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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].
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