Transcript Quadratics: Sequel Concepts to Polynomials

Quadratics: Sequel Concepts
to Polynomials
Richardson 423
Math 2
Quadratics: What’s the big deal?
• In this chapter of Math 2 we will be covering Quadratics
• In the previous lesson we learned to quantify groups of terms, exponents, and
multiple variable problems.
• We take the next step in applying these concepts of rules and calculation into
a visual application.
• (Some would say it placing a physical face on why we are doing all this mathematics
exercise)
• This lesson will define the vocabulary and necessary notes that are required
for your NC Final Exam requirements.
• It is imperative that you take 40 minutes a night to review the following
information to ensure a good grade this week.
Quadratics: What is Quadratics?
• Quadratics:
• Definition: quadratics describes something that pertains to squares, to the
operation of squaring, to terms of the second degree, or equations or
formulas that involve such terms
• i.e. the application of equations and terms that have variables of a second degree or
higher.
• Quadratics literally have atleast one term that has a
variable in the second degree
• Quadratics can have more than three terms but the
normal equations we will initially work with will have
three.
• Quadratics follow the terms:
• ax2+bx+c
So what is that we do with Quadratics?
• Quadratics are we use to create to plot Parabolas
• Parabolas-The parabola is the curve formed from all the points (x, y) that are
equidistant from the directrix and the focus. The line perpendicular to the
directrix and passing through the focus (that is, the line that splits
the parabola up the middle) is called the "axis of symmetry".
Here’s another look at the illustration close
up.
So wait, what? That was a lot of new terms!
Hey Mr. Herb! You just spat a lot of alien jibberish at me SUN!
You’re right. Aight calm down. Relax, We’ll go over everything starting with the next
slide.
Ok, cause, I’m saying…I don’t know all dat….
SHUDDUP!!!!
Mr. Herb’s
thoughts.
Quadratic vocabulary terms
• Focus- is a fixed point on the interior of a parabola used in the formal
definition of the curve.
Quadratic Vocabulary Terms
• Directrix- a line perpendicular to the axis symmetry used in the
definition of the parabola.
Quadratic vocabulary terms
• Axis of symmetry-placement line the
mirrors the motion of the parabola.
• Vertex-the point at whch a parabola
makes its sharpest turn.
• The vertex is halfway between the directrix
and the focus.
Okay so what does that have to do with all
the math stuff we have been doing?
• Good question. Well think about it like this. We have been gearing
you up for this event with proper mental tools to solve the quadratic
equations.
• So let us understand what this week’s objectives are in this case.
What is a Quadratic function?
• Quadratic function
• An equation where the variables located in the terms have a degree of 2.
• Example: 2xy2,x2
• Quadratic functions consists of 3 terms in most cases.
Quadratic Term
Ax2
+
Linear Term
Bx
Constant term
+
C
What does a quadratic function make when it
is graphed
• A parabola….no seriously…It makes a parabola….
• You may laugh now.
“Ha”
So what do we do with these Quadratic
equations?
• Man! You guys are full of good questions today!
• First we are using our skills of exponents and variables/terms to help decipher
the expressions to simplify the mathematical equation to one simple
sentence answer.
• Second from the simplified equation we have calculated we are trying to
identify if the Quadratic equation has:
• One real solution
• Two real solutions
• No real solution
• Third once we identify which of the three cases we have we want to graph
our results on a standard Cartesian graph.
Quadratic Solutions: First case scenario
Two real solutions
• In this scenario, once we finalize on our quadratic solution we look to find the
zero’s or ‘roots’ of our equation.
• This happens once we have grouped and combined like terms and have worked
out our equation to one quadratic equation/ two parentheses terms.
• Example: x2+5x+6=(x+2)(x+3)
• The roots of the equation/solution can be found by setting the parantheses terms
equal to zero.
• Example: (x+2)=0, (x+3)=0
• From here we solve to find out our 2 answers
• After we find the ‘roots’ we setup an x/y table using substitution to make us a
table of coordinates to plot on the graph.
• These problems are convenient because it honestly takes simple reduction and
the FOIL method to solve.
• Our results show that the ‘roots’ of the equation have TWO places in the bends of
the parabola that cross the x-axis.
Quadratics Solutions: Second case scenario
One real solution
• In this scenario, are given an equation that is being asked to sorted out to
final quadratic form.
• Example: 14-x2=-6x+23
• As you can see from the equation above we have to use algebra in order:
1. To get all terms to one side
2. Possible to factor out major number that will reduce the coefficients.
3. To simplify the equation to final quadratic form
1.
ax2+bx+c
• Once the equation has been simplified to standard quadratic form we use
the FOIL METHOD factor in order to find our roots.
• In the ‘one real solution’ scenario we find that after factoring, the solution
has the same number.
• In this case the zero for the graph would be the vertex of the parabola. This
means the parabola only crosses the x-axis at ONE point.
Quadratic Solutions: Third case scenario
No real solution
• Last case scenario deals with quadratic equations that have NO real
roots.
• In other words despite the reduction, combining of like terms, and
simplification, the trinomial can not be easily decoded to have a
simple root.
• In the graphing sense, the parabola still follows its shape and can be
plotted but none of the points cross the ‘x-axis’.