Transformations of Quadratic functions .

Learning objectives:

This lesson focuses on the transformations of quadratic functions, and how the vertex form of a quadratic helps in understanding the transformation.

We will also discuss the supporting topics, which include quadratic parent function, and effect of leading coefficient “a” on the quadratic transformation.

The following is the brief of key concepts and  of this article:

  • Quadratic parent function.
  • Leading coefficient and its effect on parabola transformation.
  • Transformations of quadratic functions in vertex form( Translations,Dilations,Reflections).
  • How to quickly find the transformations on a given Quadratic function.
  • Summary table for Quadratic transformations.
  • Quadratic transformation rules.
  • Practice problems
  • Free math help forum

 

1. Quadratic parent function:

Parent function is the basic function or the building block of a function.

The standard form of a quadratic function is given by   y= ax^{2}+bx+c.

The parent function of a quadratic function is given by  y=x^{2}.

The vertex of quadratic parent function is at the origin (0,0) and the axis of symmetry is the  “y” axis.

The axis of symmetry bisects the quadratic graph into two identical parts.

parent function y = x^2

Therefore.the axis of symmetry divides the graph into two equal parts as shown above.

 

Leading coefficient and its effect on the parabola transformation:

The leading coefficient “a” of the quadratic function conveys a lot of information about the orientation of the parabola and in the transformations of quadratic functions .

If the sign of  “a” is “POSITIVE”, the graph would either be “OPENING UP”  or to the “RIGHT SIDE”

If the sign of the leading coefficient is “NEGATIVE”, then the graph would be either “OPENING DOWN or “OPENING LEFT”.

The leading coefficient also conveys the Dilation factor (Vertical / Horizontal stretch).

Explained below in detail, the transformations that depend on the leading coefficient “a” and the vertex form of a quadratic

 

2.Transformations of quadratic functions in vertex form:

Transformations of a quadratic function is a change in position, or shape or the size of the quadratic parent function. These transformed functions look similar to the original quadratic parent function.

The vertex form of a parabola  y=a\left( x-h\right) ^{2}+k contains the vital information about the transformations that a quadratic functions undergoes.

The vertex coordinates (h,k) and the leading coefficient “a”, for any orientation of parabola, give rise to 3  possible transformations of quadratic functions .

They are Translations, Dilation and Reflection. Translations can be further classified into Horizontal and vertical translations.

Dilations can also be further classified as horizontal and vertical. But they can be used interchangeably.

i) Horizontal Translation (The graph moves right/left )

ii) Vertical Translation  (The graph moves up/down)

iii) Reflection about the x-axis (The mirror image of the graph about the x-axis)

iv) Dilation of the graph (Horizontal or vertical .Graph gets skinny or fatty).

 

TRANSLATIONS:

i) Horizontal Translation (The graph moves right/left )

Horizontal translation of a graph is just the shifting of the graph to the left or to the right side.

There will be no deformation of the parent function when horizontal translation is done.

The “x” – coordinate of the vertex contains the information about the horizontal translation.

The “h” (x”- coordinate of the vertex) gives the amount by which graph moves horizontally

For a Positive-h, the graph shifts “RIGHT” by “h” units.

For a Negative-h, the graph shifts “LEFT ” by h units.

Let us observe the parent function is  y=x^2 (The Red graph) shown below.

It evident that the transformed function y= (x-4)^2 (The blue graph) has has positive  4 and the graph shifts right by 4 units since “h” is positive 4.

On similar lines, the transformed function y =(x -(-4))^2 has  h=-4 and the  graph shifts left by 4 units.

Transformations of quadratic function.horizontal left and right shifts of parent function y =x^2
                                  Right shifting and Left shifting of the parent function y=x^2

 

ii) Vertical Translation  (The graph moves up/down)

Similar to horizontal translation, vertical translation is just the shifting of the graph UP or Down.

Vertical translation is the amount by which the graph moves up or down.

