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Sometimes, you may be asked to state the transformations of a parabola. This means that you must describe in words how the parabola has changed from the parent function after the vertical stretch/compression, horizontal shift, vertical shift and a reflection in the x-axis.

 

The transformations of a parabola are written in the exact order that a parabola is transformed on the graph. The order of stating the transformations of a parabola is:

 

  1. State the vertical stretch/compression by a factor of "the a-value of the vertex form"

  2. Reflection in the x-axis

  3. Horizontal translation "the h-value of the vertex form" units left/right

  4. Vertical translation "the k-value of the vertex form" units up/down

Quadratic Transformations

State the Transformation

y = 1/2(x+5)^2 + 7

EXAMPLE:

First, state the vertical stretch/compression. In this case, it is vertical compression, therefore, the transformation would be written as:

                  1. Vertical compression by a factor of 1/2

 

The next step would be state if there is a reflection in the x-axis. Since the a-value of this equation is positive, there is no reflection.

 

The third step is to state the horizontal shift. If the h value is positive in the brackets, there will a horizontal shift to the left, but if the h-value is negative in the brackets, the shift will be to the right. In this case, the horizontal shift will be written as:

                 2. Horizontal translation 5 units left

 

The last translation is to describe the vertical translation. If the k-value is positive, the parabola will move upwards, and if it is negative, the parabola will move downwards. The vertical translation of this parabola will be written as:

                3. Vertical translation 7 units up

FINAL ANSWER:

  1. Vertical Compression by a factor of 1/2

  2. Horizontal translation 5 units left

  3. Vertical translation 7 units up

Writing the Equation

Sometimes, you may be asked to work in reverse and develop an equation by reading the stated transformations. To write an equation by reading the translations, first look for any reflections in the x-axis and if there are any, then the a-value will be negative. Then put the rest of the units into the vertex form equation to get to the final answer. 

EXAMPLE:

Write the equation for the graph of f(x) = x^2 that has the following transformations applied:

  1. Vertical stretch by a factor of 10

  2. Reflection in the x-axis

  3. Horizontal translation 6 units right

  4. Vertical translation 9 units up

First write the vertex form equation:

                                                              y = a(x-h)^2 + k

 

Now, sub in all the values into the equation:

 

  • Sub in 10 for 'a' as described in the question and 10 must be negative as there is a reflection in the x-axis.

                                                              y = -10(x-h)^2 + k

 

  • Next, sub in the h-value. Be mindful that if the horizontal translation is to the right, the h value will be negative inside the brackets. If the horizontal translation is to the left, the h-value will be positive inside the brackets.

                                                              y = -10(x-6)^2 + k 

 

  • Lastly, sub in the k-value. If the vertical translation is upwards, the k-value will be positive, but if the vertical translation is downwards, the k value will be negative.

                                                               y = -10(x-6)^2 + 9

FINAL ANSWER:

y= -10(x-6)^2 + 9

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