Sketch the graph of the given function. Identify the domain, range, intercepts, and asymptotes. a). y = 9-3^x

Answer 1

To analyze and sketch the function \( y = 9 - 3^x \), let's go through the requested elements step by step:

1. Domain

The domain of the function \( y = 9 - 3^x \) is all real numbers, since there are no restrictions on the values of \( x \) for which this function is defined.

Domain: \( (-\infty, \infty) \)

2. Range

The term \( 3^x \) is an exponential function that always produces positive values. As \( x \) approaches \( \infty \), \( 3^x \) grows very large, making \( y \) approach \( -\infty \).

As \( x \) approaches \( -\infty \), \( 3^x \) approaches \( 0 \), making \( y \) approach \( 9 \).

Thus, the range of \( y = 9 - 3^x \) will be:

• The maximum value is approached as \( x \) approaches \( -\infty \), which is \( 9 \).
• There is no lower bound, as \( y \) goes to \( -\infty \).

Range: \( (-\infty, 9) \)

3. Intercepts

y-intercept:

To find the y-intercept, let \( x = 0 \): \[ y = 9 - 3^0 = 9 - 1 = 8 \] Thus, the y-intercept is \( (0, 8) \).

x-intercept:

To find the x-intercept, set \( y = 0 \): \[ 0 = 9 - 3^x \implies 3^x = 9 \implies 3^x = 3^2 \implies x = 2 \] Thus, the x-intercept is \( (2, 0) \).

4. Asymptotes

Evaluate the behavior as \( x \) approaches \( \infty \) and \( -\infty \):

• As \( x \to -\infty, 3^x \to 0 \), thus \( y \to 9 \). This indicates a horizontal asymptote at \( y = 9 \).

So, the function does not have any vertical asymptotes (since it is defined for all real \( x \)).

Horizontal Asymptote: \( y = 9 \)

Graph Summary

• Domain: \( (-\infty, \infty) \)
• Range: \( (-\infty, 9) \)
• y-intercept: \( (0, 8) \)
• x-intercept: \( (2, 0) \)
• Horizontal Asymptote: \( y = 9 \)

Sketching the Graph

The graph of \( y = 9 - 3^x \) will:

• Approach \( y = 9 \) as \( x \) goes to \( -\infty \)
• Drop from \( y = 9 \) to \( -\infty \) as \( x \) increases
• Cross the y-axis at \( (0, 8) \)
• Cross the x-axis at \( (2, 0) \)

Here’s a rough sketch of the graph:

   y
   |
  9|------------------- (Asymptote)
   |     *
   |    *
   |   *
   |  *
   | *
   |*       
   |
---+-------------------- x
   0    1    2    3    

This sketch represents the behavior of the function as described above, with the intercepts and asymptote indicated.