6.1 A ProÞle of Quadratics from a Calculus Perspective 221
Notice that we have drawn families of parabolas. This is because shifting a graph vertically does not change its slope at a given point. The derivative determines a function only up to a constant. We have indicated by a dotted line the parabolas corresponding to those given in Example6.2.Althoughthederivativegivesusinformationabouttheshapeoftheparabola,it
doesnÕtgiveusanyinformationabouttheverticalpositioningoftheparabola.Thederivative will not help us pick out any one member of the family of parabolas drawn.
The Graph of a Quadratic Function
Thederivativeofthequadraticfunction2 isthelinearfunction 2 . The linear function is not horizontal because 0.1 Any nonhorizontal linear function has exactly one -intercept; at this intercept the line cuts the -axis and changes sign. This corresponds to a turning point of ; the quadratic function either changes from increasing to decreasing or vice versa.
Theturningpointofaparabolaiscalledthevertexoftheparabola.Thederivativegives us easy access to the shape of the parabola and the -coordinate of its vertex; we could have used derivatives to solve the problem in Example 6.1 efÞciently, and exactly.
Question: How can we Þnd the -coordinate of the vertex of the parabola?
The -coordinate of the vertex is the zero of the derivative function, so . Thereisnoneedtomemorizethis.Inpractice,simplydifferentiate andÞndthezeroof.
Question: How can we determine whether the parabola opens upward or downward?
If the graph of f has a positive slope,
If the graph of f has a negative slope,
, the parabola looks like
, the parabola looks like
If 2 , then the slope of the graph of is given by Ò.Ó Therefore we know that
if !0 then the parabola opens upward;
if 0 then the parabola opens downward.
D e f i n i t i o n
The slope of is written (read Ò double primeÓ) and is called the second derivative of . In LeibnizÕ notation is or 2 .
if !0, then the derivative is increasing; the graph of is concave up;
if 0, then the derivative is decreasing; the graph of is concave down.
This observation holds in general; there is nothing particular to quadratics here.
1 If 0 then the function is no longer quadratic; it is linear. In this case its derivative is a horizontal line, because the rate of change of a linear function is constant.
222 CHAPTER 6 The Quadratics: A ProÞle of a ProminentFamily of Functions
P R O B L E M S F O R S E C T I O N 6 . 1
1. On the left below are graphs of the quadratic functions , , , and ". Match , , , and " with the lines drawn on the right.
f g y y (i) (ii)
x x x x
graph of f
graph of g
x x x x
graph of h graph of j
For each of the quadratics in Problems 2 through 4, identify the - and -coordinates of the vertex and determine whether the vertex is the highest point on the curve or the lowest point on the curve.
2. 22 3
3. 32 27 15
4. 227 31
For Problems 5 through 8, nd a quadratic or linear function whose derivative is the line specied and whose graph passes through:
(a) the origin,
(b) the point 0, 2.
5. (a) 3
6. (a) 2
7. (a) 6 2
(b) 2 8
(b) # , #0
6.2 Quadratics From A Noncalculus Perspective 223
6.2 QUADRATICS FROM A NONCALCULUS PERSPECTIVE
InthissectionweÕlllookatquadraticswithoutusingcalculus.Thisisadifferentperspective on the same material to help you get a better intuitive feel for quadratics. In addition, we will take the opportunity to discuss the zeros of a quadratic, because the derivative gives us no information about the roots.
The graph of a quadratic function is a parabola opening up or down; conversely, a parabola opening up or down is the graph of a quadratic function. 2 is an example ofaparabolaopeningupward; 2 isanexampleofaparabolaopeningdownward.
2 is an example of a parabola opening to the right, and 2 is an example of a parabola opening to the left.
y y y y
x x x x
y = x2 y = —x2 x = y2 x = —y2
For the most part we are interested in the graphs of functions of ; therefore, when studying the latter two graphs, we will usually be restricting our interest to , , , and .
y y y y
x x x x
y = x y = — x y = —x y = — —x
A parabola has one turning point, called the vertex of the parabola.
The graph of a quadratic function is symmetric about its longitudinal axis, the vertical line through its vertex. VeriÞcation is left as an exercise (see the Problems for Section 6.2). Look at the simplest quadratic function, 2, as an illustration of this symmetry.
We Þnd the -intercepts of a parabola by solving 2 0. The solutions are given by the quadratic formula:2
2 The quadratic formula is discussed in Reference Section A: Algebra, Part IIIB.
224 CHAPTER 6 The Quadratics: A ProÞle of a ProminentFamily of Functions
2 4 2 2
The number of roots depends on the sign of 2 4; there are two distinct roots if this expression is positive, one root if it is zero, and no real roots if it is negative.
If 2 0 has no solutions, then the parabola 2 has no -intercepts.
If 2 0 has one solution, then the parabola 2 has one -intercept.3
If 2 0 has two solutions, then the parabola 2 has two -intercepts.
y y y
By thinking about the symmetry of the parabola we can see the following.
If there are two distinct -intercepts, then the -coordinate of the vertex lies midway between them. Compute the average of the two distinct roots of the corresponding quadratic equation by adding them and dividing by 2; youÕll get 2. Therefore the vertex of the parabola must have an -coordinate of 2.
Note that the number midway between $ and $ is ;
$$ 22 .
If there is only one -intercept, then that intercept, 2, is the -coordinate of the vertex. (Why? Think graphically.)
If there are no -intercepts, then we use the fact that the -coordinate of the vertex is midwaybetweenthepointsofintersectionoftheparabolaandanyhorizontalline that cuts it in two points. This means the -coordinate of the vertex is midway between the solutions to 2 for some , and the assertion about the vertex being at 2 follows. (Basically, the ÒÓ in the quadratic formula is adjusted so that there are two roots.)
The parabola given by 2 opens upward if ! 0 and downward if 0.
Reasoning: For large enough in magnitude (think about and , the 2 term dominates the other two terms; by this we mean that it overpowers the
3 This is a real root with multiplicity two. If is a root of multiplicity two, then the polynomial has the factor 2 as opposed to simply .
6.2 Quadratics From A Noncalculus Perspective 225
other terms and determines the sign and behavior of . 2 is always positive, allowing us to reason as follows.
If ! 0, then 2 is positive. Therefore is positive for large enough, and the parabola opens upward.
If 0, then 2 is negative. Therefore is negative for large enough, and the parabola opens downward.
Three points determine a quadratic. If we know three points satisfying the equation 2 , we can solve a system of three simultaneous linear equations4 for the three unknown constants, , , and .5 If one of the points we know happens to be the vertex of the parabola (and we are aware of that), then the parabola is determined by just the vertex and one other point.6
4 See Reference Section B on Simultaneous Equations.
5 We know that two points determine a line. Essentially, the number of points needed to determine a particular type of equation boilsdowntothenumberofconstantsonecanÒplayaroundwithÓintheequationÕsgeneralform.Forexample,foralinearequation # , we can adjust the # and the ; two points or independent pieces of information are needed. A quadratic equation 2 has three constants that we can modify, so we need three points to determine its equation.
6 Why? Using the symmetry of the parabola over the vertical line through its vertex, from the second point we can Þnd a third point.