Algebra ( Period 5)

Transformations of Quadratic Functions  9-3

Transformations change the position or size of any figure.
In Math 8 we are doing this with geometric figures like triangles and parallelograms. In Algebra we will do this with parabolas.

TRANSLATION:
The parabola stays the same size but simply SLIDES to the left, right, up or down (or even a combination—like up and to the right)

VERTICAL TRANSLATION: A vertical translation up or down happens with you add a constant (k) to a parent function. For example the parent graph y = x² has the vertex at the origin but if you add 4 to the it,
 y = x² + 4à the vertex moves up 4 on the y axis
If you add -4  { y = x² + (-4)  or simply y = x² - 4}  it moves down 4 on the y axis.

HORIZONTAL TRANSLATION: A horizontal translation right or left happens when you add a constant (h) to the  x-value of the parent function.
We place that h value in a ( ) with the x variable:
(x –h)² would be a slide RIGHT

(x + h)² would be a slide LEFT

COMBINATION TRANSLATION: Together, both types of slides are shown by this formula ( known as the VERTEX FORMAT)  f(x) = (x –h)² +k
Notice that since the formula has –h, it moves in the opposite direction right or left. Since the formula is  +k   it moves in the same direction up or down!

DILATION: This type of transformation makes the graph either narrower or wider. As “a” is a smaller and smaller fraction/decimal the parabola gets wider and wider. AS “a” gets bigger—the parabola gets narrower.

REFLECTION: This transformation FLIPS the parabola upside down. This happens when “a” is negative.

So now the full VERTEX FORM of a parabola/quadratic is

y = a(x - h)² + k

Algebra Honors ( Period 4)

Transformations of Quadratic Functions  9-3

Transformations change the position or size of any figure.

In Math 8 we are doing this with geometric figures like triangles and parallelograms. In Algebra we will do this with parabolas.

TRANSLATION:
The parabola stays the same size but simply SLIDES to the left, right, up or down (or even a combination—like up and to the right)

VERTICAL TRANSLATION: A vertical translation up or down happens with you add a constant (k) to a parent function. For example the parent graph y = x² has the vertex at the origin but if you add 4 to the it,
y = x² + 4à the vertex moves up 4 on the y axis
If you add -4  { y = x² + (-4)  or simply y = x² - 4}  it moves down 4 on the y axis.

HORIZONTAL TRANSLATION: A horizontal translation right or left happens when you add a constant (h) to the  x-value of the parent function. We place that h value in a ( ) with the x variable:
(x –h)² would be a slide RIGHT

(x + h)² would be a slide LEFT

COMBINATION TRANSLATION: Together, both types of slides are shown by this formula ( known as the VERTEX FORMAT)  f(x) = (x –h)² +k

Notice that since the formula has –h, it moves in the opposite direction right or left. Since the formula is  +k   it moves in the same direction up or down!

DILATION: This type of transformation makes the graph either narrower or wider. As “a” is a smaller and smaller fraction/decimal the parabola gets wider and wider. AS “a” gets bigger—the parabola gets narrower.

REFLECTION: This transformation FLIPS the parabola upside down. This happens when “a” is negative.

So now the full VERTEX FORM of a parabola/quadratic is

y = a(x - h)² + k

Algebra ( Period 5)

Solving Quadratic Equations by Graphing 9-2

After putting the function in standard form ( ax² + bx + c) make the F(x) or y = 0 to find the x-intercept(s) of the parabola! We already know this!

x- intercepts = the roots = the zeros = the solutions of the quadratic.

One of three things will happen when you graph the parabola

1) it will go through the x axis twice (the vertex is below if it is a happy face or above it it’s  sad face.)  TWO REAL SOLUTIONS   TWO REAL ROOTS

2) It will have only one intercept (the VERTEX is on the x-axis) ONE REAL SOLUTION ONE REAL ROOT ( actually called a DOUBLE ROOT)

3) It will have NO x-intercepts ( the vertex is above if it is a happy face or below if it’s a sad face)  NO REAL SOLUTION   NO REAL ROOTS

If the x-intercept(s) are not an integer, we can estimate the roots OR use a calculator to find them exactly ( finding the zeros)

We sometimes just say that they are between the two integers that we find them on the graph!
We can also estimate to the tenths by making a table.

HOWEVER we will learn other methods in this chapter as we go along!

Algebra Honors ( Period 4)

Solving Quadratic Equations by Graphing 9-2

After putting the function in standard form ( ax² + bx + c) make the F(x) or y = 0 to find the x-intercept(s) of the parabola! We already know this!

x- intercepts = the roots = the zeros = the solutions of the quadratic.

One of three things will happen when you graph the parabola

1) It will go through the x axis twice (the vertex is below if it is a happy face or above it it’s  sad face.)  TWO REAL SOLUTIONS   TWO REAL ROOTS

2) It will have only one intercept (the VERTEX is on the x-axis) ONE REAL SOLUTION ONE REAL ROOT ( actually called a DOUBLE ROOT)

3) It will have NO x-intercepts ( the vertex is above if it is a happy face or below if it’s a sad face)  NO REAL SOLUTION   NO REAL ROOTS

If the x-intercept(s) are not an integer, we can estimate the roots OR use a calculator to find them exactly ( finding the zeros)

We sometimes just say that they are between the two integers that we find them on the graph!
We can also estimate to the tenths by making a table.

