Use transformations and the zeros of the quadratic formula f(x)=(x+4)(x-2) to determine the zeroes of each of the following functions:
a. y=3f(x)
f. y=f(-1/2x)
I have no idea how to start this. Any help appreciated. Thanks.
3f(x) has the same zeroes as f(x), since
3(x+4)(x-2) = 0
is zero at the same places where f(x) is.
f(-x/2) = (-x/2 + 4)(-x/2 - 2)
= 1/2(-x+8) * 1/2 (-x - 4)
= 1/4 (x-8)(x+4)
so it has zeroes at 8 and -4
Note that using the original zeroes of -4 and 2,
-1/2(8) = -4
-1/2(-4) = 2
so f(ax) has zeroes at x1/a and x2/a
To determine the zeros of each function, we first need to find the zeros of the quadratic function f(x) = (x + 4)(x - 2).
The quadratic formula states that the zeros of a quadratic function in the form ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, our quadratic function is f(x) = (x + 4)(x - 2), so a = 1, b = 2, and c = -8.
Using the quadratic formula, we can find the zeros of f(x) as follows:
x = (-2 ± √(2^2 - 4(1)(-8))) / (2(1))
x = (-2 ± √(4 + 32)) / 2
x = (-2 ± √(36)) / 2
x = (-2 ± 6) / 2
This gives us two possible solutions:
1. x = (-2 + 6) / 2 = 4 / 2 = 2
2. x = (-2 - 6) / 2 = -8 / 2 = -4
Therefore, the zeros of f(x) = (x + 4)(x - 2) are x = -4 and x = 2.
Now let's use these zeros to find the zeros of the given functions:
a. y = 3f(x)
To find the zeros of this function, we substitute x = -4 and x = 2 into the given function and solve for y:
For x = -4:
y = 3f(-4)
y = 3((-4 + 4)(-4 - 2))
y = 3(0)(-6)
y = 0
For x = 2:
y = 3f(2)
y = 3((2 + 4)(2 - 2))
y = 3(6)(0)
y = 0
Therefore, the zeros of y = 3f(x) are y = 0.
f. y = f(-1/2x)
To find the zeros of this function, we substitute x = -4 and x = 2 into the given function and solve for y:
For x = -4:
y = f((-1/2)(-4))
y = f(2)
y = (2 + 4)(2 - 2)
y = 6(0)
y = 0
For x = 2:
y = f((-1/2)(2))
y = f(-1)
y = (-1 + 4)(-1 - 2)
y = 3(-3)
y = -9
Therefore, the zeros of y = f(-1/2x) are y = 0 and y = -9.
To determine the zeroes of each of the given functions using transformations and the zeros of the quadratic function f(x) = (x + 4)(x - 2), you can follow these steps:
a. y = 3f(x):
First, let's find the zeros of the quadratic function f(x) = (x + 4)(x - 2). The zeros occur when the function equals zero, so set f(x) = 0 and solve for x.
0 = (x + 4)(x - 2)
Setting each factor equal to zero individually:
x + 4 = 0 or x - 2 = 0
Solving for x:
x = -4 or x = 2
These are the zeros of the quadratic function f(x).
Next, consider the function y = 3f(x). The number 3 outside the function acts as a vertical stretch or compression. It does not change the zeros of the function, but it scales the y-values. So the zeros of y = 3f(x) will be the same as that of f(x), which are x = -4 and x = 2.
Therefore, the zeros of y = 3f(x) are x = -4 and x = 2.
f. y = f(-1/2x):
Again, let's find the zeros of the quadratic function f(x) = (x + 4)(x - 2).
Now, consider the function y = f(-1/2x). The expression -1/2x inside the function acts as a horizontal reflection and compression. It modifies the x-values by reflection and scaling. To find the zeros of y = f(-1/2x), we need to find the x-values that make f(-1/2x) equal to zero.
Replacing x in f(x) = (x + 4)(x - 2) with -1/2x:
f(-1/2x) = (-1/2x + 4)(-1/2x - 2)
Setting this equal to zero:
0 = (-1/2x + 4)(-1/2x - 2)
Considering each factor separately:
-1/2x + 4 = 0 or -1/2x - 2 = 0
Solving for x:
-1/2x = -4 or -1/2x = 2
Multiplying both sides by -2:
x = 8 or x = -4
These are the zeros of the quadratic function f(-1/2x). So the zeros of y = f(-1/2x) are x = 8 and x = -4.
In summary:
a. The zeros of y = 3f(x) are x = -4 and x = 2.
f. The zeros of y = f(-1/2x) are x = 8 and x = -4.