Active Preparation for Calculus

Subsection 1.5.1 Properties of Quadratic Functions

Quadratic functions are likely familiar to readers of this text from experience in previous courses. Throughout, we let \(y = q(x) = ax^2 + bx + c\) where \(a\text{,}\) \(b\text{,}\) and \(c\) are real numbers with \(a \ne 0\text{.}\) From the outset, it is important to note that when we write \(q(x) = ax^2 + bx + c\) we are thinking of an infinite family of functions where each member depends on the three paramaters \(a\text{,}\) \(b\text{,}\) and \(c\text{.}\)

Because quadratic functions are familiar to us, we will quickly restate some of their important known properties.

As we can see in Figure 1.5.3, by shifting the graph of a quadratic function vertically, we can make its graph cross the \(x\)-axis \(0\) times (as in the graph of \(p\)), exactly \(1\) time (\(q\)), or twice (\(r\)). These points are the \(x\)-intercepts of the graph.

While the quadratic formula will always provide any real solutions to \(q(x) = 0\text{,}\) in practice it is often easier to attempt to factor before using the formula. For instance, given \(q(x) = x^2 - 5x + 6\text{,}\) we can find its \(x\)-intercepts quickly by factoring. Since

it follows that \((2,0)\) and \((3,0)\) are the \(x\)-intercepts of \(q\text{.}\) Note more generally that if we know the \(x\)-intercepts of a quadratic function are \((r,0)\) and \((s,0)\text{,}\) it follows that we can write the quadratic function in the form \(q(x) = a(x-r)(x-s)\text{.}\)

Every quadratic function has a \(y\)-intercept; for a function of form \(y = q(x) = ax^2 + bx + c\text{,}\) the \(y\)-intercept is the point \((0,c)\text{,}\) as demonstrated in Figure 1.5.4.

In addition, every quadratic function has a symmetric graph that either always curves upward or always curves downward. The graph opens upward if and only if \(a \gt 0\) and opens downward if and only if \(a \lt 0\text{.}\) We often call the graph of a quadratic function a parabola. Every parabola is symmetric about a vertical line that runs through its lowest or highest point.

Note particularly that due to symmetry, the vertex of a quadratic function lies halfway between its \(x\)-intercepts (provided the function has \(x\)-intercepts). In both Figures 1.5.5 and 1.5.6, we see how the parabola is symmetric about the vertical line that passes through the vertex. One way to understand this symmetry can be seen by writing a given quadratic function in a different algebraic form.

In Example 1.5.7, we saw some of the advantages of writing a quadratic function in the form \(q(x) = a(x-h)^2 + k\text{.}\) We call this the vertex form of a quadratic function.

Subsection 1.5.2 Modeling falling objects

One of the reasons that quadratic functions are so important is because of a physical fact of the universe we inhabit: for an object only being influenced by gravity, acceleration due to gravity is constant. If we measure time in seconds and a rising or falling object's height in feet, the gravitational constant is \(g = -32\) feet per second per second.

One of the fantastic consequences of calculus — which, like the realization that acceleration due to gravity is constant, is largely due to Sir Isaac Newton in the late 1600s — is that the height of a falling object at time \(t\) is modeled by a quadratic function.

If height is measured instead in meters and velocity in meters per second, the gravitational constant is \(g = 9.8\) and the function \(h\) has form \(h(t) = -4.9t^2 + v_0t + s_0\text{.}\)

Subsection 1.5.3 How quadratic functions change

So far, we've seen that quadratic functions have many interesting properties. In Preview Activity 1.5.1, we discovered an additional pattern that is particularly noteworthy.

Recall that we considered a water balloon is tossed vertically from a fifth story window whose height, \(h\text{,}\) in meters, at time \(t\text{,}\) in seconds, is modeled²Here we are using \(a = -5\) rather than \(a = -4.9\) for simplicity. by the function

We then completed two tables to investigate how both function values and averages rates of change varied as we changed the input to the function.

