a If you shift something up by 1 unit, we have performed a vertical translation.
B
b How do you perform a translation in the vertical direction?
C
c Solve the equation sin x+1=0.
D
d Does the graph intersect the x-axis more than once?
A
aDiagram:
B
b We have to add 1.
C
cDiagram:
D
d Yes, we did.
Practice makes perfect
a Let's first graph y=sin x.
If we shift this graph up by one unit, the curve will have a range of 0≤ y≤ 2.
b When a function is shifted in the vertical direction, we have performed a vertical translation. Therefore, if the graph shifted one unit up, we must have added 1 to the parent function.
Parent Function:& y=sin x
Vertical translation of +1:& y=sin x + 1
c The intercepts of the function refers to the points where the graph intersects the x-axis. To find their coordinates on the graph, we should set y equal to 0 and solve for x.
The equation has a solution at - π2. Let's mark this on our graph from Part A.
Examining the diagram, we see that there are additional intercepts. Also, the distance between consecutive intercepts is 2π, because they are one period apart.
d Yes, we should have listed more than one intercept, as the graph intersects the x-axis more than once. In fact, if we extended the graph to the left or to the right, we could show even more solutions. Since consecutive intercepts are 1 period apart, we can write them all in the following way.
