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SectionVertical and Horizontal Shifts

In this section, we explore how certain changes in the formula for a function affect its graph. In particular, we will compare the graph of \(y = f (x)\) with the graphs of

\begin{equation*} y = f (x) + k,~~~ \text{ and }~~~ y =f (x + h) \end{equation*}

for different values of the constants \(k\) and \(h\text{.}\) Such variations are called transformations of the graph.

SubsectionVertical Shifts

Figure230 shows the graphs of \(f (x) = x^2 + 4\text{,}\) \(g(x) = x^2 - 4\text{,}\) and the basic parabola, \(y = x^2\text{.}\) By comparing tables of values, we can see exactly how the graphs of \(f\) and \(g\) are related to the basic parabola.

parabolas

Vertical Shift
\(x\) \(-2\) \(-1\) \(~0~\) \(~~1~~\) \(~2~\)
\(y=x^2\) \(4\) \(1\) \(0\) \(1\) \(4\)
\(f(x)=x^2+4\) \(8\) \(5\) \(4\) \(5\) \(8\)
\(x\) \(-2\) \(-1\) \(0\) \(1\) \(2\)
\(y=x^2\) \(4\) \(1\) \(0\) \(1\) \(4\)
\(g(x)=x^2-4\) \(0\) \(-3\) \(-4\) \(-3\) \(0\)

Figure230

Each \(y\)-value in the table for \(f (x)\) is four units greater than the corresponding \(y\)-value for the basic parabola \(y=x^2 \text{.}\) Consequently, each point on the graph of \(f (x)\) is four units higher than the corresponding point on the basic parabola, as shown by the arrows. Similarly, each point on the graph of \(g(x)\) is four units lower than the corresponding point on the basic parabola.

The graphs of \(y = f (x)\) and \(y = g(x)\) are said to be translations (or shifts) of the graph of \(y = x^2\text{.}\) They are shifted to a different location in the plane but retain the same size and shape as the original graph. In general, we have the following principles.

Vertical Shifts

Let \(k\gt 0\text{.}\) Compared with the graph of \(y = f (x)\text{,}\)

  1. the graph of \(y=f(x)+k\) is shifted upward \(k\) units, and
  2. the graph of \(y=f(x)-k\) is shifted downward \(k\) units.
Example231

Graph the following functions.

  1. \(g(x) = \abs{x} + 3\) and \(y=\abs{x}\) and discuss how they are related.
  2. \(h(x) = \dfrac{1}{x}- 2\) and \(y=\frac{1}{x}\) and discuss how they are related.
Solution
  1. The table shows that the \(y\)-values for \(g(x)\) are each three units greater than the corresponding \(y\)-values for the absolute value function. The graph of \(g(x) = \abs{x} + 3\) is a translation of the basic graph of \(y = \abs{x}\text{,}\) shifted upward three units, as shown in Figure232.

    translate absolute value
    \(x\) \(-2\) \(-1\) \(0\) \(1\) \(2\)
    \(y=\abs{x}\) \(2\) \(1\) \(0\) \(1\) \(2\)
    \(g(x)=\abs{x}+3\) \(5\) \(4\) \(3\) \(4\) \(5\)
    Figure232
  2. The table shows that the \(y\)-values for \(h(x)\) are each two units smaller than the corresponding \(y\)-values for \(y = \dfrac{1}{x}\text{.}\) The graph of \(h(x) = \dfrac{1}{x} - 2\) is a translation of the basic graph of \(y = \dfrac{1}{x}\text{,}\) shifted downward two units, as shown in Figure233.

    translate reciprocal
    \(x\) \(-2\) \(-1\) \(\dfrac{1}{2}\) \(1\) \(2\)
    \(y=\dfrac{1}{x}\) \(\dfrac{-1}{2}\) \(-1\) \(2\) \(1\) \(\dfrac{1}{2}\)
    \(h(x)=\dfrac{1}{x}-2\) \(\dfrac{-5}{2}\) \(-3\) \(0\) \(-1\) \(\dfrac{-3}{2}\)
    Figure233
Example234

The function \(E = f (h)\) graphed in Figure235 gives the amount of electrical power, in megawatts, drawn by a community from its local power plant as a function of time during a 24-hour period in 2002. Sketch a graph of \(y = f (h) + 300\) and interpret its meaning.

electrical power
Figure235
Solution

The graph of \(y = f (h) + 300\) is a vertical translation of the graph of \(f\text{,}\)

as shown in Figure236. At each hour of the day, or for each value of \(h\text{,}\) the \(y\)-coordinate is 300 greater than on the graph of \(f\text{.}\) So at each hour, the community is drawing 300 megawatts more power than in 2002.

electrical power
Figure236

An evaporative cooler, or swamp cooler, is an energy-efficient type of air conditioner used in dry climates. A typical swamp cooler can reduce the temperature inside a house by 15 degrees. Figure238a shows the graph of \(T = f (t)\text{,}\) the temperature inside Kate's house \(t\) hours after she turns on the swamp cooler. Write a formula in terms of \(f\) for the function \(g\) shown in Figure238b and, give a possible explanation of its meaning.

swamp cooler graphs
Figure238

SubsectionHorizontal Shift

Now consider the graphs of \(f (x) = (x + 2)^2\) and \(g(x) = (x - 2)^2\) shown in Figure239. Compared with the graph of the basic function \(y = x^2\text{,}\) the graph of \(f (x) = (x + 2)^2\) is shifted two units to the left, as shown by the arrows. You can see why this happens by studying the function values in the table.

