Quadratics - Note and Lessons
Links to Site Lessons & Exercises: [ Graphing Exercises ] [ Graph y = a[(x-h)^2 +k] Theory ] [ Factoring Quadratics ] [ Difference of Two Squares ] [ Completing the Square ] [ Convert to Standard Form (Arith) ] [ Quadratic Formula ] [ Finding Coefficients ] [ Applications ] [ Quadratics Summary ] [ Exercises ]
Two or multiple views may be better than one. Good luck.
Links Elsewhere
Note: The excellent (ad-supported) Kyrgyz-Turkish
High Schools Mathematics Pages site (see its [lecturen
Notes] [worksheets]
[review_exercises]
) has excellent senior high school or college notes in pdf and html form for
many areas of mathematics including quadratic
equations and radical equations of quadratic form - equations that can
solved after some work via quadratic equation methods.
Graphing, equations and formulas
A guide for studying and teaching quadratics follows. In the
guide, the initial lessons have more details and examples than the later ones -
a situation to change.
1. Standard Forms for Graphing
Each quadratic expression
y = ax2+bx+c
can be rewritten in (converted to) the standard form for graphing
y = a[(x-h)2+k]
in a way that the two expressions
ax2+bx+c and a[(x-h)2+k]
have the same value, no matter what values we assign to the
numbers a, b, c and x.
The case a = 0 would lead to a linear bx +c or
constant expression c for y. That case was studied early in the site
coverage of straight lines.
The Qc alternate standard form for graphing follows by
putting q = ak and then expanding y = a[(x-h)2+k] with the aid of
the distributive law to obtain
y = a(x-h)2+ak
and then the alternate standard form for graphing, that is,
y = a(x-h)2+ q
The equivalent expression
y = ax2+bx+c and y = a[(x-h)2+k]
and y = a(x-h)2+ q
when obtained from each other through algebra, that is,
through repeated use of the distributive law and/or factorization,
all give the same value for y.
The first lesson, Quadratics
Graphing, Numerical Exercises, show how shift the graph of the parabola
y = x2
and if need be rescale the vertical axis to obtain the graph of
another parabola
y = a[(x-h)2+k]
The graph of these parabolas drawn with their axes of symmetries
look like pitchforks that point up if the coefficient a is positive and point
down if the coefficient a is negative.
The second lesson, Graphing
y = a[(x-h)2 +k], General Discussion or Theory. is best
understood after the first.
2. Factoring by Inspection
Some quadratic expressions
y = ax2+bx+c
where the coefficients a, b and c are whole numbers or integers
can be easily factored, written as a product a(x-r)(x-s) for some integers or
rational numbers r and s.
We will study the case
y = x2+bx+c
where a = 1 (so it is not written) and where b and c are integers. In
this case, factoring by inspection tries to rewrite x2+bx+c in the
form (x-r)(x-s) where r and s are integer, b = r+ s is their sum, and c = rs is
their product. The key to factoring by inspection is look at all pairs of
integers r an s whose product gives the constant term c, and see if any pair has
the property that r+s = b = the coefficient of the linear term x. You can
find the factors r and s by trial and error, or you follow the route described
in the lesson Factoring
Quadratics by "Inspection", a route that employs the prime
number factorization of the natural or whole number |c| to generate
all possible values of r and s.
3. Sign Analysis and Zeroes and of Factored Quadratics
This lesson (to be written) examines when and where expressions of the form
a(x-r)(x-s) or of the form a[(x-h)2+k] are positive, negative and
zero. The when and where are provided by intervals of the real line.
Here x is the variable and the letters a, r, s, h and k are parameters
- numbers kept constant in each example. See the site lessons on how
to talk about numbers and quantities in three skills for algebra.
Step A. See or remember these geometric
and algebraic
previews of calculus (Chapters 2 to 4 in Volume 3) for an easy informal
introduction to
- the monotonicity of functions y = f(x), that is where they are increasing
or decreasing, with the where referring to intervals,
- the location of extrema: local, end-point and absolute maxima and minima,
and
- the sign analysis not of functions, but of formulas for their slopes.
For definitions of terms (increasing, decreasing, maxima and and
minima) see the Zeroes and Sign
Analysis function properties page.
Step B Do the following exercises. Step A should make the following
definitions easy to understand and the following exercises
For
a Curve or function y
For A Quadratic (factored)
For a Cubic Factored
Un-Sign Functions etc.
easy to do or complete.
Teachers: (1) This topic,
a site innovation (?), introduces students to the zero and nonzero
product properties of real numbers through hands-on numerical experience -
locating the zeroes of factors. It also develops algebraic skills.
- If A and B are nonzero real numbers then their
product AB is nonzero.
- Contrapositive Form. If a product AB of two real
numbers is zero then at least one of the factors A and B must be zero.
