Transforming Quadratic Functions

Instructor: David Karsner

David holds a Master of Arts in Education

The basic form of a quadratic function is f(x)=x^2 . The graph is a parabola with a vertex at (0,0) opening up. All other quadratic functions are transformations of this parent function.

Drawing Rainbows with Quadratic Functions.

Suppose you have been hired to write the computer program for a video game. In this game you have to create a rainbow. You can use the graph of a quadratic function to draw it. They are shaped like rainbows. The game requires that the rainbow follows the pot of gold around. The rainbow will need to be moved. You can use transformations on quadratic functions to get that rainbow to move. This lesson will define quadratic functions and their transformations. It will also give the basic steps to transform any quadratic function.


Quadratic Function

A quadratic function is a polynomial function that has a degree of two. The largest exponent is a two. The basic form of this function would be f(x)=x2. Another example of a quadratic function would be f(x)=2x2-6x-12.

Parent Function

Parent functions are the function out of a category of functions that all of the other functions are derived from. They are the most basic form of function in that category. In the category of quadratics f(x)=x2 is the parent function.

Vertex Form

The vertex form of a quadratic function will look like f(x)=a(x-h)2+k. In vertex form the point (h,k) is the vertex of the parabola. The value of the 'a' will denote whether the graph will open up or open down (concavity).


A parabola is the shape of all quadratic functions. It resembles the letter u.

Different Quadratic Functions


The vertex of a quadratic is the point on the graph in which the graph switches from increasing to decreasing or decreasing to increasing. It is indicated with a point (h,k). The vertex of a quadratic function will either be the maximum or minimum value of that function.

Transformation of a Function

A transformation on a function is to change the parameters on the parent function. In the quadratic category the parameters are the a,h, and k. This will create another function of the same kind as the parent function. This could possibly cause the function to move around on the xy coordinate grid. It could move left or right, up or down. It could reflect across the x or y axis or be stretched horizontally or vertically.

Transforming Quadratic Functions

The parent function of the quadratic is f(x)=x2. In vertex form it would be f(x)=1(x-0)2+0 where a=1, h=0, and k=0. The graph has its vertex at (0,0) and opens up. By changing the value of a,h, and k called parameters, you can create a transformation of the function.

What the 'a' does.

The 'a' affects the height of the parabola. The 'a' will multiply the y values by a factor of 'a'. In the parent function the a=1 and is usually not written. It contains the points (-1,1), (0,0), and (1,1). F(x)=4x2 is a transformation of the parent quadratic function. The value of 'a' is 4. The points of f(x) are (-1,4), (0,0),and (1,4). Notice how all the y values of each point were multiplied by 4. The graph will be stretched by a factor of 4, making it look skinnier.

What the 'h' does.

'H' is the horizontal movement away from the origin. In the parent function h=0. In vertex form the 'h' is located inside of the parentheses. It has the opposite effect on the value of x. For example, f(x)= (x-5)2 has a h value of -5, but it moves the parabola 5 places to the right. The points (-1,1), (0,0), and (1,1) from the parent function move to (4,1), (5,0), and (6,1).

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