Worksheet for graphing exponential functions, including y = 4·2^x, y = 5·2^x, y = 4·(1/2)^x, and y = 2·(1/2)^x.

To sketch the graphs of these exponential functions, we need to find a few key points for each equation and then connect them with a smooth curve.

Here is the step-by-step breakdown for each problem:

1) $y = 4 \cdot 2^x$

This is an exponential growth function because the base (2) is greater than 1. The number 4 in front means the graph starts higher on the y-axis.

Step 1: Find key points by picking x-values.
* If $x = 0$: $y = 4 \cdot 2^0 = 4 \cdot 1 = 4$. Point: (0, 4)
* If $x = 1$: $y = 4 \cdot 2^1 = 4 \cdot 2 = 8$. Point: (1, 8)
* If $x = 2$: $y = 4 \cdot 2^2 = 4 \cdot 4 = 16$. Point: (2, 16)
* If $x = -1$: $y = 4 \cdot 2^{-1} = 4 \cdot \frac{1}{2} = 2$. Point: (-1, 2)
* If $x = -2$: $y = 4 \cdot 2^{-2} = 4 \cdot \frac{1}{4} = 1$. Point: (-2, 1)

Step 2: Sketch.
Plot these points. The graph will start low on the left side, pass through (-2, 1), (-1, 2), cross the y-axis at (0, 4), and shoot up quickly through (1, 8) and (2, 16). It gets closer and closer to the x-axis on the left but never touches it.

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2) $y = 5 \cdot 2^x$

This is also exponential growth. It looks similar to #1, but it is stretched vertically because it starts at 5 instead of 4.

Step 1: Find key points.
* If $x = 0$: $y = 5 \cdot 2^0 = 5 \cdot 1 = 5$. Point: (0, 5)
* If $x = 1$: $y = 5 \cdot 2^1 = 5 \cdot 2 = 10$. Point: (1, 10)
* If $x = 2$: $y = 5 \cdot 2^2 = 5 \cdot 4 = 20$. Point: (2, 20) *(Note: This is at the very top edge of your grid)*
* If $x = -1$: $y = 5 \cdot 2^{-1} = 5 \cdot 0.5 = 2.5$. Point: (-1, 2.5)
* If $x = -2$: $y = 5 \cdot 2^{-2} = 5 \cdot 0.25 = 1.25$. Point: (-2, 1.25)

Step 2: Sketch.
Plot the points. The curve crosses the y-axis at (0, 5). It rises steeply to (1, 10) and (2, 20). On the left, it flattens out near the x-axis, passing through roughly (-1, 2.5).

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3) $y = 4 \cdot (\frac{1}{2})^x$

This is an exponential decay function because the base ($\frac{1}{2}$) is between 0 and 1. The graph will go down from left to right.

Step 1: Find key points.
* If $x = 0$: $y = 4 \cdot (\frac{1}{2})^0 = 4 \cdot 1 = 4$. Point: (0, 4)
* If $x = 1$: $y = 4 \cdot (\frac{1}{2})^1 = 4 \cdot 0.5 = 2$. Point: (1, 2)
* If $x = 2$: $y = 4 \cdot (\frac{1}{2})^2 = 4 \cdot 0.25 = 1$. Point: (2, 1)
* If $x = -1$: $y = 4 \cdot (\frac{1}{2})^{-1} = 4 \cdot 2 = 8$. Point: (-1, 8)
* If $x = -2$: $y = 4 \cdot (\frac{1}{2})^{-2} = 4 \cdot 4 = 16$. Point: (-2, 16)

Step 2: Sketch.
Start high on the left. Plot (-2, 16) and (-1, 8). The graph drops down to cross the y-axis at (0, 4). It continues dropping more slowly through (1, 2) and (2, 1), getting closer to the x-axis on the right side.

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4) $y = 2 \cdot (\frac{1}{2})^x$

This is also exponential decay. It is lower than #3 because it starts at 2 instead of 4.

Step 1: Find key points.
* If $x = 0$: $y = 2 \cdot (\frac{1}{2})^0 = 2 \cdot 1 = 2$. Point: (0, 2)
* If $x = 1$: $y = 2 \cdot (\frac{1}{2})^1 = 2 \cdot 0.5 = 1$. Point: (1, 1)
* If $x = 2$: $y = 2 \cdot (\frac{1}{2})^2 = 2 \cdot 0.25 = 0.5$. Point: (2, 0.5)
* If $x = -1$: $y = 2 \cdot (\frac{1}{2})^{-1} = 2 \cdot 2 = 4$. Point: (-1, 4)
* If $x = -2$: $y = 2 \cdot (\frac{1}{2})^{-2} = 2 \cdot 4 = 8$. Point: (-2, 8)
* If $x = -3$: $y = 2 \cdot (\frac{1}{2})^{-3} = 2 \cdot 8 = 16$. Point: (-3, 16)

Step 2: Sketch.
Start high on the far left. Plot (-3, 16), (-2, 8), and (-1, 4). The graph drops to cross the y-axis at (0, 2). It continues dropping through (1, 1) and (2, 0.5), flattening out near the x-axis on the right.

Final Answer:
Graphing Instructions Summary:

1. $y = 4 \cdot 2^x$: Plot points (-2, 1), (-1, 2), (0, 4), (1, 8), (2, 16). Draw a smooth curve increasing from left to right.
2. $y = 5 \cdot 2^x$: Plot points (-2, 1.25), (-1, 2.5), (0, 5), (1, 10), (2, 20). Draw a smooth curve increasing from left to right (steeper than #1).
3. $y = 4 \cdot (\frac{1}{2})^x$: Plot points (-2, 16), (-1, 8), (0, 4), (1, 2), (2, 1). Draw a smooth curve decreasing from left to right.
4. $y = 2 \cdot (\frac{1}{2})^x$: Plot points (-3, 16), (-2, 8), (-1, 4), (0, 2), (1, 1), (2, 0.5). Draw a smooth curve decreasing from left to right (lower than #3).