HOWEVER we will learn other methods in this chapter as we go along!

Algebra Honors ( Period 4)

Graphing Quadratic Functions 9-1

The standard form of a quadratic ( 2^nd degree function)  is

ax² + bx + c

The shape it makes when you graph it is called a parabola
Parabolas are symmetric around a line called the axis of symmetry. (AoS)

 The axis goes through a point on the parabola called the vertex

 The vertex is the maximum if the graph is a sad face and a minimum if it’s a happy face.
The parent graph is the simplest form of the graph.
For quadratics, the parent graph is  y = x²

 Note: that while b and c can both be ZERO, “a” can NEVER be zero for a quadratic.
WHY?      Then it would be a linear equation.

if a > 0 ( positive) the parabola opens upward ( happy face)
if a < 0 ( negative) the parabola opens downward ( sad face)

What happens as the “a” coefficient gets really big or really small ( fraction / decimal)?
The larger the “a” coefficientà the narrower the parabola.
The smaller (fractional/decimal) the “a” coefficientà the wider the parabola

{THINK: what happened when the “m” (or slope) coefficient got big? The slope got steeper. So, similarly, both sides of the U get steeper at the same time!!}

{NOW THINK: what happened when the “m” (or slope) coefficient got tiny ( or fractional)? The slope became what we called a bunny slope. So, similarly, both sides of the U get to be bunny slopes at the same time!!}

The x value of the vertex as well as the axis of symmetry are the SAME:

a and be are from the standard form of the quadratic.
Notice when b is missing ( b = 0), the vertex is always on the y-axis!! and therefore the axis of symmetry ( AoS) is also the y-axis.

 The c value is the y-intercept ( when b=0) because when x = 0 you are on the y axis.
f(x) = ax² + bx+ c
f(0) = a(0)² + b(0) + c
f(0) = c = the y value  … and in this case the y-intercept!

The domain of a quadratic (parabola) that is a function (opens up or down) is ALL REAL NUMBERS
The range depends on which way it opens:
If it opens up, the range will be   

      


If it opens down, the range will be    

You can graph quadratics exactly the same way you graphed lines—plug in your choice of an x value and use the equation to find your y value

 Because it is a U shape, you should graph at least 5 points as follows:

FIRST:  make sure the equation is in standard form!
y must be isolated on one side and then you can read the a and b coefficients

 y = ax² + bx + c

Point 1) the vertex: the minimum value of the smile or the maximum value of the frown
the x value of the VERTEX is  -b/2a
Plug that into the equation and find the y value of the vertex
Next draw the AXIS OF SYMMETRY (AoS)  

Notice- this is a linear equation! It is a line through the vertex that is parallel to the y-axis!

Point 2) Pick an x value immediately to the right or the left of the AXIS OF SYMMETRY (AoS)   and find its “y” by plugging into the equation.

Point 3)  Pick another x value one step farther on that same side of the AXIS OF SYMMETRY (AoS)  as point 2—and find its “y” by plugging into the equation.

Points 4 and 5) Graph the mirror image of both Point 2 and Point 3 on the other side of the AXIS OF SYMMETRY (AoS) by counting from the of the AXIS OF SYMMETRY (AoS)

You can choose to do two more points—if you want—but your points are joined in a SMOOTH U SHAPE ( not a V shape) and extend the lines with arrows on each end!

Example:   f(x) = -3x²  or just             y = -3x²

Notice right away—the “a” coefficient is negative so this is a frown face… a sad face! You know which way the graph will open! (Downward)

The x value of the vertex is –b/2a  a = -3 but b = 0 ( it is missing!)
so the x value of the vertex is –b/2a = -0/2(-3) = 0    Plug that back into the function f(0) = -3(0²) = 0

(Or just begin to realize that if b is missing the vertex is always on the y axis and the  AXIS OF SYMMETRY (AoS) will always be x = 0. )

In this case because there is NOT a “c” in the function, the vertex is ( 0,0) the origin.
The domain is all real numbers

The range is y ≤  0

To graph this function:

1) Graph the vertex ( 0,0)

2) Draw the AXIS OF SYMMETRY (AoS) a dotted line at x = 0  ( actually this is the y-axis so use a different colored pencil to indicate the AoS)

 3) Pick an x value immediately to the right of the AoS, x = 1 works so plug that into the equation to find its y value
y = -3(1²) = -3 PLOT ( 1, -3)

 4) Pick the next x value x = 2 and  plug that into the equation to find its y value
y = -3(2²) = -12 PLOT ( 2, -12)

 5) Count the same 1 step to the LEFT of the AoS as at the same y value of ( 1, -3) That would be ( -1, -3) and plot that point!

