Step: 1

From the graph of the function, y = -2^{x} is defined for all x -values but only has y values that are less than 0.

Step: 2

The domain of y = -2^{x} is the set of all real numbers and the range is the set of all negative numbers.

Correct Answer is : Domain is set of all real numbers and the range is set of all negative numbers.

Step: 1

Make a table of values that includes both positive and negative x values. Be sure to follow the order of operations when evaluating the function.

Step: 2

[Original exponential function.]

Step: 3

Substitute the values - 2, - 1, 0, 1, 2 and 3 for x in the function y = 2( 2 ) x .

Step: 4

[Replace x with -2.]

Step: 5

= 2(1 4 ) = 0.5

[Use definition of negative exponent.]

Step: 6

[Replace x with -1.]

Step: 7

= 2(1 2 ) = 1

[Use definition of negative exponent.]

Step: 8

[Replace x with 0.]

Step: 9

= 2 × 1 = 2

[2^{0} = 1.]

Step: 10

[Replace x with 1.]

Step: 11

[Replace x with 2.]

Step: 12

[Replace x with 3.]

Step: 13

Step: 14

Draw a coordinate plane and plot the six points given by the table. Then draw a smooth curve through the points.

Step: 15

So, the graph of the exponential function y = 2( 2 ) x matches with the Graph 1.

Correct Answer is : Graph 1

Step: 1

Draw a table for the number of viewers of a sports channel in 1996, 1997, 1998, 1999, 2000, 2001, 2002. x value for these years will be - 2, - 1, 0, 1, 2, 3, 4.

- 2 | - 1 | 0 | 1 | 2 | 3 | 4 | |

62.2 | 93.3 | 140 | 210 | 315 | 472.5 | 708.8 |

Step: 2

Draw a coordinate plane and plot the points given by table. Then draw a smooth curve through the points.

Step: 3

Notice that the graph has y -intercept of 140, and that it gets closer to negative side of the x -axis as the x values decrease.

Step: 4

Step: 5

So, the graph of the number of viewers of the sports channel y = 140(1.5)^{x} matches with the Figure 1.

Correct Answer is : Figure 1

Step: 1

Make a table of values that includes both positive and negative x values. Be sure to follow the order of operations when evaluating the function.

Step: 2

[Original exponential function.]

Step: 3

Substitute the values - 2, - 1, 0, 1, 2 and 3 for x in the function y = (1 2 ) ^{- x}

Step: 4

[Replace x with -2.]

Step: 5

= 1 4 = 0.25

Step: 6

[Replace x with -1.]

Step: 7

= 1 2 = 0.5

Step: 8

[Replace x with 0.]

Step: 9

= 1

[(1 2 )^{0} = 1.]

Step: 10

[Replace x with 1.]

Step: 11

= 1 ( 1 2 ) = 2

[Use definition of negative exponent.]

Step: 12

[Replace x with 2 and use definition of negative exponent.]

Step: 13

[Replace x with 3 and use definition of negative exponent.]

Step: 14

Step: 15

Draw a coordinate plane and plot the six points given by the table. Then draw a smooth curve through the points.

Step: 16

So, the graph 1 matches with given exponential function.

Correct Answer is : Graph 1

Step: 1

From the graph of the function, y = - (4^{x}) is defined for all x values but only has y values that are less than 0.

Step: 2

The domain of y = - (4^{x}) is the set of all real numbers and the range is the set of all negative numbers.

Correct Answer is : The domain is all real numbers and the range is all negative real numbers.

Step: 1

Make the table of values for the equation y = 6 (3)^{x}.

- 3 | - 2 | - 1 | 0 | 1 | 2 | |

2 | 6 | 18 | 54 |

Step: 2

Draw a coordinate plane and plot the six points given by the table.

Step: 3

Draw a smooth curve through the plotted points.

Step: 4

The graph gets closer to the negative side of the x -axis as the x -values decrease.

Correct Answer is : Negative x -axis

Step: 1

[Original function.]

Step: 2

Take some values for x , such as - 3, - 2, - 1, 0, 1 and substitute in the given function to get the coordinates.

Step: 3

[Substitute - 3 for x .]

Step: 4

= - 0.001

[Simplify.]

Step: 5

[Substitute - 2 for x .]

Step: 6

= - 0.01

[Simplify.]

Step: 7

[Substitute - 1 for x .]

Step: 8

= - 0.07

[Simplify.]

Step: 9

[Substitute 0 for x .]

