Question

which functions represent exponential growth?
$y = f(x)$
$y = g(x)$
$y = h(x)$
$y = k(x)$

which functions have a horizontal asymptote?
$y = f(x)$
$y = h(x)$
$y = g(x)$
$y = k(x)$

Explanation:

Step1: Recall Exponential Function Properties

Exponential functions (in the form \( y = a^x + b \), \( a>0,a
eq1 \)) have a horizontal asymptote (the line \( y = b \)) and show exponential growth/decay. For growth, the base \( a>1 \), and the function increases as \( x \) increases.

Step2: Analyze Each Function's Graph

• \( y = f(x) \): The graph rises steeply as \( x \) increases, typical of exponential growth (base \( >1 \)), and has a horizontal asymptote (approaches a horizontal line as \( x\to-\infty \)).
• \( y = g(x) \): Similar to \( f(x) \), shows exponential - like growth and has a horizontal asymptote.
• \( y = h(x) \): Appears to be a linear or polynomial function (no exponential growth/horizontal asymptote behavior).
• \( y = k(x) \): Similar to \( f(x) \) and \( g(x) \), shows exponential - like growth and has a horizontal asymptote.

For the first question (exponential growth): Functions with exponential growth have a base \( > 1 \) and show increasing behavior with a horizontal asymptote. From the graph, \( y = f(x) \), \( y = g(x) \), \( y = k(x) \) show exponential growth (steep increase, horizontal asymptote).

For the second question (horizontal asymptote): Exponential functions have horizontal asymptotes. From the graph, \( y = f(x) \), \( y = g(x) \), \( y = k(x) \) have horizontal asymptotes (they approach a horizontal line as \( x\to-\infty \)), while \( y = h(x) \) does not.

Answer:

For "Which functions represent exponential growth?": \( y = f(x) \), \( y = g(x) \), \( y = k(x) \)
For "Which functions have a horizontal asymptote?": \( y = f(x) \), \( y = g(x) \), \( y = k(x) \)