There are three transformations present in the second function.
See solution.
Practice makes perfect
Both equations are simple rational functions can generally be written in the following format.
y=a( 1/x-h ) +k
The constants a, h, and k represent various transformations of the parent function, y= 1x. Let's first graph the parent function. To do that, we need to calculate a few data points. Notice that the function has a vertical asymptote at x=0 so we cannot choose this value as an input.
|c|c|c|
[-0.9em]
x & 1/x & f(x) [0.7em]
[-0.9em]
-3& 1/-3 & ≈ - 0.33 [0.7em]
[-0.9em]
-2& 1/-2 & - 0.5 [0.7em]
[-0.9em]
-1& 1/-1 & - 1 [0.7em]
[-0.9em]
-0.5& 1/-0.5 & - 2 [0.7em]
[-0.9em]
0.5& 1/0.5 & 2 [0.7em]
[-0.9em]
1& 1/1 & 1 [0.7em]
[-0.9em]
2& 1/2 & 0.5 [0.7em]
[-0.9em]
3& 1/3 & ≈ 0.33 [0.7em]
Now we can graph the function by marking the points in a coordinate plane and connecting the points.
To graph the second function, we have to transform the parent function. We can identify these transformations by analyzing the constants and coefficient on the function's right-hand side.
|c|l|
Right-hand side& Transformation
4(1/x+5)+7 & &Vertical stretch &by a factor of 4.
4(1/x+ 5)+7& &Horizontal translation &to the left by 5 units.
4(1/x+5)+ 7& &Vertical translation &up by 7 units.
Let's start with the translations. It can seem difficult to translate the graph. However, just like all of the points that fall on the graph are translated, so will the graph's horizontal and vertical asymptote be translated.
Now we can graph the translated function.
Finally, we will vertically stretch the function by a factor of 4. Notice that the vertical stretch happens with respect to the horizontal asymptote. This means the function's value stretches away from this line.
