Determine the equations of the asymptotes and all intercepts with the axes of the following exponential functions. In addition, determine whether the graph is increasing or decreasing and state whether the graph lies above or below the asymptote:

Question

Asked on July 28, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-07-28T03:26:34.456Z

Here are the properties for each exponential function: a) y = 2^x Step 1: Determine the asymptote. The general form of an exponential function is y = a · b^x + d. Here, a=1, b=2, and d=0. The horizontal asymptote is y=d. The asymptote is y = 0. Step 2: Determine the intercepts. For the y-intercept, set x=0: y = 2^0 = 1 The y-intercept is (0, 1). For the x-intercept, set y=0: 0 = 2^x There is no real value of x for which 2^x = 0. There is no x-intercept. Step 3: Determine if the graph is increasing or decreasing. Since the base b=2 > 1 and the coefficient a=1 > 0, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=1 > 0, the graph lies above the asymptote. The graph lies above the asymptote. b) y = 2^x - 1 Step 1: Determine the asymptote. Here, a=1, b=2, and d=-1. The horizontal asymptote is y=d. The asymptote is y = -1. Step 2: Determine the intercepts. For the y-intercept, set x=0: y = 2^0 - 1 = 1 - 1 = 0 The y-intercept is (0, 0). For the x-intercept, set y=0: 0 = 2^x - 1 2^x = 1 x = 0 The x-intercept is (0, 0). Step 3: Determine if the graph is increasing or decreasing. Since the base b=2 > 1 and the coefficient a=1 > 0, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=1 > 0, the graph lies above the asymptote. The graph lies above the asymptote. c) y = 2^x + 1 Step 1: Determine the asymptote. Here, a=1, b=2, and d=1. The horizontal asymptote is y=d. The asymptote is y = 1. Step 2: Determine the intercepts. For the y-intercept, set x=0: y = 2^0 + 1 = 1 + 1 = 2 The y-intercept is (0, 2). For the x-intercept, set y=0: 0 = 2^x + 1 2^x = -1 There is no real value of x for which 2^x = -1. There is no x-intercept. Step 3: Determine if the graph is increasing or decreasing. Since the base b=2 > 1 and the coefficient a=1 > 0, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=1 > 0, the graph lies above the asymptote. The graph lies above the asymptote. d) f(x) = -2^-x + 4 Step 1: Rewrite the function and determine the asymptote. The function can be written as f(x) = -1 · (2^-1)^x + 4 = -1 · ((1)/(2))^x + 4. Here, a=-1, b=(1)/(2), and d=4. The horizontal asymptote is y=d. The asymptote is y = 4. Step 2: Determine the intercepts. For the y-intercept, set x=0: f(0) = -2^-0 + 4 = -1 + 4 = 3 The y-intercept is (0, 3). For the x-intercept, set f(x)=0: 0 = -2^-x + 4 2^-x = 4 2^-x = 2^2 -x = 2 x = -2 The x-intercept is (-2, 0). Step 3: Determine if the graph is increasing or decreasing. Since the coefficient a=-1 < 0 and the base b=(1)/(2) < 1, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=-1 < 0, the graph lies below the asymptote. The graph lies below the asymptote. e) f(x) = -2(2)^x + 2 Step 1: Determine the asymptote. Here, a=-2, b=2, and d=2. The horizontal asymptote is y=d. The asymptote is y = 2. Step 2: Determine the intercepts. For the y-intercept, set x=0: f(0) = -2(2)^0 + 2 = -2(1) + 2 = -2 + 2 = 0 The y-intercept is (0, 0). For the x-intercept, set f(x)=0: 0 = -2(2)^x + 2 2(2)^x = 2 2^x = 1 x = 0 The x-intercept is (0, 0). Step 3: Determine if the graph is increasing or decreasing. Since the coefficient a=-2 < 0 and the base b=2 > 1, the function is decreasing. The graph is decreasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=-2 < 0, the graph lies below the asymptote. The graph lies below the asymptote. f) f(x) = 3((1)/(3))^x + 2 Step 1: Determine the asymptote. Here, a=3, b=(1)/(3), and d=2. The horizontal asymptote is y=d. The asymptote is y = 2. Step 2: Determine the intercepts. For the y-intercept, set x=0: f(0) = 3((1)/(3))^0 + 2 = 3(1) + 2 = 3 + 2 = 5 The y-intercept is (0, 5). For the x-intercept, set f(x)=0: 0 = 3((1)/(3))^x + 2 3((1)/(3))^x = -2 ((1)/(3))^x = -(2)/(3) There is no real value of x for which ((1)/(3))^x is negative. There is no x-intercept. Step 3: Determine if the graph is increasing or decreasing. Since the coefficient a=3 > 0 and the base b=(1)/(3) < 1, the function is decreasing. The graph is decreasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=3 > 0, the graph lies above the asymptote. The graph lies above the asymptote. What's next? 📸

