Westonci.ca offers quick and accurate answers to your questions. Join our community and get the insights you need today. Discover reliable solutions to your questions from a wide network of experts on our comprehensive Q&A platform. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
To graph the exponential function [tex]\( p(x) = 4 \cdot (0.75)^x + 3 \)[/tex], we need to understand its behavior and then plot key points:
1. Identify Key Components:
- The base exponential function is [tex]\( (0.75)^x \)[/tex].
- It is scaled by a factor of 4.
- It is vertically translated upward by 3 units.
2. Behavior of the Function:
- As [tex]\( x \)[/tex] increases, [tex]\( (0.75)^x \)[/tex] gets smaller because [tex]\( 0.75 \)[/tex] is less than 1.
- As [tex]\( x \)[/tex] decreases (negative values), [tex]\( (0.75)^x \)[/tex] grows larger because [tex]\( (0.75)^x \)[/tex] becomes a larger fraction (since you're dividing by increasingly smaller positive numbers).
3. Determine Key Points to Plot:
Here we calculate some key points to get a good sense of the function’s shape:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ p(-2) = 4 \cdot (0.75)^{-2} + 3 = 4 \cdot \left(\frac{1}{0.75}\right)^2 + 3 = 4 \cdot \left(\frac{4}{3}\right)^2 + 3 = 4 \cdot \frac{16}{9} + 3 = \frac{64}{9} + 3 \approx 10.11 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ p(-1) = 4 \cdot (0.75)^{-1} + 3 = 4 \cdot \left(\frac{1}{0.75}\right) + 3 = 4 \cdot \frac{4}{3} + 3 = \frac{16}{3} + 3 \approx 8.33 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ p(0) = 4 \cdot (0.75)^0 + 3 = 4 \cdot 1 + 3 = 4 + 3 = 7 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ p(1) = 4 \cdot (0.75)^1 + 3 = 4 \cdot 0.75 + 3 = 3 + 3 = 6 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ p(2) = 4 \cdot (0.75)^2 + 3 = 4 \cdot 0.5625 + 3 = 2.25 + 3 = 5.25 \][/tex]
4. Asymptotic Behavior:
- As [tex]\( x \to \infty \)[/tex], [tex]\( (0.75)^x \)[/tex] approaches 0, and thus [tex]\( p(x) \to 3 \)[/tex]. The horizontal line [tex]\( y = 3 \)[/tex] is a horizontal asymptote.
5. Graph the Function:
- Plot the points calculated: [tex]\( (-2, 10.11) \)[/tex], [tex]\( (-1, 8.33) \)[/tex], [tex]\( (0, 7) \)[/tex], [tex]\( (1, 6) \)[/tex], and [tex]\( (2, 5.25) \)[/tex].
- Sketch the curve that approaches 3 as [tex]\( x \)[/tex] increases.
- Remember to include the asymptotic line at [tex]\( y = 3 \)[/tex].
Here is an illustrative sketch on how you can expect the graph to look:
[tex]\[ \begin{array}{c} \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = {\(p(x)\)}, domain=-4:4, samples=100, grid=major ] \addplot [ thick, domain=-4:4, samples=100, color=blue, ] {4*(0.75)^x + 3}; \addlegendentry{\(p(x) = 4 \cdot (0.75)^x + 3\)}; \draw[dashed] (axis cs:-4,3) -- (axis cs:4,3); \node at (axis cs:2.5,3.3) {y=3 (Asymptote)}; \end{axis} \end{tikzpicture} \end{array} \][/tex]
Notice:
- As [tex]\( x \)[/tex] tends to very large positive values, the function approaches the horizontal asymptote [tex]\( y = 3 \)[/tex].
- The exponential decline is rapid for negative [tex]\( x \)[/tex]-values and gradually flattens as [tex]\( x \)[/tex] increases.
This detailed plot shows the overall trend and confirms our understanding of the exponential function [tex]\( p(x) = 4 \cdot (0.75)^x + 3 \)[/tex].
1. Identify Key Components:
- The base exponential function is [tex]\( (0.75)^x \)[/tex].
- It is scaled by a factor of 4.
- It is vertically translated upward by 3 units.
