Question

7.1 check your understanding comparing exponential functions
write at least 5 sentences to compare graph a and graph b. you may want to create a
venn diagram/double bubble map before you begin.
key terms to think about: increasing, decreasing, domain, range, y - intercept, horizontal
asymptote, what the function looks like, possible values of a and b.
(graphs of graph a and graph b are shown, with graph a on the left and graph b on the right. graph a has a curve decreasing from left to right, and graph b has a curve increasing from left to right.)

Explanation:

1. Behavior (Increasing/Decreasing): Graph A is a decreasing exponential function (as \( x \) increases, \( y \) decreases), while Graph B is an increasing exponential function (as \( x \) increases, \( y \) increases).
2. Domain: Both graphs have a domain of all real numbers (\( (-\infty, \infty) \)) since there are no restrictions on \( x \)-values for exponential functions.
3. Range: Both have a range of \( ( -2, \infty ) \) for Graph A (approaches \( y = -2 \) as a horizontal asymptote) and \( (2, \infty ) \) for Graph B (approaches \( y = 2 \) as a horizontal asymptote), so in general, their ranges are \( (h, \infty) \) where \( h \) is the horizontal asymptote.
4. Y - Intercept: Graph A intersects the \( y \)-axis at \( (0, 2) \), and Graph B also intersects the \( y \)-axis at \( (0, 2) \), so they share the same \( y \)-intercept.
5. Horizontal Asymptote: Graph A has a horizontal asymptote at \( y = -2 \), and Graph B has a horizontal asymptote at \( y = 2 \); these asymptotes are symmetric with respect to the \( x \)-axis.

Answer:

1. Graph A is decreasing (as \( x \) increases, \( y \) decreases), while Graph B is increasing (as \( x \) increases, \( y \) increases).
2. Both graphs have a domain of all real numbers (\( (-\infty, \infty) \)).
3. The \( y \)-intercept for both Graph A and Graph B is \( (0, 2) \).
4. Graph A has a horizontal asymptote at \( y = -2 \), and Graph B has a horizontal asymptote at \( y = 2 \).
5. The range of Graph A is \( (-2, \infty) \), and the range of Graph B is \( (2, \infty) \).