x-axis when \( a \) is negative. Once you can move confidently between vertex form and general form, graphing and analyzing any quadratic function becomes much quicker.

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