Graphs of Exponential Functions

Question

To get the graph of g(x), the graph of 𝑓(𝑥) = (3.1)^x  is shifted 2 units to the left, compressed vertically by a factor of 3/5 , reflected about the x-axis, and then shifted up 1units.

a. (1) Write the equation of g.

b. (1) State the y-intercept of g to the nearest thousandth.

c. (1) Determine the domain of g.

d. (1) Determine the range of g.

Answer 1

a) If we start with f(x) = 3.1^x, then when we apply the transformations:

1. shift 2 units to the left -> 3.1^x+2
2. when we perform transformations that affect the x-values, it's always the opposite of what we would think. That means to shift to the left (the negative direction), we have to add to x. The x's are sneaky!
3. compressed vertically by 3/5 -> 3/5(3.1^x+2)
4. vertical compression just means multiplying that factor by the entire function. this is considered compression here because 3/5 is less than 1
5. reflect about the x-axis -> -3/5(3.1^x+2)
6. when we reflect a graph about the x-axis, that means the y-values are the ones that changed. this means we have to multiply the entire function by -1
7. shift up by 1 -> -3/5(3.1^x+2) +1
8. a vertical shift upwards is where we add the factor to the entire function, so plus 1

then we get, g(x) = -3/5(3.1^x+2) +1

b) Whenever we want to find the y-intercept of any function, we can find that by plugging in x = 0! Why? Because by default, the y-intercept is where x is equal to zero.

y0 = g(0) = -3/5(3.1⁰⁺²) + 1 = -4.766

y-intercept: (0, -4.766)

c) The domain for all exponential functions are: all real numbers! you can plug any positive or negative number into the function and get a real answer

domain: (-∞,∞)

d) For an exponential function f(x) = a^x, ( where a > 1) the range is just y > 0 or (0, ∞). But because we flipped the graph across the x-axis and shifted it upwards by 1, the range is (-∞, 1).

1. when we flip the graph across the x-axis, the range is now (-∞, 0)
2. then we shift the graph up by 1 unit so it is now (-∞,1)

range: (-∞,1)

Hope that explains everything!