Plot the following exponential functions, identifying the horizontal asymptote, y-intercept, and…

Question

Plot the following exponential functions, identifying the horizontal asymptote, y-intercept, and domain/range.

ScanSolve AI Tutor · Accepted Answer

To plot these exponential functions, we'll identify key features like the horizontal asymptote, y-intercept, and a few points. a) f(x) = −2^(−x) + 4; This function can be rewritten as f(x) = −(1/2)^x + 4. Step 1: Identify the horizontal asymptote. The term +4 indicates a vertical shift upwards by 4 units. The horizontal asymptote is at y = 4. Step 2: Find the y-intercept. Set x=0: f(0) = −2⁻⁰ + 4 = −1 + 4 = 3 The y-intercept is (0, 3). Step 3: Calculate additional points for plotting. x — f(x) = −2^(−x) + 4 — Point-2 — −2⁻⁽⁻²⁾ + 4 = −2² + 4 = −4 + 4 = 0 — (−2, 0); −1 — −2⁻⁽⁻¹⁾ + 4 = −2¹ + 4 = −2 + 4 = 2 — (−1, 2); 0 — −2⁰ + 4 = −1 + 4 = 3 — (0, 3); 1 — −2⁻¹ + 4 = −1/2 + 4 = 3.5 — (1, 3.5); 2 — −2⁻² + 4 = −1/4 + 4 = 3.75 — (2, 3.75) Step 4: State the domain and range. The domain is all real numbers: x ∈ (−∞, ∞). The range is all real numbers less than the horizontal asymptote: y ∈ (−∞, 4). b) f(x) = −2(2)^x + 2 Step 1: Identify the horizontal asymptote. The term +2 indicates a vertical shift upwards by 2 units. The horizontal asymptote is at y = 2. Step 2: Find the y-intercept. Set x=0: f(0) = −2(2)⁰ + 2 = −2(1) + 2 = −2 + 2 = 0 The y-intercept is (0, 0). Step 3: Calculate additional points for plotting. x — f(x) = −2(2)^x + 2 — Point-2 — −2(2)⁻² + 2 = −2(1/4) + 2 = −1/2 + 2 = 1.5 — (−2, 1.5); −1 — −2(2)⁻¹ + 2 = −2(1/2) + 2 = −1 + 2 = 1 — (−1, 1); 0 — −2(2)⁰ + 2 = −2(1) + 2 = 0 — (0, 0); 1 — −2(2)¹ + 2 = −4 + 2 = −2 — (1, −2); 2 — −2(2)² + 2 = −2(4) + 2 = −8 + 2 = −6 — (2, −6) Step 4: State the domain and range. The domain is all real numbers: x ∈ (−∞, ∞). The range is all real numbers less than the horizontal asymptote: y ∈ (−∞, 2). c) f(x) = 3(1/2)^x + 2 Step 1: Identify the horizontal asymptote. The term +2 indicates a vertical shift upwards by 2 units. The horizontal asymptote is at y = 2. Step 2: Find the y-intercept. Set x=0: f(0) = 3(1/2)⁰ + 2 = 3(1) + 2 = 3 + 2 = 5 The y-intercept is (0, 5). Step 3: Calculate additional points for plotting. x — f(x) = 3(1/2)^x + 2 — Point-2 — 3(1/2)⁻² + 2 = 3(4) + 2 = 12 + 2 = 14 — (−2, 14); −1 — 3(1/2)⁻¹ + 2 = 3(2) + 2 = 6 + 2 = 8 — (−1, 8); 0 — 3(1/2)⁰ + 2 = 3(1) + 2 = 5 — (0, 5); 1 — 3(1/2)¹ + 2 = 3(1/2) + 2 = 1.5 + 2 = 3.5 — (1, 3.5); 2 — 3(1/2)² + 2 = 3(1/4) + 2 = 0.75 + 2 = 2.75 — (2, 2.75) Step 4: State the domain and range. The domain is all real numbers: x ∈ (−∞, ∞). The range is all real numbers greater than the horizontal asymptote: y ∈ (2, ∞).

$x$	$f(x) = -2^{-x} + 4$	Point
-2	$-2^{-(-2)} + 4 = -2^2 + 4 = -4 + 4 = 0$	$(-2, 0)$
-1	$-2^{-(-1)} + 4 = -2^1 + 4 = -2 + 4 = 2$	$(-1, 2)$
0	$-2^0 + 4 = -1 + 4 = 3$	$(0, 3)$
1	$-2^{-1} + 4 = -\frac{1}{2} + 4 = 3.5$	$(1, 3.5)$
2	$-2^{-2} + 4 = -\frac{1}{4} + 4 = 3.75$	$(2, 3.75)$

$x$	$f(x) = -2(2)^x + 2$	Point
-2	$-2(2)^{-2} + 2 = -2(\frac{1}{4}) + 2 = -\frac{1}{2} + 2 = 1.5$	$(-2, 1.5)$
-1	$-2(2)^{-1} + 2 = -2(\frac{1}{2}) + 2 = -1 + 2 = 1$	$(-1, 1)$
0	$-2(2)^0 + 2 = -2(1) + 2 = 0$	$(0, 0)$
1	$-2(2)^1 + 2 = -4 + 2 = -2$	$(1, -2)$
2	$-2(2)^2 + 2 = -2(4) + 2 = -8 + 2 = -6$	$(2, -6)$

$x$	$f(x) = 3\left(\frac{1}{2}\right)^x + 2$	Point
-2	$3\left(\frac{1}{2}\right)^{-2} + 2 = 3(4) + 2 = 12 + 2 = 14$	$(-2, 14)$
-1	$3\left(\frac{1}{2}\right)^{-1} + 2 = 3(2) + 2 = 6 + 2 = 8$	$(-1, 8)$
0	$3\left(\frac{1}{2}\right)^0 + 2 = 3(1) + 2 = 5$	$(0, 5)$
1	$3\left(\frac{1}{2}\right)^1 + 2 = 3(\frac{1}{2}) + 2 = 1.5 + 2 = 3.5$	$(1, 3.5)$
2	$3\left(\frac{1}{2}\right)^2 + 2 = 3(\frac{1}{4}) + 2 = 0.75 + 2 = 2.75$	$(2, 2.75)$

Plot the following exponential functions, identifying the horizontal asymptote, y-intercept, and domain/range.

Step-by-step solution

Part a· 4 steps

Identify the horizontal asymptote.

Find the y-intercept.

Calculate additional points for plotting.

State the domain and range.

Part b· 4 steps

Identify the horizontal asymptote.

Find the y-intercept.

Calculate additional points for plotting.

State the domain and range.

Part c· 4 steps

Identify the horizontal asymptote.

Find the y-intercept.

Calculate additional points for plotting.

State the domain and range.

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