Quadratic Functions

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Quadratic Functions

• are functions with a squared term.

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Introduction
We have already learned about linear functions,
for example f(x) 4x 1 g(x) -6x -2
h(x) (-2/3)x 2
The graphs of these functions are straight lines
and each equation gives us clues as to what
each line looks like f(x) 4x 1
Slope of 4
y-intercept of 1
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Quadratic Functions
Quadratic Term
Linear Term
Constant
4

• Identify the quadratic term, the linear term, and
  the constant term.
• 1) 2) 3)

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Question
What do the functions f(x) 2x2 4x 3
g(x) x2 4x


h(x) -x2 5x 6 have in common?
Answer
They are all QUADRATIC FUNCTIONS that can be
written in standard form f(x) ax2 bx c
where a ? 0
Pay close attention these coefficients a, b
and c will be clues that help us to know what the
graphs of these quadratic functions look like
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GRAPHING A QUADRATIC FUNCTION
A quadratic function has the form y ax 2 bx
c where a ? 0.
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Real Life Connection
Remember, quadratic functions represent real-life
situations such as
the height of a kicked soccer ball f(x)
-1/2(-9.8)t2 20t
the motion of falling objects pulled by gravity
height -16t2 100
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GRAPHING A QUADRATIC FUNCTION
The graph is U-shaped and is called a parabola.
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GRAPHING A QUADRATIC FUNCTION
The highest or lowest point on the parabola is
called the ver tex.
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GRAPHING A QUADRATIC FUNCTION
These are the graphs of y x 2 and y - x 2.
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GRAPHING A QUADRATIC FUNCTION
The origin is the lowest point on the graph of y
x 2, and the highest point on the graph of y
- x 2.
The origin is the vertex for both graphs.
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Graphs of Quadratic Functions
If the coefficient of the x2 term, a, is
positive, the parabola opens upward
If the coefficient of the x2 term, a, is
negative, the parabola opens downward
Quadratic Functions graph into a shape called a
Parabola
Maximum Point
Vertex
Minimum Point
f(x) x2 - 4
f(x) -x2 5
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• Use the related graph of each equation to
  determine its minimum or maximum point.
• 1) 2)

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GRAPHING A QUADRATIC FUNCTION
In general, the axis of symmetry for the
parabola is the vertical line through the vertex.
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Roots of Quadratic Functions
The roots of quadratic functions are the values
of x where the function equals zero ( f(x) 0 ).
When the function is graphed, the roots are
depicted as the places where the graph crosses
the x axis (since at this location, f(x) y 0
).
Quadratic functions can have two, one or zero
real-number roots.
Equivalently, their parabolas can have two
one
or zero real-number roots.
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Properties of the Graph of a Quadratic Function
Parabola opens up if a gt 0 the vertex is a
minimum point. Parabola opens down if a lt 0 the
vertex is a maximum point.
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Graphs of a quadratic function f(x) ax2 bx c
Axis of symmetry
Vertex is highest point
Axis of symmetry
a gt 0
a lt 0
Opens up
Opens down
Vertex is lowest point
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Graph the parabola
x y
0 0
1 3
2 12
-1 3
-2 12
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The graph of is a parabola with vertex
(0, 0)
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Locate the vertex and find the axis of symmetry
of the following parabola. Does it open up or
down? Graph the parabola.
Vertex
Since -3 lt 0 the parabola opens down.
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x y
2 13
3 10
4 1
1 10
0 1
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Determine whether the graph opens up or
down. Find its vertex and axis of symmetry.
Graph the parabola.
x-coordinate of vertex
y-coordinate of vertex
Axis of symmetry
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x y
-3 -13
-2 -11
-1 -5
-4 -11
-5 -5
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• Solve by graphing. (Find the roots)

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• Solve by graphing. (Find the roots)
• (3x 4)(2x 7) 0

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• Graph each function. Name the vertex and axis of
  symmetry.

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• Graph each function. Name the vertex and axis of
  symmetry.

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Putting it all together
Heres a problem that lets us use everything
weve learned today

• Given the function f(x) 5x x2 6
• Tell whether the function graphs into a parabola
  that opens upwards or downwards.
• Find the equation of the line of symmetry for the
  graph
• Find the coordinates of the vertex of the
  parabola
• Determine how many real-number roots the function
  has
• Graph the function

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Solve Quadratics by Square Roots
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Square Root Method

• Use if there is no linear term. (i.e. B 0)
• Get the Quadratic Term on one side and the
  Constant on the other side.
• Simply take the Square Root of Both Sides.

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Solve Quadratics by Square Roots

• Recall that
• Ex
• Thus, if we can isolate the square on one side of
  the equation, we can use square roots to remove
  the square
• When taking EVEN roots, remember that there is a
  positive a negative answer!