More generally, students can be informed, the product of
nonzero real numbers is nonzero when each of the real numbers, the product
factors, are nonzero. The contrapositive form is as follows. If a product
of real factors is zero, then at least one and possibly more of the
factors must be zero. (2) The sign and zero analysis of
linear, quadratic and rational expressions appears in the slope-based
preview of calculus where sign analysis of slopes identifies where a
function y = f(x) is increasing or decreasing, and from that identifies
the location of maxima and minima. See the first chapters of
the online volume 3, Why
Slopes and More Math, to learn more. (3) Finally, the previous
items (1) and (2) will make the the algebraic way of writing and reasoning
clearer to students, a worthwile investment in itself, that will help
below. |
4. Factorization Method for Difference of two Squares
The step used after the Conversion to Standard Form to factor a quadratic
when k is negative is given in the site lesson, the Difference
of Two Squares.
What can be done if k is zero or positive. Here we can say
the square of a negative number is undefined until you have mastered complex
numbers (easier than you think)
5. Completing the Square
Lesson: Completing
the Square
Each quadratic expression
y = ax2+bx+c
can be rewritten in (converted to) the standard form for graphing
y = a[(x-h)2+k]
by Completing the
Square. Follow the link to find how. That a first step in factoring
quadratic expressions ax2+bx+c and find the number and location of
its roots, that is, where ax2+bx+c = 0, if at all.
6. Factoring Quadratic Expressions - Arithmetic Way
Lesson: Arithmetic
Approach - working with numbers
The quadratic formula for finding roots of expressions comes follow
from (i) completing the square and then (ii) factoring if (i) results in the
difference of two squares. Numerical examples follow to illustrates the step.
The quadratic formula itself is derived in the next lesson.
In other parts of this site (see chapters 14 an 15 of the
site volume 2, Three
Skills for Algebra, arithmetic and algebraic solutions to questions have
been developed. This lesson provides the arithmetic solution for solving
quadratics equations or factoring them. The next lesson gives the algebraic
solution or route. In the next lesson, the general case in which the letters
a, b and c are not immediately replaced by numbers follows the same route and
leads to the quadratic formula.
purplemath offers two lessons
on quadratic examples
of completing the square and deriving the quadratic formula.for roots of
ax2+bx + c = 0. Multiple views are better than one.
7. Factoring Quadratic Expressions - Algebraic Way
Lesson: Algebraic
Approach - working with literals and so deriving the quadratic formula.
This lesson derives the quadratic formula by following the factorization and
equation solving roots. The lesson begins with reducible and irreducible
quadratic expressions, a brief but optional topic. Then it follows three steps.
- For completing the square, convert ax2+bx + c into the
form
a[x2+2Qx +C] - factor the coefficient a (non-zero as this is a
quadratic and not a linear expression).
- Complete the square by converting latter into the form a[(x+Q)2+k],
- If latter results in a difference of two squares, then k = -w2 <
0, and we can put a[(x+Q)2+k] = a[(x+Q)2-w2]
= a[x+ Q - w][x+Q+w] using the difference of two squares factorization
method.
Here the coefficients Q, C, k and w can be expressed in terms of a, b and c.
The question of when ax2+bx + c = a[x+ Q - w][x+Q+w] = 0 leads
to the quadratic formula for the case k = -w2 < 0.
Remark: The derivation of the quadratic formula also works if your
knowledge of complex numbers is sufficient (enough) for the deployment or use
of square roots of negative numbers.
8. Backward and Foreword use of Formulas
Lessons: Finding
Coefficients in Quadratics, yet to be written, will provides a few
examples of the following.
To draw a building which occupies a rectangle with the aid
of coordinates, we would choose or take coordinate axes parallel to the sides
of the rectangle as line segments parallel to coordinate axes are easily
described using equations of the form x = a and y =b. Then to obtain small
numbers a and b in those equations, we would place the origin of the
coordinate system at one corner of the rectangle so that two sides lie on the
coordinate axes. Alternatively, if symmetry might be advantages in our future
reasoning or calculations, the origin might be placed at the center of the
rectangle or offset, on one axis of symmetry, not both. So in choosing a
coordinate system, we have options that may be taken to aid future work, or
the determination of equations. Geometrical consideration of what
coordinate system to use or which equation to use as the first step in further
work are useful not only in drawing, but also but also in mathematics
and physics, and indeed in any situation where coefficients in equations have
to be found, etc, etc.
The equivalent expression
y = ax2+bx+c and y = a[(x-h)2+k]
and y = a(x-h)2+ q
when obtained from each other through algebra, that is,
through repeated use of the distributive law and/or factorization, all
give the same value for y. In a given problem or situation, each expression has
or may have an ease of use advantage over the other. Experience is needed to
recognize the advantages if any.
Forward or Direct Use: One or more of
the above expressions may be used to graph a quadratic or to draw conclusions
about it. That say represents the forward or direct use of these expressions.
Backward Problem: Given data about the graph of a
quadratic, find a formula for it. The or a solution is to to find the
coefficients in one of the following equivalent equations for a quadratic.
y = ax2+bx+c and y = a[(x-h)2+k]
and y = a(x-h)2+ q
So from the data, you may want to find the coefficients a, b and c (option
I), or find the coefficients a, h and k (Option II), or find the coefficients a,
h and q (Option III).
