Describing Transformations of Linear Functions

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Concept

Function Families

Functions of similar characteristics can be grouped together in what is known as a function family. One such family is the linear functions, which consists of every function with a constant rate of change. By applying one or several transformations to a parent function, it is possible to construct any function in its function family.
Rule

Types of Transformations

The table below shows different types of transformations a function can undergo. Notice the corresponding function notation, where f(x)f(x) is the original function and g(x)g(x) is the transformed function.

Transformation Function Notation
Vertical Translation g(x)=f(x)+kg(x)=f(x){\color{#0000FF}{\vphantom{ }+k}}
Horizontal Translation g(x)=f(xh)g(x)=f(x{\color{#0000FF}{\vphantom{ }-h}})
Reflection in the xx-axis g(x)=-f(x)g(x)={\color{#0000FF}{\text{-}}}f(x)
Reflection in the yy-axis g(x)=f(-x)g(x)=f({\color{#0000FF}{\text{-} }}x)
Vertical Stretch and Shrink g(x)=af(x)g(x)={\color{#0000FF}{a \cdot\vphantom{ }}}f(x)
Horizontal Stretch and Shrink g(x)=f(ax)g(x)=f({\color{#0000FF}{a \cdot\vphantom{ }}}x)
Rule

Translation

A translation is a transformation that moves a graph in some direction, without any rotation, shrinking, stretching, etc. A function's graph is vertically translated by adding some number to the function rule. Generally, this is written as g(x)=f(x)+k, g(x) = f(x) + k, where kk is some number defining the vertical translation, and g(x)g(x) is the translated function. A positive kk increases the value of the function for every x,x, moving the graph upward. Similarly, a negative kk moves the graph downward.

Translate graph upward

A function's graph is horizontally translated by instead subtracting some number from the input of the function rule. In function notation, this is written as g(x)=f(xh), g(x) = f(x - h), where hh is some number giving the horizontal translation. A positive hh reduces the input value, xh,x - h, making it as though every xx-value is smaller than it really is. Thus, greater xx-values are needed to get the same result, leading to a translation to the right. Similarly, a negative hh gives a translation to the left.

Translate graph to the right

Rule

Reflection

A reflection is a transformation that flips a graph over some line. This line is called the line of reflection, and is commonly either the xx- or yy-axis. A reflection in the xx-axis is achieved by changing the sign of the yy-coordinate of every point on the graph. Algebraically, this is expressed as g(x)=-f(x). g(x) = \text{-} f(x). The yy-coordinate of all xx-intercepts is 0.0. Thus, changing the sign of the function value at xx-intercepts makes no difference — any xx-intercepts are preserved when a graph is reflected in the xx-axis.

Reflect graph in xx-axis

A reflection in the yy-axis is instead achieved by changing the sign of every input value. g(x)=f(-x) g(x) = f(\text{-} x) When x=0,x = 0, which is at any yy-intercept, this reflection doesn't affect the input value. Thus, yy-intercepts are preserved by reflections in the yy-axis.

Reflect graph in yy-axis

Exercise

The rules of ff and hh are given such that hh is a transformation of f.f. f(x)=2x+4h(x)=f(x3) f(x)=2x+4 \qquad h(x)=f(x-3) Describe the transformation(s) ff underwent to become h.h. Then, state the slope and the yy-intercept of h.h.

Solution

To begin, we'll analyze the given function rules. Subtracting the input of ff by 33 gives the function h.h. We can recognize this as a horizontal translation, but in which direction? When the input of a function is reduced by some number, 33 in this case, greater input values are needed to achieve the same output values. Thus, this is a translation to the right, by 33 units.

From the graph above, it's possible to find the slope and yy-intercept. However, this is not a reliable method, since the graph is only supposed to be a sketch. Instead, we can find them algebraically. hh is defined as h(x)=f(x3), h(x) = f(x - 3), which means that the rule of hh can be found by reducing the input in the rule of ff by 3.3. In practice, this is done by replacing every xx in the rule of ff with x3.{\color{#0000FF}{x - 3}}. f(x)=2x+4f(x3)=2(x3)+4 f(x) = 2x + 4 \quad \Leftrightarrow \quad f({\color{#0000FF}{x - 3}}) = 2({\color{#0000FF}{x - 3}}) + 4 Substituting this into the rule for hh gives h(x)=2(x3)+4. h(x) = 2(x - 3) + 4. Simplifying the right-hand side will lead to hh being expressed in slope-intercept form, allowing us to identify its slope and yy-intercept.

h(x)=2(x3)+4h(x) = 2(x - 3) + 4
h(x)=2x6+4h(x) = 2x - 6 + 4
h(x)=2x2h(x) = 2x - 2

We can now identify the slope, 2,2, and the yy-intercept, (0,-2).(0, \text{-} 2). Notice that the slope wasn't affected by the translation. Since translations do nothing but move graphs, all translations preserve the slope of a linear function.

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Rule

Vertical Stretch and Shrink

A function graph can be vertically stretched or shrunk by multiplying the function rule by some factor a>0.a > 0. Algebraically, g(x)=af(x). g(x) = a \cdot f(x). The vertical distance between the graph and the xx-axis will then change by the factor aa at every point on the graph. If a>1,a > 1, this will lead to the graph being stretched vertically. Similarly, a<1a < 1 leads to the graph being shrunk vertically. Note that xx-intercepts have the function value 0.0. Thus, they are not affected by this transformation.

Stretch graph vertically

Rule

Horizontal Stretch and Shrink

By multiplying the input of a function by a factor a>0,a > 0, its graph can be horizontally stretched or shrunk. g(x)=f(ax) g(x) = f(a \cdot x) If a>1,a > 1, every input value will be changed as though it was further away from x=0x = 0 than it really is. This leads to the graph being shrunk horizontally — every part of the graph is moved closer to the yy-axis. In the same fashion, a<1a < 1 leads to a horizontal stretch. The horizontal distance between the graph and the yy-axis is changed by a factor of 1a.\frac 1 a.

Stretch graph horizontally

yy-intercepts have the xx-value 0,0, which is why they are not affected by this transformation.
Exercise

The graph below shows two linear functions ff and g.g. Write the rule for gg as a transformation of f.f.

Solution

From the graph we can see that f(x)f(x) and g(x)g(x) have slopes of different signs. This is an indicator that gg is some reflection of f,f, but in what line was it reflected? The xx-intercept, (2,0),(2, 0), is preserved by this transformation, which is agreeable with a reflection in the xx-axis: g(x)=-f(x). g(x) = \text{-} f(x). To confirm whether this is the case, we can choose a few xx-values to find their corresponding yy-values. If the functions have values of different signs at every x,x, then gg must be a reflection of ff in the xx-axis. Choosing x=3,x = 3, we find the function values f(3)=1f(3) = 1 and g(3)=-1,g(3) = \text{-} 1, which have different signs.

Looking at x=-1,x = \text{-} 1, we find that f(-1)=-3f(\text{-} 1) = \text{-} 3 and g(-1)=3,g(\text{-} 1) = 3, also different signs. Choosing any other xx-value would yield the same result. Thus, we know that gg is a reflection of ff in the xx-axis, and can be expressed as g(x)=-f(x). g(x) = \text{-} f(x).

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Exercises

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