Justine graphs the function f(x) = (x – 7)2 – 1. On the same grid, she graphs the function
g(x) = (x + 6)2 – 3. Which transformation will map f(x) on to g(x)?
left 13 units, down 2 units
right 13 units, down 2 units
left 13 units, up 2 units
right 13 units, up 2 units
starting with the graph of x^2,
f(x) is right 7, down 1
g(x) is left 6, down 3
so, what do you think?
huh? what do you do to go right 6 to left 7?
You go left 13.
so, ...
To determine the transformation that maps function f(x) onto g(x), we need to analyze the changes between the two functions.
The general form of a quadratic function is f(x) = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola. Looking at f(x) = (x - 7)^2 - 1 and g(x) = (x + 6)^2 - 3, we can see that the vertex (h, k) of f(x) is (-7, -1) and the vertex of g(x) is (-6, -3).
Comparing the two vertices, we can observe the following transformations:
- Horizontal Translation: f(x) has been moved 1 unit to the right (7 units to the right of the y-axis) to become g(x).
- Vertical Translation: f(x) has been moved 2 units down to become g(x).
Now, let's analyze the answer options:
1. Left 13 units, down 2 units: Based on the analysis above, the transformation involves moving the graph of f(x) to the right, not left. Hence, this is not the correct transformation.
2. Right 13 units, down 2 units: As mentioned earlier, f(x) is translated 1 unit to the right to become g(x). Therefore, moving it 13 units to the right is an accurate transformation. Additionally, the vertical translation of down 2 units is also consistent. Hence, this option seems to be the correct transformation.
3. Left 13 units, up 2 units: This transformation does not account for the horizontal translation to the right of f(x), making it an incorrect choice.
4. Right 13 units, up 2 units: This transformation does not account for the downward vertical translation of f(x), making it an incorrect choice.
In conclusion, the correct transformation that maps f(x) onto g(x) is moving the graph of f(x) 13 units to the right and 2 units down, as stated in option 2: "Right 13 units, down 2 units."