use transformations of the graph of f(x)=2^x to graph the given function. be sure to graph and give the…

use transformations of the graph of f(x)=2^x to graph the given function. be sure to graph and give the equation of the asymptote. use the graph to determine the functions domain and range. if applicable, use a graphing utility to confirm your hand - drawn graphs. g(x)=2^x + 2 graph g(x)=2^x + 2 and its asymptote. use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

use transformations of the graph of f(x)=2^x to graph the given function. be sure to graph and give the equation of the asymptote. use the graph to determine the functions domain and range. if applicable, use a graphing utility to confirm your hand - drawn graphs. g(x)=2^x + 2 graph g(x)=2^x + 2 and its asymptote. use the graphing tool to graph the function as a solid curve and the asymptote as a dashed line.

Answer

Explanation:

Step1: Analyze the parent function

The parent function is ( f(x) = 2^x ). The graph of ( f(x) = 2^x ) has a horizontal asymptote at ( y = 0 ), domain ( (-\infty, \infty) ), and range ( (0, \infty) ).

Step2: Determine the transformation

For the function ( g(x) = 2^x + 2 ), we have a vertical shift of the parent function ( f(x) = 2^x ) upward by 2 units.

Step3: Find the asymptote

A vertical shift of ( f(x) ) by ( k ) units (upward if ( k>0 )) changes the horizontal asymptote. For ( g(x) = 2^x + 2 ), the horizontal asymptote is ( y = 0 + 2 = 2 ) (dashed line).

Step4: Determine domain and range

  • Domain: Since there are no restrictions on the input ( x ) for the exponential function, the domain of ( g(x) ) is ( (-\infty, \infty) ).
  • Range: The range of ( f(x) = 2^x ) is ( (0, \infty) ). After shifting upward by 2 units, the range of ( g(x) ) becomes ( (2, \infty) ) (all values greater than 2).

Step5: Graphing

To graph ( g(x) = 2^x + 2 ), start with the graph of ( f(x) = 2^x ) (which passes through ( (0,1) ), increases exponentially). Shift each point on ( f(x) ) upward by 2 units. For example, the point ( (0,1) ) on ( f(x) ) becomes ( (0, 1 + 2)=(0,3) ) on ( g(x) ). Draw the solid curve for ( g(x) ) and the dashed line ( y = 2 ) for the asymptote.

Answer:

  • Asymptote equation: ( y = 2 ) (dashed line)
  • Domain: ( (-\infty, \infty) )
  • Range: ( (2, \infty) )
  • Graph: Solid curve for ( g(x)=2^x + 2 ) (exponential curve shifted up 2, passing through (0,3), increasing) and dashed line ( y = 2 ).