Skip to main content

Section 2.1 Translations

Consider the quadratic function \(f(x) = x^2\text{,}\) along with the graphs \(g(x) = (x - 1)^2\) and \(h(x) = x^2 + 2\text{.}\)

Notice that the graph of \(g\) is the graph of \(f\text{,}\) shifted 1 unit to the right. Similarly, the graph of \(h\) is hte graph of \(f\) shifted 2 units up.

Also notice that we could write \(g\) and \(h\) in terms of \(f\text{,}\) as,

\begin{gather*} g(x) = (x - 1)^2 = f(x - 1)\\ h(x) = x^2 + 2 = f(x) + 2 \end{gather*}

For \(h(x)\text{,}\) the addition of 2 means that the output values \(h(x)\) of the function are 2 more than the output values of \(f(x)\text{,}\) for the same input \(x\text{.}\) This corresponds to a vertical shift of 2 units up.

For \(g(x) = f(x - 1)\text{,}\) there is a subtraction of 1 inside the function argument. This means that to obtain the same output value \(f(x)\text{,}\) we need to increase the value of \(x\) by 1. For example,

\begin{equation*} \begin{array}{l} g(0) = f(-1) \\ g(1) = f(0) \\ g(2) = f(1) \\ \vdots \end{array} \end{equation*}

That is, to determine the output value of \(g\) at a particular point, determine the output value of \(f\) 1 unit to the left. This leads to \(g\) being a horizontal shift of \(f\) 1 unit to the right.

Subsection 2.1.1

Horizontal and Vertical Translations (Shifts)

In general, shifts that occur inside the function argument correspond to horizontal shifts, whereas shifts that occur “outside” the function correspond to vertical shifts.

Roughly, for a vertical shift, a positive number means the graph is shifted in the “positive” \(y\)-direction, that is, upwards. Similarly, a negative number corresponds to a shift in the “negative” \(y\)-direction, that is, downwards.

For horizontal shifts, the signs are opposite. A positive number inside the argument corresponds to a shift in the “negative” \(x\)-direction, that is, to the left. Similarly, a negative number corresponds to a shift in the “positive” \(x\)-direction, that is, to the right.

Roughly, subtracting \(h\) from \(x\) shifts the graph right because for the function \(g(x) = f(x - h)\text{,}\) to obtain \(f(0)\text{,}\) we need to evaluate \(g(h) = f(h - h) = f(0)\text{.}\) That is, the \(y\)-coordinate of the original graph at \(x = 0\) (\(f(0)\)) is now found at \(x = h\) on the new graph (\(h\) units to the right). That is, because of the subtraction of \(h\text{,}\) the inputs \(x\) must be increased by \(h\) to compensate, which leads to a rightward shift. In general, to obtain \(f(a)\text{,}\) we need to input \(a + h\text{.}\)

In summary,

\begin{align*} y = f(x) \amp \longrightarrow y = f(x - h)\\ (x,y) \amp \longrightarrow (x + h, y) \end{align*}
\begin{align*} y = f(x) \amp \longrightarrow y = f(x) + k\\ (x,y) \amp \longrightarrow (x, y + k) \end{align*}

Also, in general, with transformations, the most basic type of function in a class of functions, such as \(f(x) = x^2\text{,}\) or \(f(x) = \abs{x}\text{,}\) are called parent functions, intuitively because all other functions in the family of functions can be derived from this parent function.

Subsection 2.1.2 Quadratic Functions

You may recall from the study of quadratic functions, that the function \(f(x) = (x - h)^2 + k\) has vertex at the point \((h,k)\text{.}\) In fact, this is just an example of translations. The parent function is \(f(x) = x^2\text{,}\) and the transformation is a horizontal and vertical shift. Consider this Desmos applet, to explore the effects of the translation on the graph.

Subsection 2.1.3 Linear Functions

Translations can also be applied to perhaps the most basic function: the linear function. Consider the function \(f(x) = 2x\text{,}\) which represents the line \(y = 2x\text{,}\) a straight line with slope 2 through the origin. The graph of \(y = 2(x - 3)\) is the graph obtained by shifting the graph of \(y = 2x\) to the right 3 units. Similarly, the graph of \(y = 2x + 4\) is obtained by shifting \(y = 2x\) up 4 units. Consider this Desmos applet, to explore shifts for linear functions.

Notice that manipulating the equation can lead to equivalent transformations. For example, \(y = 2(x - 3)\) is equivalent to \(y = 2x - 6\text{.}\) The former is a horizontal shift 3 units right, while the latter is a vertical shift 6 units down. For this particular function, these two translations are equivalent.

Subsection 2.1.4 Equivalent Transformations

In general, for some kinds of functions, there will be more than one way to represent the same transformation.

Subsection 2.1.5 Polynomial Functions

Polynomial functions can be translated in a similar way. For example, for \(f(x) = x^3 - 8x\text{,}\) consider the function \(g(x) = f(x + 2) = (x + 2)^3 - 8(x + 2)\text{.}\) The graph of this function is the graph of \(f\text{,}\) shifted 2 units left. Notice that the equation of this new function is somewhat complicated, \(g(x) = (x + 2)^3 - 8(x + 2)\text{.}\) However, we know it is the translated version of the original graph. Consider this Desmos applet, to explore shifts for any polynomial function.

Subsection 2.1.6 Systematic Method

\begin{equation*} \begin{array}{cc} y = f(x) \amp y = f(x - h) + k \\ (x,y) \amp (x + h, y + k) \\ \hline \vdots \amp \vdots \end{array} \end{equation*}

Choose a few points on the parent function to use to transform into points on the transformed function.

Subsection 2.1.7 Domain and Range after Translations

These translations shift the graph of a function, so they also intuitively shfit their domain and range.

Subsection 2.1.8 Rational Functions and Asymptotes

Similarly, rational functions can be translated.

Horizontal asymptotes get shifted vertically. Vertical asymptotes get shifted horizontally.