4.3 Graphing Quadratic Equations and its Transformations

Ms. Lilian Albarico

Quadratic Function

The graph produced by y = x2 is a curve called a parabola.

The parabola is a symmetrical curve that has one axis of symmetry.

The axis of symmetry is a vertical line that passes through the turning point of the parabola. The turning point is called the vertex.

The equation of the axis of symmetry is x = the value of the x co-ordinate of the vertex. Every parabola that is graphed is this standard curve with transformations performed on it.

Quadratic Function

x

y

Transformations:Vertical Reflection (Reflection in the x-axis)

Vertical Stretch

Vertical Translation/Shift

Horizontal Translation/Shift

Key Terms:Mapping notation – a notation that describes how a graph and its image are related

x2 -------> (x, y)

reflection in x-axis – a transformation that vertically flips the graph of a curve in the x-axis

vertical stretch – a transformation that describes how the y-values of a curve are stretched by a scale factor 

vertical translation – a transformation that describes the number of units and the direction that a curve moves vertically from the origin

horizontal translation – a transformation that describes the number of units and the direction that a curve moves horizontally from the origin

Vertical Reflection

Vertical Reflection (V.R.) – a transformation in which a

plane figure flips over vertically and has a

horizontal axis of reflection. Also called as Reflection in

x-axis.

Vert.ical Reflection

Vertical Reflection

y=x2 (x, y)

y=-x2 (x, -y)

Vertical Reflection

Draw the following quadratic equations, then write the mapping notations and include a table with 5 values:

1)f(x) = x2 and -f(x) = x2

2) f(x) = 4x2 and -f(x) = 4x2

3) f(x) = 1/4x2 and - f(x) = 1/4x2

Vertical StretchVertical Stretch (V.S.) –

a transformation that describes how the y – value is stretched by a scale factor (the

number times y)

Vertical Stretch

Vertical Stretch

1/2x2 (x, 2y)

2x2 (x, 1/2y)

Vertical StretchDraw the following quadratic equations, then write the mapping notation and include a table with 5 data values:

1)f(x) = x2 2y = x2 1/3y= x2

2)3y = x2 4y= x2 1/4y = x2 3/4y = x2

Homework

CYU # 5 on page 175

CYU # 8, 9, 10,11 ,12 , and 13 on pages 177-178.

Vertical Translation

Vertical Translation (V.T.) - a transformation that

describes the number of units and direction that a

curve has moved vertically.

Vertical Translation

Vertical Translation

x2 + 1 (x, y-1)

x2-1 (x, y+1)

Vertical Translation

Draw the following quadratic equations, then write the mapping notation and include a table with 5 data values:

1)y = x2 y + 2 = x2 y-2 = x2

2)y-3 = x2 (y+3)= -x2 y-1/2 = x2 y+1/2 = 2x2

Horizontal Translation

Horizontal Translation (H.T.) – a transformation that describes the number of units and direction that

a curve has moved horizontally.

Horizontal Translation

Horizontal Translation

(x+1)2 (x-1, y)

(x-1)2 (x+1, y)

Horizontal Translation

Draw the following quadratic equations, then write the mapping notation and include a table with 5 data values:

1)y = x2 y = (x+3)2 y = (x-4)2

2)y = 2(x+3)2 -y= (x-2)2 2y = (x-2)2 y-1 = (x+1)2

HOMEWORK

CYU #14 AND 14 ON PAGE 179

CYU # 18, 20, 21, 22, and 23 on page 181 – 182.

SUMMARY

Learning Check:When you see a –y in the equation then the image is a _________________ in the x-axis.

“When I see a 3y in the equation, then I know that there is a vertical stretch of ___________.”

The vertical stretch factor is the _________ of the coefficient of y.

“When I see y+2 in the equation, then I know that there is vertical translation of ____ units _____”.

“When I see y-3 in the equation, then I know that there is vertical translation of ____ units _____”.

The vertical translation is the ___________of the number being added to the y.

“When I see x+3 in the equation, then I know that there is horizontal translation by ____ units to the _____”.

“When I see x-4 in the equation, then I know that there is horizontal translation by ____ units to the _____”.

The horizontal translation is the ___________ of the number being added to the x.”

Let’s combine all transformations!

1) Graph the following quadratic functions and write down their mapping notations:

-y = (x+2)2

-2y = (x-1)2

y = (x-3)2

2) Based from the mapping notation below, write the quadratic equation and the draw its graph transformation:

(x , y) (x+2, y)

(x, y) (x+6, y-1)

(x, y) (x-1/2, y – 3/2)

(x, y) (x-2, -2y)

Absolute Value Functions

Absolute Value Functions

absolute value – the absolute value of a number is the number without its sign

|n| = n, |-n| = n.

If absolute value is related to a number line, then it indicates the measure of the distance between a point and its origin without reference to direction. Absolute value functions are graphed using y = |x| after the transformations have been applied.

VERTICAL TRANSLATIONy – q = |x| (x, y+q)

For example:

y-3 = |x| y+5= |x|

HORIZONTAL TRANSLATIONy = |x-p| (x+p, y)

For example:

y = |x-3| y= |x+4|

VERTICAL STRETCH

cy = |x| (x, 1/3y)

For example:

1/2y = |x| 2y= |x|

REFLECTION in X-AXIS

-y = |x| (x, -y)

For example:

y = -1/10|x| 1/10y= |x|

Putting it All Together

1) Graph the following absolute value functions and write its mapping notation:

-1/2 (y+3) = |x+5|

y-3 = |x+3|

1/4y= |x-2|

2) Write the absolute value functions of the following mapping notation and graph its transformation:

(x , y) (x+1, y)

(x, y) (x, y-4)

(x, y) (x-3, y + 3)

(x, y) (-x, y+5)

AssignmentYou will draw a shape using different kinds of graphs – linear, quadratic, exponential, and absolute value functions. There should only be one point of origin. Make sure your graphs are clear and accurate. Be as creative as you can.

Below your graph:

A) State the domain and range of your different graphs.

B) If it is either quadratic or absolute value functions, write its mapping notations.

Deadline : December 26, 2012

Homework

FOCUS QUESTIONS # 30 and 31 on page 186.

CYU # 33, 34, 35, 36 and 37 on pages 187-189.