In 1.5.9, we see an interesting pattern in the average velocities of the ball. Indeed, if we remove the “\(AV\)” notation and focus on the starting value of each interval, viewing the resulting average rate of change, \(r\text{,}\) as a function of the starting value, we may consider the related table seen in Table 1.5.10, where it is apparent that \(r\) is a linear function of \(a\text{.}\)

\(a\)	\(r(a)\)
\(0\)	\(r(0) = 15\) m/s
\(1\)	\(r(1) = 5\) m/s
\(2\)	\(r(2) = -5\) m/s
\(3\)	\(r(3) = -15\) m/s
\(4\)	\(r(4) = -25\) m/s

Indeed, viewing this data graphically as in Figure 1.5.11, we observe that the average rate of change of \(h\) is itself changing in a way that seems to be represented by a linear function. While it takes key ideas from calculus to formalize this observation, for now we will simply note that for a quadratic function there seems to be a related linear function that tells us something about how the quadratic function changes. Moreover, we can also say that on the downward-opening quadratic function \(h\) that its average rate of change appears to be decreasing as we move from left to right³Provided that we consider the average rate of change on intervals of the same length. Again, it takes ideas from calculus to make this observation completely precise..

A key closing observation here is that the fact the parabola “bends down” is apparently connected to the fact that its average rate of change decreases as we move left to right. By contrast, for a quadratic function that “bends up”, we can show that its average rate of change increases as we move left to right (see Exercise 1.5.5.2). Moreover, we also see that it's possible to view the average rate of change of a function on \(1\)-unit intervals as itself being a function: a process that relates an input (the starting value of the interval) to a corresponding output (the average rate of change of the original function on the resulting \(1\)-unit interval).

For any function that consistently bends in either exclusively upward or exclusively downward on a given interval \((a,b)\text{,}\) we use the following formal language⁴Calculus is needed to make Definition 1.5.12 rigorous and precise. to describe it.

Thus, we now call a quadratic function \(q(x) = ax^2 + bx + c\) with \(a \gt 0\) “concave up”, while if \(a \lt 0\) we say \(q\) is “concave down”.

Subsection 1.5.4 Summary

• Quadratic functions (of the form \(q(x) = ax^2 + bx + c\) with \(a \ne 0\)) are emphatically not linear: their average rate of change is not constant, but rather depends on the interval chosen. At the same time, quadratic functions appear to change in a very regimented way: if we compute the average rate of change on several consecutive \(1\)-unit intervals, it appears that the average rate of change itself changes at a constant rate. Quadratic functions either bend upward (\(a \gt 0\)) or bend downward (\(a \lt 0\)) and these shapes are connected to whether the average rate of change on consecutive \(1\)-unit intervals decreases or increases as we move left to right.

• For an object with height \(h\) measured in feet at time \(t\) in seconds, if the object was launched vertically at an initial velocity of \(v_0\) feet per second and from an initial height of \(s_0\) feet, the object's height is given by

  \begin{equation*} h = q(t) = -16t^2 + v_0t + s_0\text{.} \end{equation*}

  That is, the object's height is completely determined by the initial height and initial velocity from which it was launched. The model is valid for the entire time until the object lands. If \(h\) is instead measured in meters and \(v_0\) in meters per second, \(-16\) is replaced with \(4.9\text{.}\)

• A quadratic function \(q\) can be written in one of three familiar forms: standard, vertex, or factored⁵It's not always possible to write a quadratic function in factored form involving only real numbers; this can only be done if it has \(1\) or \(2\) \(x\)-intercepts.. Table 1.5.13 shows how, depending on the algebraic form of the function, various properties may be (easily) read from the formula. In every case, the sign of \(a\) determines whether the function opens up or opens down.

  	standard	vertex	factored 6 Provided \(q\) has \(1\) or \(2\) \(x\)-intercepts. In the case of just one, we take \(r = s\text{.}\)
  form	\(q(x) = ax^2 + bx + c\)	\(q(x) = a(x-h)^2 + k\)	\(q(x) = a(x-r)(x-s)\)
  \(y\)-int	\((0,c)\)	\((0,ah^2 + k)\)	\((0,(0,ars)\)
  \(x\)-int 7 Provided \(b^2 - 4ac \ge 0\) for standard form; provided \(-\frac{k}{a} \ge 0\) for vertex form.	\(\left(\frac{-b \pm \sqrt{b^2 - 4ac}}{2a} , 0 \right)\)	\(\left(h \pm \sqrt{-\frac{k}{a}} , 0 \right)\)	\((r,0)\text{,}\) \((s,0)\)
  vertex	\(\left(-\frac{b}{2a}, q\left( -\frac{b}{2a} \right) \right)\)	\((h,k)\)	\(\left( \frac{r+s}{2}, q\left( \frac{r+s}{2} \right) \right)\)