Consider the function f(x)=x[(2x+1)^2](x-4), without graphing determine if f(x) has line symmetry about the y axis, point symmetry about the origin or neither. Explain.
if even (y-axis symmetry), then f(-x) = f(x)
if odd (symmetry about origin), then f(-x) = -f(x)
Clearly this function fails both tests.
To determine if a function has line symmetry about the y-axis, we need to check if the function remains the same when we replace x with -x.
Let's substitute -x for x in the given function f(x):
f(-x) = (-x)[2(-x)+1]^2(-x-4)
Simplifying this expression:
f(-x) = -x(-2x+1)^2(-x-4)
Reordering the terms:
f(-x) = (-x)(-1)(2x-1)^2(x+4)
Simplifying further:
f(-x) = x(2x-1)^2(x+4)
Comparing f(x) = x[(2x+1)^2](x-4) with f(-x) = x(2x-1)^2(x+4), we can see that they are not identical. Therefore, the function f(x) does not have line symmetry about the y-axis.
To determine if a function has point symmetry about the origin, we need to check if the function remains the same when we replace x with -x and y with -y.
Let's check if f(x) = -f(-x):
f(x) = x[(2x+1)^2](x-4)
-f(-x) = -(-x)[(2(-x)+1)^2](-x-4)
Simplifying this expression:
-f(-x) = x(2x+1)^2(x+4)
Comparing f(x) = x[(2x+1)^2](x-4) with -f(-x) = x(2x+1)^2(x+4), we can see that they are not identical. Therefore, the function f(x) does not have point symmetry about the origin.
In conclusion, the function f(x) = x[(2x+1)^2](x-4) has neither line symmetry about the y-axis nor point symmetry about the origin.
To determine if a function has line symmetry about the y-axis or point symmetry about the origin, we need to examine the symmetry of the function with respect to the y-axis and origin.
Line symmetry about the y-axis occurs if replacing x with -x in the function results in an equivalent expression. This means that if f(-x) = f(x), the function has line symmetry about the y-axis.
Point symmetry about the origin occurs if replacing x with -x and simplifying the expression leads to the exact same equation. This means that if f(-x) = -f(x), the function has point symmetry about the origin.
Let's determine if the function f(x) = x[(2x+1)^2](x-4) has line symmetry about the y-axis:
1. Replace x with -x:
f(-x) = -x[(2(-x)+1)^2](-x-4)
= -x[(-2x+1)^2](-x-4)
= -x(4x^2 - 4x + 1)(-x-4)
2. Simplify the expression:
f(-x) = (-4x^3 + 4x^2 - x)(-x-4)
= 4x^3 - 4x^2 + x(x+4)
= 4x^3 - 4x^2 + x^2 + 4x
Now, let's compare f(-x) with f(x) to determine if they are equivalent:
f(-x) = 4x^3 - 4x^2 + x^2 + 4x
f(x) = x[(2x+1)^2](x-4)
The two expressions (f(-x) and f(x)) are not equivalent, so the function f(x) = x[(2x+1)^2](x-4) does not have line symmetry about the y-axis.
Now, let's determine if the function has point symmetry about the origin:
1. Replace x with -x:
f(-x) = -x[(2(-x)+1)^2](-x-4)
= -x(4x^2 - 4x + 1)(-x-4)
2. Simplify the expression:
f(-x) = (-4x^3 + 4x^2 - x)(-x-4)
= 4x^3 - 4x^2 + x(x+4)
= 4x^3 - 4x^2 + x^2 + 4x
Again, let's compare f(-x) with -f(x) to determine if they are equivalent:
-f(x) = -[x[(2x+1)^2](x-4)]
= -x[(2x+1)^2](x-4)
= (-x)(4x^2 + 4x + 1)(-x-4)
= -4x^3 - 4x^2 - x
The expression -f(x) is not equivalent to f(-x), so the function f(x) = x[(2x+1)^2](x-4) does not have point symmetry about the origin.
Therefore, the function f(x) = x[(2x+1)^2](x-4) neither has line symmetry about the y-axis nor point symmetry about the origin.