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Transformations of Quadratic Functions

Quadratic functions are no exception when it comes to the types of transformations they can undergo. Applying a translation, reflection, stretch, or shrink to a quadratic function always leads to another quadratic function.
Rule

Transformations of Quadratic Functions

Rule

Translation

By adding some number to every function value, its graph is translated vertically. To instead translate it horizontally, a number is subtracted from the input of the function rule. The number is subtracted and not added, so that a positive translates the graph to the right.

Translate graph to the right

Translate graph upward

Notice that if the quadratic function is translated both vertically and horizontally, the resulting function is This is exactly the vertex form of a quadratic function. The vertex of is located at When the graph is then translated units horizontally and units vertically, the vertex moves to

Rule

Reflection

A function is reflected in the -axis by changing the sign of all function values: Graphically, all points on the graph move to the opposite side of the -axis, while maintaining their distance to the -axis.

Reflect graph in -axis

A graph is instead reflected in the -axis, moving all points on the graph to the opposite side of the -axis, by changing the sign of the input of the function. Note that the -intercept is preserved.

Reflect graph in -axis

Rule

Stretch and Shrink

A function graph is vertically stretched or shrunk by multiplying the function rule by some constant : All vertical distances from the graph to the -axis are changed by the factor Thus, preserving any -intercepts.

Stretch graph vertically

By instead multiplying the input of a function rule by some constant its graph will be horizontally stretched or shrunk by the factor Since the -value of -intercepts is they are not affected by this transformation.

Stretch graph horizontally

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Exercise


Reflecting in the -axis, and then adding to the input of the resulting function, gives Determine which graph, I or II, corresponds to

Show Solution
Solution

By applying the transformations to the graph of it will overlap with either I or II. This way, we can determine which graph is that of The reflection in the -axis moves every point on the graph to the other side of the -axis.

Adding to the input of the resulting function gives us : We can recognize this as a translation of by unit to the left. Graphing this translation, we see that the graph of coincides with graph II.

Thus, II is the graph of

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Exercise

The rules of and are given such that is a transformation of Describe the transformation(s) underwent to become Then, write the rule of in vertex form and plot its graph.

Show Solution
Solution

Notice that subtracting from the input of and adding to the output gives the function We can recognize the subtraction from the input as a translation to the right by unit. Adding to the output corresponds to a translation upward by units. Thus, has been translated unit to the right and units upward. The function is defined by so we have to find to be able to state the rule of This is done by replacing every in the rule of with Substituting this into the rule of gives us Notice that this function is already written in vertex form. To graph the function, we'll start by plotting the vertex, and the axis of symmetry.

Substituting into the rule gives us the -intercept.

We can now plot the point and reflect it in the axis of symmetry at

Now, connecting the points with a parabola gives the desired graph.


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