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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, g(x)=f(x)+k, 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) g(x) = f(x - h) The number hh is subtracted and not added, so that a positive hh translates the graph to the right.

Translate graph to the right

Translate graph upward

Notice that if the quadratic function f(x)=ax2f(x) = ax^2 is translated both vertically and horizontally, the resulting function is g(x)=a(xh)2+k. g(x) = a(x - h)^2 + k. This is exactly the vertex form of a quadratic function. The vertex of f(x)=ax2f(x) = ax^2 is located at (0,0).(0,0). When the graph is then translated hh units horizontally and kk units vertically, the vertex moves to (h,k).(h, k).

Rule

Reflection

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

Reflect graph in xx-axis

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

Reflect graph in yy-axis

Rule

Stretch and Shrink

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

Stretch graph vertically

By instead multiplying the input of a function rule by some constant a>0,a > 0, g(x)=f(ax), g(x) = f(a \cdot x), its graph will be horizontally stretched or shrunk by the factor 1a.\frac 1 a. Since the xx-value of yy-intercepts is 0,0, they are not affected by this transformation.

Stretch graph horizontally

Exercise


Reflecting ff in the xx-axis, and then adding 11 to the input of the resulting function, gives g.g. Determine which graph, I or II, corresponds to g.g.

Solution

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

Adding 11 to the input of the resulting function gives us gg: g(x)=-f(x+1). g(x) = \text{-} f(x + 1). We can recognize this as a translation of -f(x)\,\text{-} \hspace{-1pt} f(x) by 11 unit to the left. Graphing this translation, we see that the graph of gg coincides with graph II.

Thus, II is the graph of g.g.

info Show solution Show solution
Exercise

The rules of ff and gg are given such that gg is a transformation of f.f. f(x)=2x2g(x)=f(x1)+2 f(x) = 2x^2 \qquad g(x) = f(x - 1) + 2 Describe the transformation(s) ff underwent to become g.g. Then, write the rule of gg in vertex form and plot its graph.

Solution

Notice that subtracting 11 from the input of ff and adding 22 to the output gives the function g.g. We can recognize the subtraction from the input as a translation to the right by 11 unit. Adding 22 to the output corresponds to a translation upward by 22 units. Thus, ff has been translated 11 unit to the right and 22 units upward. The function gg is defined by g(x)=f(x1)+2, g(x) = f(x - 1) + 2, so we have to find f(x1)f(x - 1) to be able to state the rule of g.g. This is done by replacing every xx in the rule of ff with x1.x - 1. f(x)=2x2f(x1)=2(x1)2 f(x) = 2x^2 \quad \Leftrightarrow \quad f(x - 1) = 2(x - 1)^2 Substituting this into the rule of gg gives us g(x)=2(x1)2+2. 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),(1,2), and the axis of symmetry.

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

g(x)=2(x1)2+2g(x) = 2(x - 1)^2 + 2
g(0)=2(01)2+2g({\color{#0000FF}{0}}) = 2({\color{#0000FF}{0}} - 1)^2 + 2
g(0)=2(-1)2+2g(0) = 2(\text{-} 1)^2 + 2
g(0)=21+2g(0) = 2 \cdot 1 + 2
g(0)=2+2g(0) = 2 + 2
g(0)=4g(0) = 4

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

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


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