<< Chapter < Page | Chapter >> Page > |
Given a tabular function, create a new row to represent a horizontal shift.
A function $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ is given in [link] . Create a table for the function $\text{\hspace{0.17em}}g(x)=f(x-3).$
$x$ | 2 | 4 | 6 | 8 |
$f(x)$ | 1 | 3 | 7 | 11 |
The formula $\text{\hspace{0.17em}}g(x)=f(x-3)\text{\hspace{0.17em}}$ tells us that the output values of $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ are the same as the output value of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ when the input value is 3 less than the original value. For example, we know that $\text{\hspace{0.17em}}f(2)=1.\text{\hspace{0.17em}}$ To get the same output from the function $\text{\hspace{0.17em}}g,\text{\hspace{0.17em}}$ we will need an input value that is 3 larger . We input a value that is 3 larger for $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ because the function takes 3 away before evaluating the function $\text{\hspace{0.17em}}f.$
We continue with the other values to create [link] .
$x$ | 5 | 7 | 9 | 11 |
$x-3$ | 2 | 4 | 6 | 8 |
$f(x\u20133)$ | 1 | 3 | 7 | 11 |
$g(x)$ | 1 | 3 | 7 | 11 |
The result is that the function $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ has been shifted to the right by 3. Notice the output values for $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ remain the same as the output values for $\text{\hspace{0.17em}}f(x),\text{\hspace{0.17em}}$ but the corresponding input values, $\text{\hspace{0.17em}}x,\text{\hspace{0.17em}}$ have shifted to the right by 3. Specifically, 2 shifted to 5, 4 shifted to 7, 6 shifted to 9, and 8 shifted to 11.
[link] represents a transformation of the toolkit function $\text{\hspace{0.17em}}f(x)={x}^{2}.\text{\hspace{0.17em}}$ Relate this new function $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}f(x),\text{\hspace{0.17em}}$ and then find a formula for $\text{\hspace{0.17em}}g(x).$
Notice that the graph is identical in shape to the $\text{\hspace{0.17em}}f(x)={x}^{2}\text{\hspace{0.17em}}$ function, but the x- values are shifted to the right 2 units. The vertex used to be at (0,0), but now the vertex is at (2,0). The graph is the basic quadratic function shifted 2 units to the right, so
Notice how we must input the value $\text{\hspace{0.17em}}x=2\text{\hspace{0.17em}}$ to get the output value $\text{\hspace{0.17em}}y=0;\text{\hspace{0.17em}}$ the x -values must be 2 units larger because of the shift to the right by 2 units. We can then use the definition of the $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ function to write a formula for $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ by evaluating $\text{\hspace{0.17em}}f(x-2).$
The function $\text{\hspace{0.17em}}G(m)\text{\hspace{0.17em}}$ gives the number of gallons of gas required to drive $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ miles. Interpret $\text{\hspace{0.17em}}G(m)+10\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}G(m+10).$
$G(m)+10\text{\hspace{0.17em}}$ can be interpreted as adding 10 to the output, gallons. This is the gas required to drive $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ miles, plus another 10 gallons of gas. The graph would indicate a vertical shift.
$G(m+10)\text{\hspace{0.17em}}$ can be interpreted as adding 10 to the input, miles. So this is the number of gallons of gas required to drive 10 miles more than $\text{\hspace{0.17em}}m\text{\hspace{0.17em}}$ miles. The graph would indicate a horizontal shift.
Given the function $\text{\hspace{0.17em}}f(x)=\sqrt{x},\text{\hspace{0.17em}}$ graph the original function $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ and the transformation $\text{\hspace{0.17em}}g(x)=f(x+2)\text{\hspace{0.17em}}$ on the same axes. Is this a horizontal or a vertical shift? Which way is the graph shifted and by how many units?
The graphs of $\text{\hspace{0.17em}}f(x)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g(x)\text{\hspace{0.17em}}$ are shown below. The transformation is a horizontal shift. The function is shifted to the left by 2 units.
Now that we have two transformations, we can combine them. Vertical shifts are outside changes that affect the output ( y -) values and shift the function up or down. Horizontal shifts are inside changes that affect the input ( x -) values and shift the function left or right. Combining the two types of shifts will cause the graph of a function to shift up or down and left or right.
Notification Switch
Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?