2. Behavior of the Function:
- As [tex]\( x \)[/tex] increases, [tex]\( (0.75)^x \)[/tex] gets smaller because [tex]\( 0.75 \)[/tex] is less than 1.
- As [tex]\( x \)[/tex] decreases (negative values), [tex]\( (0.75)^x \)[/tex] grows larger because [tex]\( (0.75)^x \)[/tex] becomes a larger fraction (since you're dividing by increasingly smaller positive numbers).
3. Determine Key Points to Plot:
Here we calculate some key points to get a good sense of the function’s shape:
- For [tex]\( x = -2 \)[/tex]:
[tex]\[ p(-2) = 4 \cdot (0.75)^{-2} + 3 = 4 \cdot \left(\frac{1}{0.75}\right)^2 + 3 = 4 \cdot \left(\frac{4}{3}\right)^2 + 3 = 4 \cdot \frac{16}{9} + 3 = \frac{64}{9} + 3 \approx 10.11 \][/tex]
- For [tex]\( x = -1 \)[/tex]:
[tex]\[ p(-1) = 4 \cdot (0.75)^{-1} + 3 = 4 \cdot \left(\frac{1}{0.75}\right) + 3 = 4 \cdot \frac{4}{3} + 3 = \frac{16}{3} + 3 \approx 8.33 \][/tex]
- For [tex]\( x = 0 \)[/tex]:
[tex]\[ p(0) = 4 \cdot (0.75)^0 + 3 = 4 \cdot 1 + 3 = 4 + 3 = 7 \][/tex]
- For [tex]\( x = 1 \)[/tex]:
[tex]\[ p(1) = 4 \cdot (0.75)^1 + 3 = 4 \cdot 0.75 + 3 = 3 + 3 = 6 \][/tex]
- For [tex]\( x = 2 \)[/tex]:
[tex]\[ p(2) = 4 \cdot (0.75)^2 + 3 = 4 \cdot 0.5625 + 3 = 2.25 + 3 = 5.25 \][/tex]
4. Asymptotic Behavior:
- As [tex]\( x \to \infty \)[/tex], [tex]\( (0.75)^x \)[/tex] approaches 0, and thus [tex]\( p(x) \to 3 \)[/tex]. The horizontal line [tex]\( y = 3 \)[/tex] is a horizontal asymptote.
5. Graph the Function:
- Plot the points calculated: [tex]\( (-2, 10.11) \)[/tex], [tex]\( (-1, 8.33) \)[/tex], [tex]\( (0, 7) \)[/tex], [tex]\( (1, 6) \)[/tex], and [tex]\( (2, 5.25) \)[/tex].
- Sketch the curve that approaches 3 as [tex]\( x \)[/tex] increases.
- Remember to include the asymptotic line at [tex]\( y = 3 \)[/tex].
Here is an illustrative sketch on how you can expect the graph to look:
[tex]\[ \begin{array}{c} \begin{tikzpicture} \begin{axis}[ axis lines = middle, xlabel = \(x\), ylabel = {\(p(x)\)}, domain=-4:4, samples=100, grid=major ] \addplot [ thick, domain=-4:4, samples=100, color=blue, ] {4*(0.75)^x + 3}; \addlegendentry{\(p(x) = 4 \cdot (0.75)^x + 3\)}; \draw[dashed] (axis cs:-4,3) -- (axis cs:4,3); \node at (axis cs:2.5,3.3) {y=3 (Asymptote)}; \end{axis} \end{tikzpicture} \end{array} \][/tex]
Notice:
- As [tex]\( x \)[/tex] tends to very large positive values, the function approaches the horizontal asymptote [tex]\( y = 3 \)[/tex].
- The exponential decline is rapid for negative [tex]\( x \)[/tex]-values and gradually flattens as [tex]\( x \)[/tex] increases.
This detailed plot shows the overall trend and confirms our understanding of the exponential function [tex]\( p(x) = 4 \cdot (0.75)^x + 3 \)[/tex].
Thank you for visiting. Our goal is to provide the most accurate answers for all your informational needs. Come back soon. We hope this was helpful. Please come back whenever you need more information or answers to your queries. We're glad you visited Westonci.ca. Return anytime for updated answers from our knowledgeable team.
