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Which exponential function passes through the points (0’21)and (1’3)

Sagot :

Answer:

[tex]y=21\left(\dfrac{1}{7}\right)^x[/tex]

Step-by-step explanation:

Exponential Equations

Exponential functions or equations are expressions that show either exponential growth or decay and have the format of,

                                                 [tex]y=a(b)^x[/tex],

where a is the y-intercept, b is the growth/decay factor (if it's growth then, b > 1; if it's decay then, 0 < b < 1).

[tex]\hrulefill[/tex]

Solving the Problem

Algebraic Solution:

We're given two solutions to our mystery exponential function: (0, 21) and (1, 3). We can plug them into the general formula and solve for a and b!

So,

                                       [tex]21=a(b)^0[/tex]

and

                                        [tex]3=a(b)^1[/tex].

Solving the first equation we get,

                                         [tex]21=a(1)\\\Longrightarrow a=21[/tex].

We can plug the a value into the second equation to get b,

                                           [tex]3=21(b)^1\\\\3=21b\\\Longrightarrow \dfrac{1}{7}=b[/tex].

So our function is,

                                         [tex]y=21\left(\dfrac{1}{7}\right)^x[/tex].

Geometric Solution:

Knowing that all y-intercepts have an x-value of 0, we can identify that one of the points or solutions given in the problem is our y-intercept: (0, 21). Thus, a = 21.

From the two points given, we can see that as x gets bigger in the positive direction, the y-value decreases indicating that this is a decay function.

To be specific, we see that the y-value decreases from 21 to one-seventh of its value: 3. So, b = 1/7.

Plugging in our a and b values into the function we get,

                                            [tex]y=21\left(\dfrac{1}{7}\right)^x[/tex].

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