Consider the function f(x) = x(x-4).
If the point (2 + c, y) is on the graph of f(x), the following point will also be on the graph of f(x)
A) (c -2,y)
B) (2 - c,y)
Show your work
To find the points on the graph of f(x) given the point (2 + c, y), we need to substitute this x-value into the function and find the corresponding y-value.
We are given:
f(x) = x(x-4)
x = 2 + c
Substituting x = 2 + c into f(x), we get:
f(2 + c) = (2 + c)(2 + c - 4)
= (2 + c)(-2 + c)
= -2(2 + c) + c(2 + c)
= -4 - 2c + 2c + c^2
= c^2 - 4
Thus, the point (2 + c, y) on the graph of f(x) corresponds to the point (2 + c, c^2 - 4).
Now let's substitute the x-value of (2 + c) into the options given to find the corresponding y-value:
A) (c - 2, y)
Substituting x = c - 2 into f(x), we get:
f(c - 2) = (c - 2)(c - 6)
= c^2 - 2c - 6c + 12
= c^2 - 8c + 12
Thus, the point (c - 2, y) corresponds to the point (c - 2, c^2 - 8c + 12).
B) (2 - c, y)
Substituting x = 2 - c into f(x), we get:
f(2 - c) = (2 - c)(2 - c - 4)
= (2 - c)(-c - 2)
= -2(2 - c) - c(2 - c)
= -4 + 2c + 2c - c^2
= -c^2 + 4c - 4
Thus, the point (2 - c, y) corresponds to the point (2 - c, -c^2 + 4c - 4).
Comparing the y-values of the two options, we have:
For option A: y = c^2 - 8c + 12
For option B: y = -c^2 + 4c - 4
Since these y-values are not the same, option A) (c - 2, y) is the point that will also be on the graph of f(x) given the point (2 + c, y).