Which option requires the least amount of effort or work? That question
itself, requires familiarity (experience) with graphing quadratics using these
three equivalent expressions for them, and in graphing to understand the
geometric & numeric meaning or use or significance of the coefficients in
option I, II and/or III.
Backward and Forward Problem Combined: Solve the backward problem and
the answer a question about the quadratic using the result of the backward
problem.
Remark: There is a forth option, a variant of the
second perhaps. If quadratic y = ax2+bx+c has roots (zeroes)
when x = u and x = v then another equivalent expression for it is y =
a(x-u)(x-v). This fourth option is available only when there are roots
(zeroes), that is when and only when the graph of the quadratic touches or
crosses the x-axis. Then in the equation for axis of symmetry x =
h, the parameter h = ½ (u+v).
Remark: Having so many options for the determination
of a quadratic from geometric data may be confusing, but experience will
or would help provide familarity with the options and so build your algebraic
skills and concepts. The forward and backward use of quadratics is a
frequent part of calculus and beyond that mathematics in physics and
engineering,
9. Quadratics: Applications in Geometry Physics Etc
Lesson: Quadratic
Applications
Problems may come from several sources.
- Solving Systems of Equations - one quadratic, one linear.
- Examples from Physics.
- Constant Velocity Motion
- Quadratic in Time implies Constant Acceleration
- Constant speed and constant acceleration motion (enriched topic)
- Examples from Economics (do, but view with suspicion)
Apart from lip service to applications, mastery of quadratics is needed for
calculus and beyond in science, engineering, mathematics and other quantitative
disciplines based on calculus (or special functions such as logarithms and
exponentials.)
Remark: Applications in economics of quadratics exist,
and you may meet them, but those I have met seem more unreal, contrived,
or artificial than the physic applications.
Remark: The quadratic formula may be used to solve ax2+bx
+ c = 0 directly. Or, factoring by inspection and factoring by completing the
square and using the difference of two squares can be use to say ax2+bx
+ c = a(x-r)(x-s) for some real numbers r and s
Problem Type: Intersection of a line and a parabola.
The intersection is found by solving a systems of Equations - one
quadratic, one linear.
The intersection of a line y = Ax + B and parabola y = ax2+bx
+ c may be found by solving Ax +B = ax2+bx + c. The latter
yields ax2+(b-B)x + (c-B) = 0 which can be solved for x by the most
convenient you see, say by inspection, by completing the square or by the
quadratic formula. The latter quadratic in x may have two, one or no solutions.
For each x solving the quadratic, there is a y = Ax+B to be computed in order to
obtain the coordinates (x,y) of an intersection point.
Problem Type: Projectile Motion
Let t denote time. Let y denote height. Then quadratics y = at2+bt+c
may be used to describe or approximate the height of thrown or
free-falling projectiles such as bullets, rocks and balls when air resistance is
negligible or neglected. For such projectiles, equations of the form x = pt+q
may describe the projective movement in a horizontal direction.
If we express t in terms of x, we see that time t is given
by a linear expression in x. That expression can be use to eliminate t in y =
at2+bt+c to obtain a quadratic relation y = Ax2+Bx+C
between the y and x coordinates of the projectile. So we conclude, the
projectile follows a quadratic path in the xy plane. In the foregoing, the
upper case letters A, B and C The letters A, B and C depend on the
coefficients p, q, a, b and c. The do not have the same meaning or same value
as the lower case letters a, b and c unless x = t in a unit-free description
of the physical situation.
In practice, you may meet
- Vertical projectile motion - the position of a falling
object subject to the constant pull of gravity at or near the earth's
surface can be described using quadratics expressions y = at2+bt
+ c with time t in place of horizontal coordinate x as the independent
variable. The direct use of this equation is to calculate coordinate y
given the value of time t. One indirect use of this equation gives the value
of y and asks for the value or possible values of t. You will need to
solve a quadratic equation for t and if there are two numerical solutions,
decide which one is required or selected by the information at hand. Further
indirect uses of the formula may give you values of y and t, clearly or not,
and ask you find the values of the coefficients a, b and c, before using y =
at2+bt + c directly, or indirectly again.
- Projectile Motion in the Plane: Here y = at2+bt
+ c and x = Bt + C describes a falling body in the vertical xy plane near
the earths surface. You may be asked to analyze these equations forwards and
backwards.
- Free Sliding Object on a slanted plane. y = at2+bt + c
and x = At2+ Bt + C but simplifications may follow, will follow,
from using a slanted coordinate system with x or y coordinate in the
plane. The equations for situation B or C may reappear.
Remark: The description of projectile motion is
provides a war-like qualitative idea of the flight of projectiles.
Suffice it to say, I do not like the connection of mathematics to the arts of
war, past and present. Mathematics skills and concepts have been driven by
various motivations in consumer life, business, construction, science (planetary
movements included), technology and war. Projectile motion provides the
application of quadratics most easily visualized and so most useful for the
development of mathematical skills - ouch.
10. Summary
Lesson: Summary of Main Ideas
gives an algebraic viewpoint.
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