Analysis

What do we want to know when we analyze functions?

We want to know the

• domain
• range
• zeros
• discontinuities
• singularities
• intervals of increasing and decreasing
• global maximum and minimum
• local maximums and minimums
• symmetry
• endbehavior
  
• and we would like a nice graph

For a quadratic function much of this information is connected to vertex of the graph, which is why we like the vertex form, which is why we like completing the square.

Domain

Quadratic functions are defined for all real numbers. Their natural domain is $\mathbb {R}$.

Range

$\blacktriangleright$ Vertex Form

The graph of a quadratic function is a parabola, which is easily connected to the completed square form of the formula.

Below is the graph of $y = f(x) = a (x - h)^2 + k$, with $a$, and $h$, and $k$ all real numbers and $a > 0$. The extreme point is called the vertex. If $a<0$, then the parabola would have opened downward and the extreme point would be at the top.

The vertex visually encodes the minimum (or maximum) value of the function.

If $a<0$, then everything is reveresed.

We can see from the formula $f(x) = a (x - h)^2 + k$, that since $(x - h)$ is squared, and thus nonnegative, the range of $f$ depends on the sign of $a$.

When $a>0$, the values of $f(x)$ are greater than or equal to $k$. This corresponds to the graph opening up. The only way to get the least value possible for $f$ is to select $x = h$. That corresponds to the vertex $(k, h)$.

When $a<0$, the values of $f(x)$ are less than or equal to $k$. This corresponds to the graph opening down. The only way to get the greatest value possible for $f$ is to select $x = h$. That corresponds to the vertex $(k, h)$.

We can see that the implied range of a quadratic comes in two types.

• The range could be all real numbers greater than or equal to some particular number: $\{ r \in \textbf {R} \, | \, r \geq k \}$.
• The range could be all real numbers less than or equal to some particular number: $\{ r \in \textbf {R} \, | \, r \leq k \}$.

If there is a stated domain, then the range will be restricted appropriately.

$\blacktriangleright$ Symmetry

If we take two domain numbers equidistant from $h$, like $h - \epsilon$ and $h + \epsilon$, and evaluate $f(x)$ at these two numbers we get the same value.

$$\begin{align*} f(h - \epsilon) & = a (h - \epsilon - h)^2 + k \\ & = a (-\epsilon)^2 + k \\ & = a (\epsilon)^2 + k \\ & = a (h + \epsilon - h)^2 + k \\ & = f(h + \epsilon) \end{align*}$$

The graph is symmetric about the line $x = h$.

When developing the quadratic formula, we had the opportunity to complete the square and that gave us a squared term that looked like

$$a \, \left (x + \frac {b}{2a}\right )^2$$

The domain coordinate of the vertex is $h = \frac {-b}{2a}$.

The graph is symmetric about the line $x = \frac {-b}{2a}$.

The zeros must also be symmetric about $\frac {-b}{2a}$, which we can see in the quadratic formula.

Zeros

The quadratic formula is

$$t = \frac {-b \pm \sqrt {b^2 - 4 a c}}{2a}$$

which we can separate into

$$t = \frac {-b}{2a} \pm \frac {\sqrt {b^2 - 4 a c}}{2a}$$

The $\pm$ shows that the zeros are symmetric about $\frac {-b}{2a}$ and the intercepts are symmetric about the line $x = \frac {-b}{2a}$.

Working backwards, we can see that if we have the zeros, like from factoring, then we have the intercepts, and the line of symmetry must run in the middle.

Continuity

Quadratic functions are continuous functions. They have no discontinuities or singularities.

Behavior

Increasing and Decreasing

The graph vividly suggests that quadratic functions switch from increasing to decreasing (or vice versa) at the symmetric/vertex number in the domain.

• If the graph is opening up, then the quadratic function is
  

  decreasing on $\left ( -\infty , \frac {-b}{2a} \right )$ and increasing on $\left ( \frac {-b}{2a}, \infty \right )$

• If the graph is opening down, then the quadratic function is
  

  increasing on $\left ( -\infty , \frac {-b}{2a} \right )$ and decreasing on $\left ( \frac {-b}{2a}, \infty \right )$

Increasing and decreasing refer to the rate of change.

• Increasing is a positive rate of change.
• Decreasing is a negative rate of change.

Now, we can replace our graphical intuition with algebraic rigor.

We have seen if we write a quadratic function as $f(x) = a (x - h)^2 + k$, then the instantaneous rate of change of $f$ is the linear function $iRoC_f(x) = 2 a (x - h)$. The values of $iRoC$ are the slopes of the lines tangent to the parabola.

Since $iRoC_f$ is a linear function, its graph is a line.

Here is a graph of both the parabola for $f$ and the line for $iRoC_f$.

Our linear rate of change function now informs us about the behavior of $f$.

As we can see, the behavior of our function, $f(x)$, can change drastically where $iRoC_f(x) = 0$. Such domain numbers deserve a special name.

Note: Domain numbers where $iRoC_f$ doesn’t exist are also places where a function’s behavior can change drastically.

We have a procedure for obtaining the $iRoC$ of a quadratic function, when the formula is in vertex form (completed square form).

What about standard form?

$$\begin{align*} Q(x) & = a (x - h)^2 + k \\ & = a \, x^2 - 2 \, a \, h \, x + a \, h^2 \, k \\ & = a \, x^2 + (- 2 \, a \, h) x + (a \, h^2 \, k) \end{align*}$$ $$\begin{align*} iRoC_Q(x) &= 2 a (x - h) \\ & = 2 \, a \, x - 2 \, a \, h \\ \end{align*}$$

There is probably an overall pattern going on here.

Maximum and Minimum

The maximum and minimum values of a quadratic function, $f$, are visually encoded in the highest and lowest points on the graph, which is the vertex of the parabola.

Depending on the sign of $a$, the maximum or minimum value of $f(x) = a \, x^2 + b \, x + c$ occurs at $\frac {-b}{2a}$. The maximum or minimum value is $f\left ( \frac {-b}{2a} \right )$

Depending on the sign of $a$, the maximum or minimum value of $f(x) = a \, (x - h)^2 + k$ is $k$ and occurs at $h = \frac {-b}{2a}$.

We can also look at the linear $iRoC_f(x)$ function. Where $iRoC_f(x) = 0$ is where the vertex is located, which encodes the maximum or minimum value of $f$.

$$\begin{align*} iRoC_Q(x) &= 0 \\ 2 \, a \, x + b & = 0 \\ x &= \frac{-b}{2a} \end{align*}$$

$\blacktriangleright$ $\frac {-b}{2a}$ is the critical number for a quadratic function given in standard form: $a \, x^2 + b \, x + c$.

$\blacktriangleright$ $h$ is the critical number for a quadratic function given in vertex form: $a \, (x - h)^2 + c$.

$\blacktriangleright$ $\frac {r_1 + r_2}{2}$ is the critical number for a quadratic function given in factored form: $a \, (x - r_1) (x - r_2)$.