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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.

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(x−h)

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

g(x)=a(x−h)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).
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

g(x)=f(-x)

Note that the y-intercept is preserved.
Reflect graph in y-axis

g(x)=a⋅f(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

g(x)=f(a⋅x),

its graph will be horizontally stretched or shrunk by the factor $a1 .$ Since the x-value of y-intercepts is 0, they are not affected by this transformation.
Stretch graph horizontally

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.

Show Solution

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 $-f(x)$ 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.

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

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
Substituting this into the rule of g gives us

g(x)=f(x−1)+2,

so we have to find f(x−1) to be able to state the rule of g. This is done by replacing every x in the rule of f with x−1.
g(x)=2(x−1)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(x−1)2+2

Substitute

x=0

g(0)=2(0−1)2+2

SubTerm

Subtract term

g(0)=2(-1)2+2

CalcPow

Calculate power

g(0)=2⋅1+2

Multiply

Multiply

g(0)=2+2

AddTerms

Add terms

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.

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