Question

describe a function g(x) in terms of f(x) if the graph of g is obtained by vertically stretching f by a factor of 5, then shifting the graph of f to the right 6 units and upward 8 units.
g(x)=Af(x+B)+C, where
A=
B=
C=

Answer 1

To describe the function \( g(x) \) in terms of \( f(x) \), we need to account for two transformations: a vertical stretch and translations (shifting).

1. **Vertical Stretch**: Stretching \( f(x) \) vertically by a factor of 5 means we multiply the function by 5. So this gives us:
\[
5f(x)
\]

2. **Horizontal Shift**: Shifting the graph to the right by 6 units means we replace \( x \) with \( x - 6 \). Now our function looks like:
\[
5f(x - 6)
\]

3. **Vertical Shift**: Shifting the graph upward by 8 units means we add 8 to the function. Hence, we have:
\[
5f(x - 6) + 8
\]

Putting it all together, we get:
\[
g(x) = 5f(x - 6) + 8
\]

Now we can identify \( A \), \( B \), and \( C \) from the expression \( g(x) = Af(x + B) + C \):
- \( A = 5 \) (the vertical stretch factor)
- \( B = -6 \) (since we shift to the right, we use \( x + B \) with \( B = -6 \))
- \( C = 8 \) (the upward shift)

Thus, the final values are:
- \( A = 5 \)
- \( B = -6 \)
- \( C = 8 \)