There will be no deformation of the parent function when there is a vertical translation of the graph.

The “k”  value ( the y -coordinate of the vertex )gives the amount of vertical up-shift or down-shift of the quadratic parent function.

For a positive-k, the graph shifts “UP” by “k” units.

For a negative-K, the graph shifts “DOWN ” by “k” units.

Let us consider the below parent graph y=x^2 (In RED colour) and transformed graph y =x^2 +3(In BLUE colour).

It may be noted that the transformed function y= (x)^2 +3, shifts the parent function y =x^2 up by 3 units (Since “k” is positive 3).

On similar lines, the transformed function y =(x )^2-3 shifts the parent function y =x^2  by 3 units down (Since “k” is negative 3).

Vertical transformation of quadratic function

REFLECTION:

 Reflection about the x-axis (The mirror image of the graph about the x-axis)

To check the reflection of the parent function about the “x” axis, we need to inspect the leading coefficient “a”.

The sign of the leading coefficient “a” gives the reflection of the parent function y =x^2 about the “x” axis. For “negative a”, the graph reflects about the “x” axis.

The graph y= -x^2  opens down which is the reflection of the parent function.

Reflection about the "x"-Axis
      Reflection of the parent function y=x^2

 

DILATION:

Horizontal and Vertical dilation (Graph gets skinny or fatty).

Dilation of a graph leads to a deformation in the shape of the parent graph.

The dilated graph looks similar to the parent function but will a scaling factor applied to the parent function.

The Scaling factor or Dilation factor is the factor by which parent graph is enlarged or compressed,that is, every coordinate point on the quadratic parent graph is multiplied or divided by a specific number.

There is a close relationship between vertical and horizontal dilation.

If are able to find the vertical dilation, we can find the equivalent horizontal dilation with no extra effort.

We can find the vertical dilation factor easily from the vertex form of the  Quadratic function. Let us see how to find the vertical dilation factor from the vertex form of a quadratic.

All we need is to observe the leading coefficient “a” of the vertex form of quadratic which gives the vertical dilation factor or the scaling factor by which the parabola stretches or compresses vertically.

If  0\leq \left| a\right| \leq 1 , then the graph of the quadratic or the parabolic shape “ VERTICAL COMPRESS“(fatty)

If  \left| a\right| >1, then the graph of the quadratic gets a “ VERTICAL STRETCH” (skinny) as the  “a” value increases.

Dilation, the transformations of quadratic parent function y =x^2.

                                                    The fatty and skinny graphs of the Quadratic parent function.

On close observation, the parent function y =x^2 on vertical compression results in the function  y=0.2x^2

The vertical dilation factor is given by the leading coefficient “a” which is 0.2.

The same function y = 0.2x^2 can be rewritten as y=\left( \sqrt {0.2}x\right) ^{2}

Therefore this would be the  horizontal stretch by a factor  \sqrt {0\cdot 2}.

Similar way, considering the function y=5x^2 it can be written as  y=\left( \sqrt {5}x\right) ^{2} .

The graph stretches vertically by a factor 5 and at the  same time  we can say that the graph is compressed horizontally by a factor  \sqrt {5}

In conclusion Vertical compression by factor say “a” is equivalent to Horizontal stretch  by a factor \sqrt {a}

Vertical stretch by factor say “a” is equivalent to Horizontal compression  by a factor \sqrt {a}.

 

Summary table for Quadratic transformations

Function Example Transformation Description Graph
F(x)=x^2F(x)=x^2 NIL
(Parent function)
This is the parent function with Vertex at (0,0).
Axis of symmetry is"y"-Axis.
Leading coefficient is "a"= 1
Transformations on quadratic parent function
F(x)=a(x^2)
Given "a" is positive and greater than 1
F(x)=4(x^2)
"VERTICAL STRETCH" by a factor a =4.

OR

"HORIZONTAL
COMPRESSION" by a factor (square root 4) =2
The graph stretches vertically by a factor 4 .The Parent function has y =1 corresponding to x =1 .