 6) Count the same 2 steps to the LEFT of the AoS at the same y value of ( 2 -12) That would be ( -2, -12) and plot that point

 7) Connect with a smooth U shape and extend with arrows!

Algebra (period 5)

Graphing Quadratic Functions 9-1

The standard form of a quadratic ( 2^nd degree function)  is

ax² + bx + c

The shape it makes when you graph it is called a parabola
Parabolas are symmetric around a line called the axis of symmetry. (AoS)

 The axis goes through a point on the parabola called the vertex

 The vertex is the maximum if the graph is a sad face and a minimum if it’s a happy face.
The parent graph is the simplest form of the graph.
For quadratics, the parent graph is  y = x²

 Note: that while b and c can both be ZERO, “a” can NEVER be zero for a quadratic.
WHY?      Then it would be a linear equation.

if a > 0 ( positive) the parabola opens upward ( happy face)
if a < 0 ( negative) the parabola opens downward ( sad face)

What happens as the “a” coefficient gets really big or really small ( fraction / decimal)?
The larger the “a” coefficientà the narrower the parabola.
The smaller (fractional/decimal) the “a” coefficientà the wider the parabola

{THINK: what happened when the “m” (or slope) coefficient got big? The slope got steeper. So, similarly, both sides of the U get steeper at the same time!!}

{NOW THINK: what happened when the “m” (or slope) coefficient got tiny ( or fractional)? The slope became what we called a bunny slope. So, similarly, both sides of the U get to be bunny slopes at the same time!!}

The x value of the vertex as well as the axis of symmetry are the SAME:

a and be are from the standard form of the quadratic.
Notice when b is missing ( b = 0), the vertex is always on the y-axis!! and therefore the axis of symmetry ( AoS) is also the y-axis.

 The c value is the y-intercept ( when b=0) because when x = 0 you are on the y axis.
f(x) = ax² + bx+ c
f(0) = a(0)² + b(0) + c
f(0) = c = the y value  … and in this case the y-intercept!

The domain of a quadratic (parabola) that is a function (opens up or down) is ALL REAL NUMBERS
The range depends on which way it opens:
If it opens up, the range will be   

      


If it opens down, the range will be    

You can graph quadratics exactly the same way you graphed lines—plug in your choice of an x value and use the equation to find your y value

 Because it is a U shape, you should graph at least 5 points as follows:

FIRST:  make sure the equation is in standard form!
y must be isolated on one side and then you can read the a and b coefficients

 y = ax² + bx + c

Point 1) the vertex: the minimum value of the smile or the maximum value of the frown
the x value of the VERTEX is  -b/2a
Plug that into the equation and find the y value of the vertex
Next draw the AXIS OF SYMMETRY (AoS)  

Notice- this is a linear equation! It is a line through the vertex that is parallel to the y-axis!

Point 2) Pick an x value immediately to the right or the left of the AXIS OF SYMMETRY (AoS)   and find its “y” by plugging into the equation.

Point 3)  Pick another x value one step farther on that same side of the AXIS OF SYMMETRY (AoS)  as point 2—and find its “y” by plugging into the equation.

Points 4 and 5) Graph the mirror image of both Point 2 and Point 3 on the other side of the AXIS OF SYMMETRY (AoS) by counting from the of the AXIS OF SYMMETRY (AoS)

You can choose to do two more points—if you want—but your points are joined in a SMOOTH U SHAPE ( not a V shape) and extend the lines with arrows on each end!

Example:   f(x) = -3x²  or just             y = -3x²

Notice right away—the “a” coefficient is negative so this is a frown face… a sad face! You know which way the graph will open! (Downward)

The x value of the vertex is –b/2a  a = -3 but b = 0 ( it is missing!)
so the x value of the vertex is –b/2a = -0/2(-3) = 0    Plug that back into the function f(0) = -3(0²) = 0

(Or just begin to realize that if b is missing the vertex is always on the y axis and the  AXIS OF SYMMETRY (AoS) will always be x = 0. )

In this case because there is NOT a “c” in the function, the vertex is ( 0,0) the origin.
The domain is all real numbers

The range is y ≤  0

To graph this function:

1) Graph the vertex ( 0,0)

2) Draw the AXIS OF SYMMETRY (AoS) a dotted line at x = 0  ( actually this is the y-axis so use a different colored pencil to indicate the AoS)

 3) Pick an x value immediately to the right of the AoS, x = 1 works so plug that into the equation to find its y value
y = -3(1²) = -3 PLOT ( 1, -3)

 4) Pick the next x value x = 2 and  plug that into the equation to find its y value
y = -3(2²) = -12 PLOT ( 2, -12)

 5) Count the same 1 step to the LEFT of the AoS as at the same y value of ( 1, -3) That would be ( -1, -3) and plot that point!

 6) Count the same 2 steps to the LEFT of the AoS at the same y value of ( 2 -12) That would be ( -2, -12) and plot that point

 7) Connect with a smooth U shape and extend with arrows!