Question

For the graph below, which of the following is a possible function for g? 

Answer 1

The general form of an exponential function is defined by f (x ) = a ^x

where a  is positive constant , a  not equals to 1 and a is called base. x is any real number.

f (x ) = 2^x

Choose values for x which matches the gragh above and find the corresponding values for y.
 

x	y = 2x	(x, y )
1	y = 21 = 2	(1,2)
0	y  = 20 = 1	(0,1)
-1	y = 2-1 = 1/2  =0.5	(-1,0.5)

 

The general form of an exponential function is defined by g (x ) = a ^x

where a  is positive constant , a  not equals to 1 and a is called base. x is any real number.

g(x ) = (1/2)^x

Choose values for x which matches the graph above and find the corresponding values for y.

 

x	y = (1/2)x	(x, y )
1	y = (1/2)1 = 1/2 = 0.5	(1,0.5)
0	y  = (1/2)0 = 1	(0,1)
-1	y = (1/2)-1 = 2	(-1,2)

Graph:

1.Draw a coordinate plane.

2.Plot the coordinate points.

3.Then sketch the graph, connecting the points with a smooth curve.

Therefore the curves are f(x)=2^x and g(x)=(1/2)^x.

 

Answer 2

Most of exponential function graphs resemble the same shape.

Observe the graph 

Function g  graph passes through (0,1) , (1,0.5), (-1,2)

for function g

The graph is very very small on right side and extremely close to x  - axis but do not touch it or cross it.

As the graph progress to the left . it starts grow faster and faster and shoots off the  top of the graph very quickly, as seen at the left.

From the characteristics of exponential function

The general form of an exponential function is defined by f (x ) = c*a ^x-h + k

where a  is positive constant , a  not equals to 1 and a is called base. x is any real number.

The graph has horizontal asympotote of y = k . and passes through (h , c +k )

a  is positive and a not equls to 1. here a  is base.

If the possible function for g  = (1/2) ^x

Compare to general form of exponential function.

c = 1 , h = 0 , k = 0

Horizontal asympotote is y  = k

passes through (h , c +k ) = (0, 1 + 0) = (0 , 1)

The graph crosses y  axis at (0,1)

In this case 0 < a  < 1, so the graph decreases.

a  = 1/2 .

Choose values for y and find the corresponding values for x.

x	y  = (1/2)x	(x, y )
-1	y  = (1/2)-1 =1/(1/2)1 = 2	(-1, 2)
0	y  = (1/2)0 =1	(0, 1)
1	y  = (1/2)1 =1/2 = 0.5	(1, 0.5)

Coordinates found in the table are the same to coordinates passes through the g (x ).

Graph

1.Draw a coordinate plane.

2.Plot the coordinate points and dash the horizontal asympotote.

3.Then sketch the graph, connecting the points with a smooth curve.

So the possible function for g  = (1/2)^x

g (x ) = (0.5)^x .

Answer 3

Note :

The general equation for any exponential function is g( x ) = a * b^x , where a and b are constants.

• Observe the graphs, They have only one y - intercept and one horizontal asymptote (the x - axis), and they are continuous.
• These are the basic characteristics of an exponential function.
• So, the functions f and g are an exponential functions.

The function g has an y - intercept at (0, 1).

This means that, x = 0 and y = f (0) = 1.

Substitute the values of x = 0 and f (0) = 1 in g( x ) = a * b^x .

f( 0 ) = ab⁰

1 = ab⁰

1 = a * 1

⇒ a = 1.

From the graph, the point (- 1, 2) lies on the graph of function g.

This means that, x = - 1 and y = f (- 1) = 2.

Substitute the values of x = - 1 and f (- 1) = 2 in g( x ) = a * b^x .

f( - 1 ) = a * b^{- 1}

2 = ab^{- 1}

Substitute The value a = 1 in 2 = ab^{- 1}.

2 = 1 * b^{- 1}

2 = 1 / b

⇒ b = 1 / 2 = 0.5 .

Substitute the values of a = 1 and b = 0.5 in g(x) = y  = a * b^x .

g(x) = y = 1 * (0.5)^x .

Therefore, The possible function for g is y = 1 * (0.5)^x .

Answer 4

CHECK :

Check by plugging original x - values to get the g (x) value.

The function is g(x) = y = 1 * (0.5)^x

Make the table of values to find the solutions that satisfy the function.

Choose values for x and find the corresponding values for y.

x	y = 1 * (0.5)x	(x, y )
- 2	y = 1 * (0.5)(- 2)  = 4	(- 2, 4)
-1	y = 1 * (0.5)(- 1) = 2	(- 1, 2)
0	y = 1 * (0.5)(0) = 1	(0, 1)
1	y = 1 * (0.5)(1) = 0.5	(1, 0.5)
2	y = 1 * (0.5)(2) = 0.25	(2, 0.25)

From the table the solutions are y = 4, 2, 1, 0.5, and 0.25, when x = - 2, - 1, 0, 1, and 2.

The points are (- 2, 4), (- 1, 2), (0, 1), (1, 0.5), and (2, 0.25).

The function is y = 1 * (0.5)^x.

Graph :

• Draw the coordinate plane.
• Plot the above points.
• Connect the ploted points.

Then, the formed line indcating given equation.

This is the graph of function g through the points (- 2, 4), (- 1, 2), (0, 1), (1, 0.5), and (2, 0.25).