The transformed function has y=4 corresponding to x =1 making it stretch by factor 4.

OR

The graph compresses horizontally by a factor sqrt 4 =2.
The Parent function has y =1 corresponding to x =1.
The transformed function has y =1 corresponding to x=0.5,
compressing by a factor 2 there by.
F(x)=a(x^2)
Given "a" is positive and 0 < a < 1
F(x)=0.25(x^2)
"VERTICAL COMPRESSION" by a factor a =0.25.

OR

"HORIZONTAL
STRETCH" by a factor (square root 0.25) =0.5
The graph compresses vertically by a factor 0.25 .In the Parent function when x =1 y =1.
In transformed function when x =1 y=0.25 making it compress by factor 4.

OR

The graph stretch horizontally by a factor sqrt 4 =2.
In the Parent function when x =1 y =1.
In transformed function for y =1 we have corresponding x=0.5.
Thus compressing by a factor 2.
F(x) =-(x^2)
Given "a" is Negative
F(x) =-(x^2)
"REFLECTION" about the "x"- axisThe parent graph reflects about the "x" axis.
Mirror image  transformation
F(x) =( x-h)^2
Given
"h"is the x -coordinate of the Vertex
F(x) =( x-3)^2Horizontal Translation
"RIGHT SIDE"
The graph
"SHIFTS RIGHT" by 3 units .
F(x) =(x+h)^2
Given
"h"is the x -coordinate of the Vertex
F(x) =(x+3)^2
Horizontal Translation
"LEFT SIDE".
The graph
"SHIFTS LEFT" by 3 units.
Horizontal left shift by 3 units
F(x) =(x)^2+k
Given
"k"is the Y-coordinate of the Vertex
F(x) =(x)^2+4
Vertical Translation
"UP WARDS"
The Graph moves
"UP by 4 units
Shift up by 3 transformation
F(x) =(x)^2-k
Given
"k"is the Y coordinate of the Vertex
F(x) =(x)^2-4
Vertical Translation
"DOWN WARDS"
The Graph moves
DOWN by 4 units
Shift down transformation
F(x) =a(x-h)^2 +k
Given (h,k) is the vertex and "a" is the leading coefficient.
F(x)=2(x-3)^2 +4
Horizontal Compression OR
Vertical Stretch

Horizontal "Right Shift"

Vertical "Up shift"
The graph strectches vertically by 2 units

Moves horizontally by 3 units

Moves vertically by 4 units.
Multiple Transformations on parent  function

 

3. Quadratic transformation rules.

Strategy Step By Step for transformations of quadratic functions :-

Step 1: First of all, transform the given function into vertex form of the quadratic using the formulas.

Step 2: Look out the sign of the coefficient “a” for the orientation of the graph. For positive “a”the graph would have “U” shape and for negative “a” graph would have inverted “U” shape.

Step 3: Examine the magnitude of the coefficient “a”.If IaI > 1 then graph gets Skinny.If 0<IaI<1, then the graph gets fatty. If Leading coefficient is  1, the graph will not have any deformation.

Step 4: The sign of “h” gives the horizontal translation to the left side or right side. The magnitude of “h” gives the amount of shift.

Step 5: The Sign of “k” gives the vertical translation upwards or Downwards. The magnitude of “k” gives the amount of shift.

 

4. Practice problems

i)Find the transformations of quadratic functions y=x^{2}+2x+4

Solution:- 

Step 1: Find the vertex form of this quadratic y=a\left( x-h\right) ^{2}+k .

An easy approach is to use the formulas to find the vertex, and the vertex form of quadratic thereby.

The “x”-coordinate is given by the formula \dfrac {-b}{2a}.

On comparison with the standard form, we have the a =1, b=2 and c =4. Therefore the  “x”- Coordinate of the vertex  h= \dfrac {-2}{2*1} =-1.

