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Section 1.5 Vertex Form and Transformations

A slightly different variation of completing the square provides an alternate form of a quadratic which gives insight into its graph.

Subsection 1.5.1 Vertex Form (Standard Form)

Definition 1.5.1.

The standard form (or vertex form) of a quadratic function is,

\begin{equation*} y = a(x - h)^2 + k \qquad \text{or} \qquad f(x) = a(x - h)^2 + k \end{equation*}

The vertex of this parabola is \((h,k)\text{.}\) The parabola opens up if \(a > 0\) and opens down if \(a \lt 0\text{.}\)

It is easier to graph a quadratic function by-hand, if it is in standard form, because the vertex can be read off the equation directly.

Subsection 1.5.2 Converting to Vertex Form from General Form

Consider the quadratic function \(f(x) = ax^2 + bx + c\text{.}\) To convert to vertex form, we complete the square.

\begin{align*} f(x) \amp = ax^2 + bx + c\\ \amp = a\brac{x^2 + \frac{b}{a}x} + c \amp\amp \text{factoring out $a$ from the first two terms}\\ \amp = a\brac{x^2 + \frac{b}{a}x + \frac{b^2}{4a^2}} + c - \frac{b^2}{4a} \amp\amp \text{adding and subtracting $a \cdot \brac{\frac{b}{2a}}^2 = \frac{b^2}{4a}$}\\ f(x) \amp = a\brac{x + \frac{b}{2a}}^2 + c - \frac{b^2}{4a} \end{align*}

Then, this shows that the vertex has \(x\)-coordinate is \(x = -\frac{b}{2a}\text{,}\) and \(y\)-coordinate \(c - \frac{b^2}{4a}\text{.}\) More conveniently, the \(y\)-coordinate can be determined by evaluating \(f\) at \(x = -\frac{b}{2a}\text{.}\) It can be verified that,

\begin{align*} f\brac{-\frac{b}{2a}} \amp = a \underbrace{\brac{-\frac{b}{2a} + \frac{b}{2a}}^2}_{=0} + c - \frac{b^2}{4a}\\ \amp = c - \frac{b^2}{4a}\\ \amp = \frac{4ac}{4a} - \frac{b^2}{4a}\\ \amp = \frac{4ac - b^2}{4a} \end{align*}

In summary,

Also, the axis of symmetry is then the vertical line \(x = -\frac{b}{2a}\text{.}\)

Subsection 1.5.3 Graphing a Quadratic Function in Standard Form

Consider the quadratic function \(f(x) = a(x - h)^2 + k\text{.}\)

  1. Plot the vertex, given by \((h,k)\text{.}\)

  2. Determine the orientation of the parabola. If \(a > 0\text{,}\) the parabola opens up, and if \(a \lt 0\text{,}\) the parabola opens downward.

  3. Plot some additional points. Some possible convenient points include the \(x\)-intercepts (by solving \(a(x - h)^2 + k = 0\)), and the \(y\)-intercept (by evaluating \(f(0)\)). If necessary, points at \(x\)-values near the vertex can also be useful.

  4. Connect the points with a smooth parabola curve. Also recall that the curve should be symmetric about the axis of symmetry \(x = h\text{.}\)

Subsection 1.5.4 Determining the Equation of a Parabola Given Vertex and One Point

Subsection 1.5.5 Examples

Consider the quadratic function \(f(x) = ax^2 + c\text{.}\) In each case, what must be true about \(a\) and \(c\text{?}\)

  • \(f\) has no \(x\)-intercepts.

  • \(f\) has one \(x\)-intercept.

  • \(f\) has two \(x\)-intercepts.

Subsection 1.5.6 Completing the Square and Factoring

\begin{align*} x^2 + 2x - 3 \amp = (x + 1)^2 - 4 \amp\amp \text{completing the square}\\ \amp = \brac{(x + 1) - 2} \brac{(x + 1) + 2} \amp\amp \text{factoring as a difference of squares}\\ \amp = (x - 1)(x + 3) \amp\amp \text{simplifying} \end{align*}

In general,

\begin{align*} (x - h)^2 + k \amp = (x - h - \sqrt{k})(x - h + \sqrt{k})\\ \amp = \brac{x - (h + \sqrt{k})} \brac{x - (h - \sqrt{k})} \end{align*}

so the roots are \(x = h + \sqrt{k}, h - \sqrt{k}\text{.}\)