Concept explainers
To find: Whether af(x) and f(ax) are equivalent or not.
Answer to Problem 52HP
Yes, if a=1
No, if a≠1
Explanation of Solution
Given information:
af(x)
f(ax)
Concept:
A Translation is a movement of the graph either horizontally parallel to x-axis or vertically parallel to y-axis.
The graph of y=f(x) where f(x)=x is same as the graph of y=x . Writing the equations as functions in the form of f(x) is useful when applying translations and reflections to graphs.
If g(x)=f(x)+k it is known as vertical shift.
If k>0 it moves upwards.
If k<0 it moves downwards.
If g(x)=f(x+l) it is known as horizontal shift.
If l<0 moves it right.
If l>0 moves it left.
If g(x)=−f(x) it is known as Vertical reflection it means reflects about X-axis
If g(x)=f(−x) it is known as Horizontal reflection it means reflects about Y-axis
If g(x)=af(x)
If a>1 it is known as Vertical stretch.
If 0<a<1 it is known as Vertical compression.
If g(x)=f(bx)
If b>1 it is known as Horizontal compression.
If 0<b<1 it is known as Horizontal Stretch.
Calculation:
Consider a=1 ,
af(x)=1×f(x)
⇒af(x)=f(x)
f(ax)=f(1×x)
⇒f(ax)=f(x)
Therefore, Alex assumption is True, i.e. af(x) and f(ax) are equivalent.
Consider a≠1 ,
From the above concept of,
If g(x)=af(x)
If a>1 it is known as Vertical stretch.
If 0<a<1 it is known as Vertical compression.
Therefore, the graph of the function af(x) is same as f(x) which is vertically stretched or compressed depending on value of a .
Whereas for the function f(ax) , using the concept of,
If g(x)=f(bx)
If b>1 it is known as Horizontal compression.
If 0<b<1 it is known as Horizontal Stretch.
Therefore, the graph of the function f(ax) is same as f(x) which is horizontally stretched or compressed depending on the value of a .
So, for a≠1 the graphs of functions f(ax) and af(x) are not same, therefore both of are not equivalent.
We conclude that,
Functions f(ax) and af(x) are equivalent if a=1 and not equivalent if a≠1 .
Chapter 3 Solutions
Glencoe Algebra 1, Student Edition, 9780079039897, 0079039898, 2018
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