Question

How do you graph $ y = - {3^{ - x}} $ using a table of values?

Answer 1

Hint: For graphing a function we have to find the coordinates of some of the points that can be plotted on the graph. We can let $ y = f(x) $ , so by putting random values of one variable we can find the value of the other value from the given equation. Using this approach, we find out the coordinates of some points lying on the curve of the given exponential function and get the graph of the function by joining these points.

Complete step-by-step answer:
The given function is $ f(x) = - {3^{ - x}} $
As the function is equal to -3 raised to the power -x, so x can take both positive and negative values but the value of the function will always come out to be negative. Thus, the graph will lie in the third and the fourth quadrant.
 $
  f(0) = - {3^{ - 0}} = - 1 \\
  f(1) = - {3^{ - 1}} = - 0.3333.. \\
  f(2) = - {3^{ - 2}} = - 0.1111... \\
  f(3) = - {3^{ - 3}} = - 0.037037... \;
  $
To plot a proper graph of the given function, we need to consider its negative values also –
 $
  f( - 1) = - {3^{ - ( - 1)}} = - 3 \\
  f( - 2) = - {3^{ - ( - 2)}} = - 9 \\
  f( - 3) = - {3^{ - ( - 3)}} = - 27 \;
  $
Joining all these points and extending the obtained curve, we can trace the graph of the function $ f(x) = - {3^{ - x}} $ .

Note: We see that as we increase the value of x, the value of y gets closer and closer to zero but doesn’t become equal to 0. So, the slope of this curve is near to zero in the fourth quadrant. We see that as we decrease the negative value of x, the value of y becomes greater and greater. So, the slope of the graph will be extremely steep in the third quadrant. In the fourth quadrant, the curve seems to be touching the x-axis, but note that it is not actually touching the axis, the values of y in the fourth quadrant are so close to zero that the curve appears to touch the x-axis.