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Between x = 2 and x = 3, which function has the smallest average rate of change?
the options are:

A)y = 6^(x-1)
B)y = 7^(x-2)
C)y = 4^(x + 1)
D)y = 4^(x - 1)

Respuesta :

bcalle
The function with the smallest rate of change is B.
Rate of change is like slope, the vertical change over the horizontal change.
The horizontal change for each of these choices will be ONE since 3-2 = 1.

Plug 2 in for x to find y, then plug 3 in for x and find y. Subtract the y values and you will get the rate of change as follows:
A) 30
B) 7
C) 192
D) 12
If you need more explaining just let me know.
ardni313

Function [tex]\displaystyle B)y=7^{x-2}[/tex] has the smallest average rate of change

Further explanation

Set A to set B is said to be a function if each member of set A pairs is exactly one member of set B

So, one value of x is only assigned to one value of y

A function can be expressed in the form of a cartesian diagram, sequential pairs, or arrow diagram

If y is a function of x, then y = f (x)

where x is the independent variable and y is the dependent variable

x is the domain and y is the result area or range

There are several forms of function including:

  • 1. linear function
  • 2. quadratic function
  • 3. trigonometric functions
  • 4. polynomial function
  • 5. fraction function etc.

The Average Rate of Change function shows the comparison between 2 objects that change

If we use function notation, the Average rate of Change of a function from a to b can be formulated as:

[tex]\large{\boxed{\boxed{\bold{ROC=\frac{f(b)-f(a)}{b-a}}}}[/tex]

ROC = rate of change

Let's check the function choices that have the smallest average rate of change (values ​​a = 2 and b = 3 we input as the value of x)

  • [tex]\displaystyle y=6^{x-1}[/tex]

[tex]\displaystyle x=2\\\\y=6^{2-1}\\\\y=6[/tex]

[tex]\displaystyle x=2\\\\y=6^{3-1}\\\\y=6^2=36[/tex]

[tex]\displaystyle ROC=\frac{36-6}{3-2}=30[/tex]

  • [tex]\displaystyle y=7^{x-2}[/tex]

[tex]\displaystyle x=2\\\\y=7^{2-2}\\\\y=7^0=1[/tex]

[tex]\displaystyle x=3\\\\y=7^{3-2}\\\\y=7^1=7[/tex]

[tex]\displaystyle ROC=\frac{7-1}{3-2}=6[/tex]

  • [tex]\displaystyle y=4^{x+1}[/tex]

[tex]\displaystyle x=2\\\\y=4^{2+1}\\\\y=4^3=64[/tex]

[tex]\displaystyle x=3\\\\y=4^{3+1}\\\\y=4^4=256[/tex]

[tex]\displaystyle ROC=\frac{256-64}{3-2}=192[/tex]

  • [tex]\displaystyle y=4^{x-1}[/tex]

[tex]\displaystyle x=2\\\\y=4^{2-1}\\\\y=4^1=4[/tex]

[tex]\displaystyle x=3\\\\y=4^{3-1}\\\\y=4^2=16[/tex]

[tex]\displaystyle ROC=\frac{16-4}{3-2}=12[/tex]

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Keywords: average rate of change, function

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