Accepted Answer · 2026-07-28T03:26:34.456Z

Here are the properties for each exponential function: a) y = 2^x Step 1: Determine the asymptote. The general form of an exponential function is y = a · b^x + d. Here, a=1, b=2, and d=0. The horizontal asymptote is y=d. The asymptote is y = 0. Step 2: Determine the intercepts. For the y-intercept, set x=0: y = 2^0 = 1 The y-intercept is (0, 1). For the x-intercept, set y=0: 0 = 2^x There is no real value of x for which 2^x = 0. There is no x-intercept. Step 3: Determine if the graph is increasing or decreasing. Since the base b=2 > 1 and the coefficient a=1 > 0, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=1 > 0, the graph lies above the asymptote. The graph lies above the asymptote. b) y = 2^x - 1 Step 1: Determine the asymptote. Here, a=1, b=2, and d=-1. The horizontal asymptote is y=d. The asymptote is y = -1. Step 2: Determine the intercepts. For the y-intercept, set x=0: y = 2^0 - 1 = 1 - 1 = 0 The y-intercept is (0, 0). For the x-intercept, set y=0: 0 = 2^x - 1 2^x = 1 x = 0 The x-intercept is (0, 0). Step 3: Determine if the graph is increasing or decreasing. Since the base b=2 > 1 and the coefficient a=1 > 0, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=1 > 0, the graph lies above the asymptote. The graph lies above the asymptote. c) y = 2^x + 1 Step 1: Determine the asymptote. Here, a=1, b=2, and d=1. The horizontal asymptote is y=d. The asymptote is y = 1. Step 2: Determine the intercepts. For the y-intercept, set x=0: y = 2^0 + 1 = 1 + 1 = 2 The y-intercept is (0, 2). For the x-intercept, set y=0: 0 = 2^x + 1 2^x = -1 There is no real value of x for which 2^x = -1. There is no x-intercept. Step 3: Determine if the graph is increasing or decreasing. Since the base b=2 > 1 and the coefficient a=1 > 0, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=1 > 0, the graph lies above the asymptote. The graph lies above the asymptote. d) f(x) = -2^-x + 4 Step 1: Rewrite the function and determine the asymptote. The function can be written as f(x) = -1 · (2^-1)^x + 4 = -1 · ((1)/(2))^x + 4. Here, a=-1, b=(1)/(2), and d=4. The horizontal asymptote is y=d. The asymptote is y = 4. Step 2: Determine the intercepts. For the y-intercept, set x=0: f(0) = -2^-0 + 4 = -1 + 4 = 3 The y-intercept is (0, 3). For the x-intercept, set f(x)=0: 0 = -2^-x + 4 2^-x = 4 2^-x = 2^2 -x = 2 x = -2 The x-intercept is (-2, 0). Step 3: Determine if the graph is increasing or decreasing. Since the coefficient a=-1 < 0 and the base b=(1)/(2) < 1, the function is increasing. The graph is increasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=-1 < 0, the graph lies below the asymptote. The graph lies below the asymptote. e) f(x) = -2(2)^x + 2 Step 1: Determine the asymptote. Here, a=-2, b=2, and d=2. The horizontal asymptote is y=d. The asymptote is y = 2. Step 2: Determine the intercepts. For the y-intercept, set x=0: f(0) = -2(2)^0 + 2 = -2(1) + 2 = -2 + 2 = 0 The y-intercept is (0, 0). For the x-intercept, set f(x)=0: 0 = -2(2)^x + 2 2(2)^x = 2 2^x = 1 x = 0 The x-intercept is (0, 0). Step 3: Determine if the graph is increasing or decreasing. Since the coefficient a=-2 < 0 and the base b=2 > 1, the function is decreasing. The graph is decreasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=-2 < 0, the graph lies below the asymptote. The graph lies below the asymptote. f) f(x) = 3((1)/(3))^x + 2 Step 1: Determine the asymptote. Here, a=3, b=(1)/(3), and d=2. The horizontal asymptote is y=d. The asymptote is y = 2. Step 2: Determine the intercepts. For the y-intercept, set x=0: f(0) = 3((1)/(3))^0 + 2 = 3(1) + 2 = 3 + 2 = 5 The y-intercept is (0, 5). For the x-intercept, set f(x)=0: 0 = 3((1)/(3))^x + 2 3((1)/(3))^x = -2 ((1)/(3))^x = -(2)/(3) There is no real value of x for which ((1)/(3))^x is negative. There is no x-intercept. Step 3: Determine if the graph is increasing or decreasing. Since the coefficient a=3 > 0 and the base b=(1)/(3) < 1, the function is decreasing. The graph is decreasing. Step 4: Determine if the graph lies above or below the asymptote. Since the coefficient a=3 > 0, the graph lies above the asymptote. The graph lies above the asymptote. What's next? 📸

Determine the equations of the asymptotes and all intercepts with the axes of the following exponential functions. In addition, determine whether the graph is increasing or decreasing and state whether the graph lies above or below the asymptote:

Need help with your own homework?

More Mathematics Questions

Determine the equations of the asymptotes and all intercepts with the axes of the following exponential functions. In addition, determine whether the graph is increasing or decreasing and state whether the graph lies above or below the asymptote:

Need help with your own homework?

More Mathematics Questions