We know that g(x)=4f(x) can also be written as g(x)=12x+8. This is much easier to graph, as it is in slope-intercept form. As with the first graph, we have to enter the equation on 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 g(x) has a steeper slope than f(x). In fact, the slope of g(x) is 4 times greater than that of f(x). The y-intercept of f(x) is 2, and the y-intercept of g(x) is 8, which another change by a factor of 4. However, their x-intercepts are the same.
b
This time, we will compare h(x)=f(4x) and f(x) by following the same steps as we did in Part A. Let's rewrite f(x) first.
f(4x)=12x+2 ⇒ h(x)=12x+2
Let's enter h(x) and g(x) on a graphing calculator. To do that we press the Y= button and enter the equations in two separate rows.
Having entered the equations, we can plot them by pressing GRAPH.
As we can see, the slope of h(x) is also steeper than f(x) by a factor of 4. Recall that this is the same it was in Part A. However, in this case, their x-intercepts are different and the y-intercepts are the same.
c
Finally, let's make some generalized statements based on the comparisons we made in Parts A and B.
Multiplying an Entire Function by a Number
When we multiply an entire function by a number, the slope of the function is changed by a factor of that same number.
When we multiply an entire function by a number, the y-intercept of the graph changes while the x-intercept remains the same.
Multiplying the Input of a Function by a Number
When we multiply only the x value of a function by a number, the slope of the function is changed by a factor of that same number.
When we multiply only the x value of a function by a number, the x-intercept of the graph changes while the y-intercept remains the same.
