Shifting Functions

The graph of $y=f(x)+k$

The table to the right shows the values of $f(x)$ for a certain function $f$ and a few values of $x$. What are the values of $f(x)+2$ at these values of $x$?

$x$ $f(x)$ $f(x)+2$

Where on the grid do the blue points (coming from the $f(x)+2$ column) appear, relative to the corresponding red points (coming from the $f(x)$ column)? Are they above, below, to the left, or to the right?

Click to see the graphs of $y_1=f(x)$ and $y_2=f(x)+2$.

Are the shapes of the red and the blue graphs the same or different?

Is the graph of $f(x)+2$ above, below, to the left, or to the right of the graph of $f(x)$?

The equation $y=f(x)+k$ is now graphed in green, with a slider for $k$.

As $k$ increases, does the green graph move up, down, to the left, or to the right?

We’ll now look at shifting some functions whose formulas are specified.

Let $g(x)=x^2$. The equation $y_1=g(x)$ is graphed on the grid to the left, along with its shifted version $y=g(x)+k$.

Set $k$ to 3. Notice that the blue graph is now above the red graph. How many units above the red graph is it?

What is the equation for $y$ in terms of $x$ when $k=3$? (Use the ‘^’ key to write powers. That is, write $x^2$ as “x^2”.)

If one graph is $d$ units below another graph, we say that it’s $-d$ units “above” the other graph.

Set $k$ to $-2$ to move the blue graph. How many units above the red graph is it now?

What is the equation for $y$ in terms of $x$ when $k=-2$?

In general:

The graph of $y=f(x)+k$ has the same shape as the graph of $y_1=f(x)$, but shifted up by $k$ units. (If $k$ is negative, this means it’s shifted down by $|k|$ units.)

Now let $e(x)=x^2-2$. The equations $y_1=e(x)$ and $y=e(x)+k$ are graphed on the grid to the left, with a slider for $k$.

The graph which has the same shape as the graph of $y_1=e(x)$, but is shifted up by 3 units, has the equation $y=x^2-2+3$, or $y=x^2+1$. You can see this on the grid by sliding $k$ to 3.

What is an equation for the graph which has the same shape as the graph of $y_1=e(x)$, but is shifted up by 5 units?

What is an equation for the graph which has the same shape as the graph of $y_1=e(x)$, but is shifted up by $-3$ units?

The graph of $y=f(x-h)$

The left-hand table below shows you the values of $f(x)$ for a certain function $\cl"red"f$, at a few values of $x$. To find the value of $f(x-5)$ when $x=1$, we can first notice that $1-5=-4$, and then find the value of $f(-4)$ by looking at the $-4$ row in the left-hand table. Since $f(-4)=1$, $f(x-5)=1$ when $x=1$.

By using the left-hand table in this way, find the value of $f(x-5)$ at each value of $x$ in the right-hand table below.

$x$ $f(x)$

$x$ $x-5$ $f(x-5)$

Where do the blue points given by $f(x-5)$ appear on the grid to the left, compared to the red points given by $f$? Are the blue points above, below, to the left, or to the right of the red points?

Click to see the graphs of $y_1=f(x)$ and $y_2=f(x-5)$.

Does the blue graph of $y_2=f(x-5)$ have the same shape as the red graph of $y_1=f(x)$, or a different shape?

How does the placement of the graph of $y_2=f(x-5)$ compare to the placement of the graph of $y_1=f(x)$? Is the blue graph above, below, to the left, or to the right of the red graph?

The equation $y = f(x-h)$ is now graphed in green, with a slider for $h$.

As $h$ changes, does the shape of the graph change?

As $h$ increases, does the green graph move up, down, to the left, or to the right?

We’ll now look again at $g(x)=x^2$. The equation $y_1=g(x)$ is graphed on the grid to the left, along with its shifted version $y=g(x-h)$.

Set $h$ to 4. Notice that the blue graph is to the right of the red graph. How many units to the right of the red graph is it?

What is the equation for $y$ in terms of $x$ when $h=4$?

Just as a graph that is $d$ units below another graph is said to be $-d$ units “above” the other graph, a graph that is $d$ units to the left of another graph is said to be $-d$ units “to the right” of the other graph.

Set $h$ to $-3$ to move the blue graph. How many units to the right of the red graph is it?

What is the equation for $y$ in terms of $x$ when $h=-3$?

In general:

The graph of $y=f(x-h)$ has the same shape as the graph of $y_1=f(x)$, but shifted right by $h$ units. (If $h$ is negative, this means it’s shifted left by $|h|$ units.)

Now let $e(x)=(x+1)^2$. The equations $y_1=e(x)$ and $y=e(x-h)$ are graphed on the grid to the left, with a slider for $h$.

The graph which has the same shape as the graph of $y_1=e(x)$, but is shifted right by 3 units, has the equation $y=(x+1-3)^2$, or $y=(x-2)^2$. You can see this by sliding $h$ to 3.

What is an equation for the graph which has the same shape as the graph of $y_1=e(x)$, but is shifted right by 2 units?

What is an equation for the graph which has the same shape as the graph of $y_1=e(x)$, but is shifted right by $-3$ units?

What is an equation for the graph which has the same shape as the graph of $y_1=e(x)$, but is shifted right by 3 units and up by $-4$ units? (Remember that the graph of $y=f(x)+k$ is shifted up by $k$ units from the graph of $y=f(x)$.)

Periodic Functions

A phenomenon that always repeats after a certain period of time is called “periodic.” For example, the phases of the moon repeat after roughly 29.5 days.

If $f$ is a function and you can find some number $h$ (other than $0$) so that $f(x-h)=f(x)$ for every $x$, then $f$ is called a periodic function. We’ll now see this means that the graph of $f$ repeats whenever $x$ is changed by $h$.

A new function $\cl"red"f$ is now graphed on the grid to the left, along with the graph of $\cl"blue"{y=f(x-h)}$.

Slide $h$ to $-4$, and notice that the graphs of $y_1=f(x)$ and $y=f(x-h)$ then line up perfectly. This means that $f(x+4)=f(x)$ for every $x$.

Using the slider, find three other values of $h$ where the graphs of $y_1=f(x)$ and $y=f(x-h)$ line up perfectly.

Is $f$ a periodic function?

If $f$ is periodic, the smallest possible positive $h$ where $f(x-h)=f(x)$ for every $x$ is called the period of $f$.

What is the period of the function $f$?

Suppose $x$ represents an amount of time passing in hours, and $\cl"red"{f(x)}$ is a measurement taken at time $x$. After what period of time does $\cl"red"{f(x)}$ first start to repeat itself?