We have that j(x)=f(x)+3 can also be written as j(x)=2x-2. This is much easier to graph, as it is in slope-intercept form. As with the first graph, we have to enter the equation in the second row by pressing the Y= button.
Having entered the equations, we can plot them by pressing GRAPH.
When we look at the graph we can see that f(x) and j(x) have the same slope. The difference is the vertical position of j(x), which is 2 units above f(x). This means it has both a different y-intercept and a different x-intercept.
b This time we will compare k(x)=f(x+3) and f(x) by following the same steps as we did in Part A. Let's rewrite f(x) first.
f(x+3)=2x-1 ⇒ k(x)=2x-1
Let's enter f(x) and k(x) on a graphing calculator. To do that we press the Y= button and enter the equations on two separate rows.
Having entered the equations, we can plot them by pressing GRAPH.
When we look at the graph we can see that f(x) and k(x) have the same slope. The difference is the vertical position of k(x), which is 4 units above f(x). This means it has both a different y-intercept and a different x-intercept. Note that this is achieved by translating f(x) to the left by 3.
c Finally, let's make some generalized statements based on the comparisons we have made in Parts A and B.
Adding a Number to an Entire Function
When we add an entire function by a number, the slope of the new function remains unchanged.
When we add an entire function by a number, the graph is translated up or down depending on the number. This causes both the x- and y-intercept of the graph to change.
Adding a Number to the Input of a Function
When we add the x value of a function by a number, the slope of the function remains unchanged.
When we add the x value of a function by a number, the graph is translated to the left or right depending on the number. This causes both the x- and y-intercept of the graph to change.
