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# Exponential Functions

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Plz help with this

Select all the correct statements. (For this problem, we assume that an exponential function is of the form ka^x where a > 0)

(A) The following function is exponential: y = 2^x

(B) The following function is exponential: y = x^2

(C) The following function is exponential: y = sqrt(x)

(D) The following function is exponential: y = cube root of x

(E) The following function is exponential: y = 1/x

(F) The following function is exponential: y = 8x - 4

Also, solve these exponential equations:

(A) 3^(2y) = 3^5

(B) 9^(3u + 5) = 3^8

(C) 8^(4x - 1) = 64^(5x)

Also, prove the following statements:

(A) If the graph of an exponential function is reflected in the x-axis, then we obtain the graph of another exponential function

(B) If the graph of an exponential function is reflected in the y-axis, then we obtain the graph of another exponential function

(C) If the graph of an exponential function is translated vertically, then we obtain the graph of another exponential function.

(D) If the graph of an exponential function is translated horizontally, then we obtain the graph of another exponential function

Feb 12, 2023

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Number 1: Only A can be expressed in the form ka*x, the others cannot therefore only A is an exponential function.

Number 2:

A) 3^(2y) = 3^5, so 2y =5 y= 5/2

B) 9^(3u+5) = 3^8, 3^(6u+10) = 3^8, 6u + 10 = 8, u = -1/3

C) 8^(4x-1) = 8^10x, 4x -1 = 10x, x= -1/6

Number 3:

A) Reflecting a point over the x axis on a graph, (x,y) becomes (x, -y). We can use this to find that reflecting a function over the x axis we get that f(x) becomes -f(x). So after reflecting a exponential function over the x axis y = ka^x becomes y=-ka^x which is an exponential function.

B) Reflecting over the y axis, a point on a graph, (x,y) becomes, (-x,y). We can use this to fine the rule that to reflect a function over the y axis we apply the rule f(x) becomes f(-x). So an exponential function, y=ka^x becomes y=ka^(-x), which is k(1/a)^x, which is also an exponential function.

C)  Like the solutions above, we first translate a point vertically so (x,y) becomes (x,y+h). Applying that to functions in general we see that y or f(x) moves up h, and x stays the same, so a function transformation up h would result in f(x) becoming f(x)+h. Applying this to exponetial functions, a tranlation up h would be y=ka^x + h which is an exponential function.

D) If a point is translated horizontally by g, then (x,y) would become (x+g, y) . Applying this to this situation, f(x) = ka^x, then moving right by g units would be k(a-g)^x (remember its minus g, minus moves right, plus moves left), which, by out definition is also an exponential function.

Feb 12, 2023