Compare these two

Question

Describe the Transformation Rules of Square Root Function compared to the Transformation Rules of Exponential Function.

Answer 1

This answer may be more intensive than required by your assignment. To best understand and compare how the transformation rules apply to each of these types of functions, we must (A) recall what the transformation rules are, and (B) acknowledge that square root functions and exponential functions are vastly different shapes.

A. The Transformation Rules:

Please note that "x" refers to the variable of the function, while "f(x)" refers to the function as

a whole, including the parent function and transformations.

1. f(x) + b shifts the function "b" units upward
2. f(x) - b shifts the function "b" units downward
3. f(x+b) shifts the function "b" units to the left
4. f(x-b) shifts the function "b" units to the right
5. -f(x) reflects the function across the x-axis (flips upside-down)
6. f(-x) reflects the function across the y-axis (flips left/right)

B. Square Root Functions vs. Exponential Functions

S Q U A R E R O O T F U N C T I O N S

https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/graphing-square-root-functions/graph-1.gif

The above link is an image of the shape of the parent function of all square root functions (parent function refers to the shape of the function with no transformations).

The transformation rules can be applied to square root functions through the rules listed above, or through the following formula:

f(x) = a √( x - h ) + k

1. The value of "a" determines the stretch/shrink of the graph (if a1, the graph is shrunk)
2. The sign (+/-) of "a" determines the flip across the x-axis
3. The value of "h" determines how much the graph is shifted left or right
4. The sign (+/-) of "h" determines which direction the graph is shifted ("-"left or "+"right)
5. The value of "k" determines how much the graph is shifted up or down
6. The sign (+/-) of "k" determines which direction the graph is shifted ("+"up or "-"down)
7. The sign (+/-) of "x" determines the flip across the y-axis

See the following figures as an example of a transformation of a square root function:

https://www.varsitytutors.com/assets/vt-hotmath-legacy/hotmath_help/topics/graphing-square-root-functions/graph-4.gif

https://www.geogebra.org/resource/KT3Xx9Fp/UXabm8zYJp2EcIsg/material-KT3Xx9Fp.png

E X P O N E N T I A L F U N C T I O N S

https://www.mechamath.com/wp-content/uploads/2021/06/graph-of-two-exponential-functions.png?ezimgfmt=ngcb83/notWebP

In the above link, you will see two examples of an exponential function. The "parent function" of the exponential function is y=b^x, but that cannot be graphed, as a base (b) is a required variable.

There are two ways to understand the reasoning behind the different graph shapes shown above:

1. The variable base "b" determines the range of the function.
2. If 0<b≤ y ≤ ∞ }, similar to the red graph shown above.
3. If 1<b
4. The variable base "b", when 0<b
5. y=0.5^x or y=(½)^x can be re-written as y=2^-x
6. Using our knowledge of transformations, we know that if the "x" of the function is made negative (f(-x)), the function is mirrored across the y-axis (flipped left/right). We can see this occur in the change from the blue graph to the red graph given above.

Outside of mirroring across the y-axis, the transformation rules can be applied to exponential functions through the rules listed above, or through the following formula:

f(x) = ab^(x-h) + k

1. The same identities apply to each letter as in the formula for the square root functions
2. EX: f(x) = (2)2^(x-1) + 3
3. The parent function of this graph (just f(x) = b^x) is f(x) = 2^x
4. The graph is then shrunk by a factor of 2 (a = 2)...
5. ...then shifted right by a factor of 1...
6. ... then shifted up by a factor of 3.

Feel free to comment with any more questions! Hope this helped!