The point P(5,4) is on the curve y=f(x). State the coordination of the image of P when the graph of y=f(x) is transformed to the graph of i) y= f(x-5) ii) y= f(x) +7
i) you want x-5 = 5, so (10,4). See how that shifts the graph to the right?
ii) well, f(5) = 4, so f(5)+7 = ___
To find the coordinates of the image of point P(5,4) after the given transformations, we will use the rules for each transformation.
i) For the transformation y = f(x - 5):
1. The value inside the parentheses, in this case, x - 5, determines the horizontal shift of the graph.
2. To find the new x-coordinate of the point P, subtract 5 from the x-coordinate of P: 5 - 5 = 0.
3. The y-coordinate of the point P remains unchanged.
Therefore, the coordinates of the image of P after the transformation are (0,4).
ii) For the transformation y = f(x) + 7:
1. The value added outside the function, in this case, +7, determines the vertical shift of the graph.
2. To find the new y-coordinate of the point P, add 7 to the y-coordinate of P: 4 + 7 = 11.
3. The x-coordinate of the point P remains unchanged.
Therefore, the coordinates of the image of P after the transformation are (5,11).
To find the image of point P(5,4) when the graph of y = f(x) is transformed to y = f(x-5):
i) When the graph of y=f(x) is transformed to y = f(x-5), it means that the function is translated horizontally by 5 units to the right. This can be achieved by substituting (x-5) into the function.
Substituting (x-5) into the function, we get:
y = f(x-5)
= f(5-5)
= f(0)
So, the new x-coordinate of the image of P(5,4) would be 0. Since the y-coordinate remains unaffected, the image would have the coordinates (0,4).
ii) When the graph of y=f(x) is transformed to y = f(x) +7, it means that the function is translated vertically by 7 units upward. This can be achieved by adding 7 to the original function.
Adding 7 to the function, we get:
y = f(x) + 7
Since the x-coordinate remains unaffected, the image would have the same x-coordinate, which is 5. The new y-coordinate of the image of P(5,4) would be 4 + 7 = 11. Hence, the image would have the coordinates (5,11).