A car backs out of a driveway. Before the car started moving, it had certain characteristics: 4 tires, windows, a steering wheel, an engine, gears, a certain color paint job. At the end of the driveway it has these same characteristics.
Moving the car down the driveway doesn’t change the nature or appearance of the car.
The same is true for functions. A function has a graph with characteristics that don’t change when the graph is moved around, or shifted, from one part of the coordinate system to another.
The easiest way to shift a function is by changing the y-value. This can be done by adding a constant value to, or subtracting a constant value from, the function.
Consider what happens to the graph of a function when a constant value is added. For the same x-values, the y-values are uniformly larger.
The effect is to lift the graph by the amount that was added.
Similar logic reveals that subtracting a constant value lowers the graph by the amount subtracted.
The graph may also be shifted to the left or right by adding to, or subtracting from, the x-value before using the result in the function.
For instance, to graph y equal the square of 5 less than x, 5 has to be subtracted from the x-value before squaring. Using the same x-values as before produces y-values way off the grid.
A better choice for x-values would give y-values not so far out of bounds. It would be better to square –2, –1, 0, 1, and 2 as was done earlier. This means we want the middle column to have those values. Let’s try again.
These are the values we would like to square to have reasonable y-values. What x-values can we start with to get them?
If we want those values in the second column, then the values in the first column need to be 5 larger.
Now, squaring gives us the reasonable y-values we want.
Plotting the x- and y-values from the first and third column produces a graph with the same shape as the graph of y equal x squared, but shifted 5 units to the right.
The simulation allows you to experiment with shifting graphs.
Select one of the parent functions to experiment with. Its unshifted graph will be displayed
You can shift the graph to the left or right by using the Horizontal control.
Sliding the control to the left will shift the graph to the left, and sliding the control to the right will shift the graph to the right.
The algebraic expression for the shifted function will also be displayed in the Modified Function box.
You can also type a value for a horizontal shift in the box.
You can shift the graph up or down by using the Vertical control.
Sliding the control to the left will shift the downward, and sliding the control to the right will shift the graph upward.
The algebraic expression for the shifted function will also be displayed in the Modified Function box.
You can also type a value for a vertical shift in the box.
Experiment with the different parent functions. Look for patterns and try to predict what will happen before doing it. Good learning and good luck!