Since we got the x- coordinate, y coordinate  is given by

.\begin{aligned}y=\left( -1\right) ^{2}+2\left( -1\right) +4\\ y=1-2+4\\ y=3\end{aligned}.

Therefore the  vertex form is

y=1*\left( x-h\right) ^{2}+k

=>y=\left( x-(-1)\right) ^{2}+3

=>y=\left( x+1)\right) ^{2}+3

 

Step 2: Lookout the sign of the leading coefficient “a” for the orientation of the graph

We had the leading coefficient as 1and the sign is positive. Therefore we can conclude that the  parabola would be an Up-Parabola

 

Step 3: Lookout for the magnitude of the leading coefficient.

Since the magnitude of the leading coefficient is 1, there is no deformation of the graph.

 

Step 4: Find the sign of the  “h” for the left or Right shift and magnitude of the “h” for the amount of shift.

We arrived for the value of “h” as -1. Therefore the sign is negative and magnitude is 1.Hence we can conclude that the graph shifts to the “LEFT SIDE” with a “Magnitude of 1 unit”.

 

Step 5: Find the sign of the  “k” for the “UP or Down Shift” and magnitude of the “k” for the amount of shift.

The value of “k” we got is +3. Therefore the sign is positive and magnitude is 3.Hence we can conclude that the Graph shifts to “UP ” with a magnitude of  3  Units.

 

Finally, here are the transformations of the quadratic function. There is a Horizontal translation the  “LEFT SIDEby 1 unit and VerticalTranslation “UP side” by 3 units

 

Transformations of quadratics y=(x+1)^2+3

 

 

 

ii )Find the transformations of quadratic functions  y=-3x^{2}+6x-4

Solution:- 

Step 1: Find the vertex form of the quadratic y=a\left( x-h\right) ^{2}+k .

Let us compare the given equation with the standard form of quadratic  y=ax^{2}+bx+c .

we observe that the leading coefficient a = -3, b= 6  and c=-4.

Let us use the formula h= \dfrac {-b}{2a}.

=>  h=   \dfrac {-6}{2\left( -3\right) }=\dfrac {-6}{-6}=1

Since we got the value of “h” , we can find the value  of “k” by substituting x =1 in the original function.

\begin{aligned}k=-3\left( 1\right) ^{2}+6\left( 1\right) -4\\ \Rightarrow k=-3+6-4\\ \Rightarrow k=-1\end{aligned}.

Therefore the vertex form is  y=a\left( x-h\right) ^{2}+k

=>y=-3\left( x-1\right) ^{2}-1

Step 2: Lookout the sign of the leading coefficient “a” for the orientation of the graph.

We have the leading coefficient as – 3. Since the sign of the leading coefficient is negative, we can conclude that the graph  “opens downwards“.

 

Step 3: Lookout for the magnitude of the leading coefficient.

Since the magnitude of the leading coefficient is 3, there will a vertical stretch of the  graph by a factor 3 or Horizontal compression by a factor  \sqrt {3}

 

Step 4: Find the sign of the  “h” for the left or Right shift and magnitude of the “h” for the amount of shift.

We found that the value of  “h” is +1. Since the sign is positive, we can conclude that the graph moves towards the right side.

Since the magnitude of “h” is 1, the graph shifts to the right side by 1 unit.

 

Step 5: Find the sign of the  “k” for the “UP or Down Shift” and magnitude of the “k” for the amount of shift.

We arrived at the “k” value as -1. Since the sign is negative, we can conclude that the graph “shifts down”.The magnitude of “k” is 1.Therefore the graph would shift down by 1 unit.

 

Finally,here are the steps for transformations of quadratic function given  y=-3x^{2}+6x-4 from the parent function  y=x^{2}

Quadratic transformations example 2

i) Reflection about  “x-axis“,

ii)Vertical stretch “by factor 3

iii)”Horizontal translation”  “RIGHT SIDE ” by 1 unit.

iv)”Vertical translation”  “Down side” by 1 unit.

 

 

 

 

 

 

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