AM
Animated Mathematics

Quadratic Transformations: Part II

We have studied the transformations vertical shift, horizontal stretch, and reflection in an earlier section, and horizontal shift was described in the last section. You are recommended to review these sections before continuing. In this section we transform the simplest quadratic equation y=x² into y=a(x-h)²+k. This is a very important form of quadratic equations:

Viewing some examples shall help you understand the above equation.

Multiple Transformations

Starting with y=x², we apply vertical stretch, horizontal shift, and then vertical shift.

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x y

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interactive example

The solutions are not necessarily integer values. The form of the quadratic equation allows for one to easily find the solutions, the algebra is shown below:

Exercise: Solutions of a Transformed Quadratic Equation

The quadratic equation y=x² has been transformed to the equation below. Enter the axis of symmetry, vertex coordinate, and solutions to two decimal places.

Symmetry axis
Vertex
Solution

Solution


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y axis
x axis


You can check your answers and working out below:

Exercise: Determine Quadratic Equation From The Graph

The quadratic equation y=x² has been transformed into the equation:

The graph shows the transformed quadratic equation. Determine the constants h,k, and a. Hint: move the point on the line and use the coordinate.

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y axis
x axis

Move Point
Enter h
Enter k
Enter a

The correct answers and working out is shown below:

Summary

We have an extensive set of transformations that can be applied to a quadratic equation. The vertex and axis symmetry can be placed by vertical and horizontal transformations. The transformation stretch and reflection can be used to place the solutions.