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# Describing Transformations of Quadratic Functions

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### Direct messages

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.

### Translation

By adding some number to every function value,
g(x)=f(x)+k,
its graph is translated vertically. To instead translate it horizontally, a number is subtracted from the input of the function rule.
g(x)=f(xh)
The number h is subtracted and not added, so that a positive h translates the graph to the right.
Translate graph to the right

Translate graph upward

Notice that if the quadratic function f(x)=ax2 is translated both vertically and horizontally, the resulting function is
g(x)=a(xh)2+k.
This is exactly the vertex form of a quadratic function. The vertex of f(x)=ax2 is located at (0,0). When the graph is then translated h units horizontally and k units vertically, the vertex moves to (h,k).

### Reflection

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

A graph is instead reflected in the y-axis, moving all points on the graph to the opposite side of the y-axis, by changing the sign of the input of the function.
g(x)=f(-x)
Note that the y-intercept is preserved.
Reflect graph in y-axis

### Stretch and Shrink

A function graph is vertically stretched or shrunk by multiplying the function rule by some constant a>0:
g(x)=af(x).
All vertical distances from the graph to the x-axis are changed by the factor a. Thus, preserving any x-intercepts.
Stretch graph vertically

By instead multiplying the input of a function rule by some constant a>0,
g(x)=f(ax),
its graph will be horizontally stretched or shrunk by the factor Since the x-value of y-intercepts is 0, they are not affected by this transformation.
Stretch graph horizontally

## Determine the transformed quadratic function

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Reflecting f in the x-axis, and then adding 1 to the input of the resulting function, gives g. Determine which graph, I or II, corresponds to g.

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

Adding 1 to the input of the resulting function gives us g:
g(x)=-f(x+1).
We can recognize this as a translation of by 1 unit to the left. Graphing this translation, we see that the graph of g coincides with graph II.

Thus, II is the graph of g.

## Describe the transformation of the quadratic function

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The rules of f and g are given such that g is a transformation of f.
Describe the transformation(s) f underwent to become g. Then, write the rule of g in vertex form and plot its graph.
Show Solution expand_more
Notice that subtracting 1 from the input of f and adding 2 to the output gives the function g. We can recognize the subtraction from the input as a translation to the right by 1 unit. Adding 2 to the output corresponds to a translation upward by 2 units. Thus, f has been translated 1 unit to the right and 2 units upward. The function g is defined by
g(x)=f(x1)+2,
so we have to find f(x1) to be able to state the rule of g. This is done by replacing every x in the rule of f with x1.
Substituting this into the rule of g gives us
g(x)=2(x1)2+2.
Notice that this function is already written in vertex form. To graph the function, we'll start by plotting the vertex, (1,2), and the axis of symmetry.

Substituting x=0 into the rule g(x) gives us the y-intercept.

g(x)=2(x1)2+2
g(0)=2(01)2+2
g(0)=2(-1)2+2
g(0)=21+2
g(0)=2+2
g(0)=4

We can now plot the point (0,4), and reflect it in the axis of symmetry at (2,4).

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