y=10(0.5)^(t/19.74)
My question: is this a horizontal stretch by a factor of 19.74?
y=10(0.5)^(8t)
My question: is this a horizontal compression by a factor of 1/8?
y=10(0.5)^x + 1
My question: is this a vertical translation of up 1 unit?
y=10(0.5)^(x-1)
My question: is this a horizontal translation one unit to the right?
Thank you
To determine whether there is a horizontal stretch or compression and vertical translation, we can analyze the general form of the exponential function:
y = a * b^(cx + d)
Where:
- a represents the base value or amplitude,
- b represents the rate of change or growth factor,
- cx represents the horizontal transformation,
- d represents the vertical transformation.
1. y = 10(0.5)^(t/19.74):
In this equation, there is no horizontal transformation present. The variable 't' is in the exponent and is not inside any additional function or operation. Therefore, t/19.74 does not affect the horizontal stretch or compression. The base value is 10, and the growth factor is 0.5. There is no vertical translation indicated by adding a constant. So, this equation does not involve any horizontal stretch, compression, or vertical translation.
2. y = 10(0.5)^(8t):
In this equation, there is a horizontal compression by a factor of 1/8. The variable 't' is multiplied by 8, which makes the exponential function more sensitive to changes in 't'. The base value is 10, and the growth factor is 0.5. There is no vertical translation indicated by adding a constant.
3. y = 10(0.5)^x + 1:
In this equation, there is no horizontal transformation present either. The variable 'x' is in the exponent, but there is no multiplier affecting it. So, there is no horizontal stretch or compression. The base value is 10, and the growth factor is 0.5. However, there is a vertical translation indicated by adding 1 to the function. It shifts the graph one unit up compared to the original exponential function.
4. y = 10(0.5)^(x-1):
In this equation, there is a horizontal translation one unit to the right. The variable 'x' inside the exponent is subtracted by 1, causing the graph to shift to the right. The base value is 10, and the growth factor is 0.5. However, there is no vertical translation indicated by adding a constant.
In summary:
- Equation 1 has no horizontal stretch, compression, or vertical translation.
- Equation 2 has a horizontal compression by a factor of 1/8 and no vertical translation.
- Equation 3 has no horizontal stretch or compression but has a vertical translation up 1 unit.
- Equation 4 has a horizontal translation one unit to the right and no vertical translation.