Step: 10

= - 0.5

[Simplify.]

Step: 11

[Substitute 1 for x .]

Step: 12

= - 3.5

[Simplify.]

Step: 13

Plot these values in a graph.

Step: 14

Among the graphs, Graph 1 represents the exponential function f (x ) = - 1 2 (7)^{x}.

Correct Answer is : Graph 1

Step: 1

Make a table of values that includes both positive and negative x values.

- 1 | 0 | 1 | 2 | 3 | 4 | |

0.004 | 0.016 | 0.0625 | 0.25 | 1 | 4 |

Step: 2

Graph the ordered pairs in the table shown and connect the points with a smooth curve.

Step: 3

Then the graph obtained is as shown in Graph 4.

Step: 4

So, Graph 4 represents the equation y = 4^{x - 3}.

Correct Answer is : Graph 4

Step: 1

Make a table of values that includes both positive and negative x values.

- 1 | 0 | 1 | 2 | 3 | |

- 4 | - 1 | - 0.25 | - 0.0625 | - 0.016 |

Step: 2

Graph the ordered pairs in the table shown and connect the points with a smooth curve.

Step: 3

Then the graph obtained matches with Graph 2.

Step: 4

So, Graph 2 represents the equation y = - (1 4 )^{x}.

Correct Answer is : Graph 2

Step: 1

[Original function.]

Step: 2

Take some values for x , such as - 3, - 2, - 1, 0, 1 and substitute in the given function to get the coordinates.

Step: 3

[Substitute - 3 for x .]

Step: 4

= 171.5

[Simplify.]

Step: 5

[Substitute - 2 for x .]

Step: 6

= 24.5

[Simplify.]

Step: 7

[Substitute - 1 for x .]

Step: 8

= 3.5

[Simplify.]

Step: 9

[Substitute 0 for x .]

Step: 10

= 0.5

[Simplify.]

Step: 11

[Substitute 1 for x .]

Step: 12

= 0.07

[Simplify.]

Step: 13

Plot these values in a graph.

Step: 14

Among the graphs, Graph 2 represents the exponential function f (x ) = 1 2 (7)^{-x}.

Correct Answer is : Graph 2

Step: 1

The graph of the function, y = 4( 2 ) x is defined for all x values, but only has y values that are greater than 0.

Step: 2

So, the domain of y = 4( 2 ) x is all real numbers and the range is all positive real numbers.

Correct Answer is : domain is all real numbers and the range is all positive real numbers.

Step: 1

Make a table of values that includes both positive and negative x values.

- 3 | - 2 | - 1 | 0 | |

0.003 | 0.02 | 0.143 | 1 |

Step: 2

Graph the ordered pairs in the table shown and connect the points with a smooth curve.

Step: 3

Then the graph obtained matches with Graph 3.

Step: 4

So, Graph 3 represents the equation y = 7^{x}.

Correct Answer is : Graph 3

Step: 1

Let the exponential equation be y = y _{0}( 1 + r )^{x}

Step: 2

6 = y _{0}(1 + r )^{0} and hence y _{0} = 6

[From the graph, substitute the point (0,6) in the general equation]

Step: 3

( 1 + r )^{ -1} = 2 y 0

Step: 4

= y _{0}(1 + r )^{ - 1 }

[From the graph,substitute the point ( - 1,2) in the general equation]

Step: 5

(1 + r )^{ -1} = 2 6

[ From step 2, y _{0} = 6]

Step: 6

1 + r = (0.3333)^{ -1} = 3.000 = 3

[Simplify]

Step: 7

Therefore, the equation is y = 6(3) ^{x}

[Step 1, 2 and 5]

Correct Answer is : y = 6(3^{x})

Step: 1

Let the exponential equation be y = y _{0} (1 + r )^{x}

Step: 2

- 1 = y _{0}(1 + r )^{0} and hence y _{0} = -1

[ From the graph, substitute the point ( 0 , -1 ) in the general equation]

Step: 3

- 2 = y _{0}(1 + r )^{1}

[From the graph, substitute the point ( 1,- 2 ) in the general equation]

Step: 4

(1 + r )^{1} = - 2 y 0

Step: 5

(1 + r )^{1} = - 2 - 1

[From step 2, y _{0} = - 1]

Step: 6

1 + r = 2

[Simplify]

Step: 7

Therefore , the equation is y = - (2^{x})

[Step 1,2 and 5]

Correct Answer is : y = - (2^{x})

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