swamp cooler graphs
Figure239

Locate a particular \(y\)-value for \(y = x^2\text{,}\) say, \(y = 4\text{.}\) You must move two units to the left in the table to find the same \(y\)-value for \(f (x)\text{,}\) as shown by the arrow. In fact, each \(y\)-value for \(f (x)\) occurs two units to the left when compared to the same \(y\)-value for \(y = x^2\text{.}\)

table for left translation
Figure240
table for right translation
Figure241

Similarly, the graph of \(g(x) = (x - 2)^2\) is shifted two units to the right compared to the graph of \(y = x^2\text{.}\) In the table for \(g\text{,}\) each \(y\)-value for \(g(x)\) occurs two units to the right of the same \(y\)-value for \(y = x^2\text{.}\) In general, we have the following principle.

Horizontal Shifts

Let \(h\gt 0\text{.}\) Compared with the graph of \(y = f (x)\text{,}\)

  1. the graph of \(y = f (x + h)~ ~\) is shifted \(h\) units to the left, and
  2. the graph of \(y = f (x - h)~ ~\) is shifted \(h\) units to the right.
Note242

At first, the direction of a horizontal translation may seem counterintuitive. Look again at the tables above to help you see how the shift occurs.

Example243

Graph the following functions.

  1. \(g(x) =\sqrt{x + 1}\) and \(y=\sqrt{x}\) and discuss how they are related.
  2. \(h(x) = \dfrac{1}{(x - 3)^2}\) and \(y=\dfrac{1}{x^2}\) and discuss how they are related.
Solution
  1. Consider the table of values for the function.

    \(x\) \(-1\) \(0\) \(1\) \(2\) \(3\)
    \(y=\sqrt{x}\) undefined \(0\) \(1\) \(1.414\) \(1.732\)
    \(y=\sqrt{x+1}\) \(0\) \(1\) \(1.414\) \(1.732\) \(2\)

    The table shows that each \(y\)-value for \(g(x)\) occurs one unit to the left of the same \(y\)-value for the graph of \(y=\sqrt{x}\text{.}\) Consequently, each point on the graph of \(y = g(x)\) is shifted one unit to the left of \(y =\sqrt{x}\text{,}\) as shown in Figure244.

    horizontal shift of square root
    Figure244
  2. Consider the table of values for the function.

    \(x\) \(-1\) \(0\) \(1\) \(2\) \(3\) \(4\)
    \(y=\dfrac{1}{x}\) \(1\) undefined \(1\) \(\dfrac{1}{4}\) \(\dfrac{1}{9}\) \(\dfrac{1}{16}\)
    \(y=\dfrac{1}{(x-3)^2}\) \(\dfrac{1}{16}\) \(\dfrac{1}{9}\) \(\dfrac{1}{4}\) \(1\) undefined \(1\)

    The table shows that each \(y\)-value for \(h(x)\) occurs three units to the right of the same \(y\)-value for the graph of \(y =\dfrac{1}{x^2}\text{.}\) Consequently, each point on the graph of \(y = h(x)\) is shifted three units to the right of \(y =\dfrac{1}{x^2}\text{,}\) as shown in Figure245.

    horizontal shift of inverse square
    Figure245
Example246

The function \(N = f (p)\) graphed in Figure247 gives the number of people who have a given eye pressure level \(p\) measured in millimeters of mercury (mm Hg) from a sample of 100 people with healthy eyes, and the function \(g\) gives the number of people with pressure level \(p\) in a sample of 100 glaucoma patients.

shift of a bell-shaped curve
Figure247
  1. Write a formula for \(g\) as a transformation of \(f\text{.}\)
  2. For what pressure readings could a doctor be fairly certain that a patient has glaucoma?
Solution
  1. The graph of \(g\) is translated \(10\) units to the right of \(f\text{,}\) so \(g(p) = f (p - 10)\text{.}\)
  2. Pressure readings above \(40\)mm Hg are a strong indication of glaucoma. Readings between \(10\) and \(40\)mm Hg cannot conclusively distinguish healthy eyes from those with glaucoma.

The function \(C = f (t)\) in Figure249 gives the caffeine level in Delbert's bloodstream at time \(t\) hours after he drinks a cup of coffee, and \(g(t)\) gives the caffeine level in Francine's bloodstream. Write a formula for \(g\) in terms of \(f\text{,}\) and explain what it tells you about Delbert and Francine.

shift of caffeine surge curve
Figure249
Example250

Graph \(f (x) = (x + 4)^3 + 2\text{.}\)

Solution

We identify the basic graph from the structure of the formula for \(f (x)\text{.}\) In this case, the basic graph is \(y = x^3\text{,}\) so we begin by locating a few points on that graph, as shown in Figure251.

We will perform the translations separately, following the order of operations. First, we sketch a graph of \(y = (x + 4)^3\) by shifting each point on the basic graph four units to the left. We then move each point up two units to obtain the graph of \(f (x) = (x + 4)^3 + 2\text{.}\) All three graphs are shown in Figure251.

translations of cubic curve
Figure251
  1. Graph the function \(f (x) = \abs{x - 2} - 1\text{.}\)
  2. How is the graph of \(f\) different from the graph of \(y=\abs{x}\text{?}\)