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Approximate vs. Exact Answers

• Most radicals are irrational numbers
• Exact answers contain the radicals in their
  simplest form
• Ex
• Approximate answers contain a decimal
  approximations of the radicals
• Ex
• For this class, unless otherwise indicated, give
  exact answers

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Solving Quadratics by Square Roots (Example)

• Solve by square roots give exact answers 5x2
  180

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Solving Quadratics by Square Roots (Example)

• Solve by square roots give exact answers x2
  15 -69

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Square Root Method
Be sure to give both the positive and negative
answers!
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Square Root Method
Be sure to give both the positive and negative
answers!
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Solving Quadratics by Factoring
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Quadratic Equation - Definition

• Up to now, we have studied polynomial expressions
• Quadratic Equation a polynomial equation of
  degree 2
• Standard Quadratic Form a quadratic equation
  written in the form ax2 bx
  c 0 where a, b, and c are constants

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Zero Factor Property

• Zero Factor Property If a b 0, then either
  a 0 or b 0
• Holds for real numbers or factors
• Thus, if a quadratic in standard form can be
  factored, we can apply the Zero Factor Property
  to solve the quadratic!
• Other methods must be employed when a standard
  quadratic cannot be factored

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Solving Quadratics by Factoring (Example)

• Solve by factoring 3x2 9x 30 0

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Solving Quadratics by Factoring (Example)

• Solve by factoring 4x2 9 12x

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Solving Quadratics by Factoring (Example)

• Solve by factoring 6x2 4 5x

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Solving Quadratics by Completing the Square
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Solving Quadratics by Completing the Square

• Often the case a quadratic cannot be factored
• Object is to transform the quadratic into a
  perfect square trinomial which condenses to the
  square of a binomial
• Ex
• We can solve the resulting equation using square
  roots

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How to Complete the Square

• Move the variable terms to the left side and the
  constant term to the right
• The coefficient in front of the square term MUST
  be a 1
• If it is anything else, both sides must be
  divided by the coefficient of the square term!
• Ex
• Ex

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Steps to Complete the Square

• The coefficient of x2 must be one. (If not divide
  both sides of the equation by the coefficient.) .
• Constant needs to be on right hand side of
  equation.
• Complete the square on x
• Take half of the coefficient of x, square the
  result, and add that number to both sides of the
  equation.
• Factor the perfect square trinomial.
• Solve the equation using square root property.

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Complete the square to make each binomial a
perfect trinomial square.

• x2 8x
• a2 5a

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Solving Quadratic Equations by Completing the
Square

• Solve the following equation by completing the
  square
• Step 1 Move quadratic term, and linear term to
  left side of the equation

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Solving Quadratic Equations by Completing the
Square

• Step 2 Find the term that completes the square
  on the left side of the equation. Add that term
  to both sides.

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Solving Quadratic Equations by Completing the
Square
Step 3 Factor the perfect square trinomial on
the left side of the equation. Simplify the
right side of the equation.
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Solving Quadratic Equations by Completing the
Square

• Step 4 Take the square root of each side

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Solving Quadratic Equations by Completing the
Square

• Step 5 Set up the two possibilities and solve

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Solving Quadratic Equations by Completing the
Square

• Solve x2 10x 21 0

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Solving Quadratic Equations by Completing the
Square

• Solve 3x2 1 - 4x

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Solving Quadratics by Completing the Square
(Example)

• Solve by completing the square give exact
  answers x2 10x 3 0

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Solving Quadratics by Completing the Square
(Example)

• Solve by completing the square give exact
  answers 4x2 8x 5

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Solving Quadratics by Completing the Square
(Example)

• Solve by completing the square give exact
  answers 3x2 x 1 0

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Solve by Completing the Square

• x2 10x 8 0

(x2 10x ) -8
(x2 10x (5)2) -8 25
(5)2 25
(x 5)2 17
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Solve by Completing the Square

• 3x2 24x 12 0

3
x2 8x 4 0
(x2 8x ) -4
(x2 8x (4)2) -4 16
(4)2 16
(x 4)2 12
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Solve by Completing the Square

• 5x2 - 30x 45 0

5
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Convert to vertex form

• y x2 10x 8

y - 8 (x2 10x )
(5)2 25
y - 8 25 (x2 10x (5)2)
y 17 (x2 10x (5)2)
y 17 (x 5)2 - 17
x 5 0
Axis of symmetry x -5
Vertex (-5, -17)
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Solve by Completing the Square

• 5x2 - 30x 75 0

5
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Solving Quadratics by the Quadratic Formula
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Solving Quadratics by the Quadratic Formula

• Essentially completing the square in formula
  format
• The quadratic must be in standard form before
  applying the quadratic formula!

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Solving Quadratics by the Quadratic Formula
(Example)

• Solve using the Quadratic Formula give exact
  answers 4x2 5x 1

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Solving Quadratics by the Quadratic Formula
(Example)

• Solve using the Quadratic Formula give exact
  answers -2x 7 x2

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Discriminant
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Discriminant

• Sometimes we like to know whether a given
  quadratic will produce real or complex solutions
• Discriminant the portion of the quadratic
  formula underneath the radical
• If b2 4ac gt 0, the quadratic has two different
  REAL solutions
• If b2 4ac 0, the quadratic has one REAL
  solution
• If b2 4ac lt 0, the quadratic has two different
  COMPLEX solutions

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Discriminant (Example)

• For each of the following, state the value of the
  discriminant and the number of real solutions
• a) 4x2 20x 25 0
• b) 2x2 7x 4 0
• c) 10x2 3x 1 0