1.Transform the graph of f(x) = 3^x to sketch g(x) = 3^-(x+1) -2. Show table of values and each transformation clearly.
2.Write two equations to represent the same exponential function with a y-intercept of 5 and an asymptote at y = 3. Investigate whether other exponential functions have the same properties. Use the transformations to explain your observations.
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Given y = f(x),
(y-k) = f(x-h) is the same graph, shifted up k units and right h units.
Think of drawing a new set of axes at (h,k). Measuring from there, the new x values are reduced by h.
Now, you know what 3^x looks like.
3^-x is the same graph, flipped around the y-axis.
g(x)+2 = f(-(x+1))
is 3^x
(a) flipped over the y-axis
(b) shifted left by 1
(x) shifted down by 2
1. To transform the graph of f(x) = 3^x to sketch g(x) = 3^-(x+1) - 2, we can apply the following transformations step by step:
a) Horizontal Translation: To shift the graph one unit to the left, we subtract 1 from x in the original function f(x). So g(x) = 3^-(x+1).
b) Vertical Reflection: To flip the graph upside down, we take the reciprocal of the entire function. So g(x) = 1/(3^-(x+1)).
c) Vertical Translation: To move the graph downward by 2 units, we subtract 2 from the function. So g(x) = 1/(3^-(x+1)) - 2.
Now let's create a table of values to compare f(x) and g(x):
| x | f(x) | g(x) |
|-------|---------|---------|
| 0 | 1 | -1 |
| 1 | 3 | -3 |
| 2 | 9 | -9 |
| 3 | 27 | -27 |
| 4 | 81 | -81 |
Plotting these points on a graph will give you the sketch of g(x) = 3^-(x+1) - 2.
2. To represent the same exponential function with a y-intercept of 5 and an asymptote at y = 3, we can start with the general form of an exponential function: f(x) = a * b^x + c, where a, b, and c are constants.
The equation with a y-intercept of 5 can be written as f(x) = a * b^x + 5. By substituting x = 0 into this equation, we find that a * b^0 + 5 = 5. This simplifies to a + 5 = 5, meaning a must be 0.
So the equation becomes f(x) = 0 * b^x + 5, which simplifies to f(x) = 5.
Now, let's consider the asymptote at y = 3. An asymptote represents a value that the function does not cross or get arbitrarily close to. In an exponential function, the asymptote occurs when b = 1. Therefore, we need b = 1 to achieve an asymptote at y = 3.
From the equation f(x) = 0 * b^x + 5, if we set b = 1, we get f(x) = 0 * 1^x + 5, which further simplifies to f(x) = 0 + 5, or f(x) = 5.
By comparing the two equations, f(x) = 5 and f(x) = 5, we observe that they are the same equation. This means that any exponential function with a y-intercept of 5 and an asymptote at y = 3 will be the constant function f(x) = 5.
The transformations help us understand this observation. Applying a vertical translation by adding a constant to an exponential function does not affect the asymptote, only the y-intercept. Likewise, changing the base of the exponential function (in this case, from b ≠ 1 to b = 1) does not change the y-intercept or the asymptote. Thus, all exponential functions with a y-intercept of 5 and an asymptote at y = 3 will be the constant function f(x) = 5.