1. Which of the following best describes how the graph of f can be obtained from the graph of y = 1/x^2? f(x) = 4/x^2

a. vertical stretch by a factor of -4
b. horizontal shift of 4 units to the right
c. reflection over the y-axis, vertical shrink by a factor of 4
d. reflection over the x-axis, vertical stretch by a factor of 4
2. Determine the quadratic function f whose vertex is (3, -2) and passes through (2,1).
I have to type the answer in the form of f(x) = ax^2 + bx + c.
I would gladly appreciate the help.

1. you must have a typo. If f(x) = -4/x^2, then (d). Otherwise, none of the above. (a) would also be technically correct, but we don't usually denote stretching by negative values.

2. You know that

y = a(x-3)^2 - 2
so, plug in your point:
1 = a(2-3)^2 - 2
a = 3

so, y = 3(x-3)^2 - 2
= 3(x^2 - 6x + 9) - 2
= 3x^2 - 18x + 25

1. To obtain the graph of f(x) = 4/x^2 from the graph of y = 1/x^2, we need to apply transformations to the original graph.

First, we have f(x) = 4/x^2, which means the function is a vertical stretch by a factor of 4 compared to y = 1/x^2.

So, the correct answer is a. vertical stretch by a factor of 4.

2. To determine the quadratic function f whose vertex is (3, -2) and passes through (2, 1), we can use the vertex form of a quadratic function:

f(x) = a(x - h)^2 + k

where (h, k) represents the vertex of the parabola.

In this case, we have the vertex (3, -2), so h = 3 and k = -2. Substituting these values into the vertex form, we get:

f(x) = a(x - 3)^2 - 2

Now we need to find the value of 'a' by substituting the coordinate (2, 1) which lies on the graph of the quadratic function:

1 = a(2 - 3)^2 - 2

We simplify this equation:

1 = a(-1)^2 - 2
1 = a - 2
a = 1 + 2
a = 3

Now, we substitute this value of 'a' back into the vertex form of the quadratic function:

f(x) = 3(x - 3)^2 - 2

Therefore, the quadratic function f(x) is f(x) = 3x^2 - 18x + 25.

Note: The given form f(x) = ax^2 + bx + c is known as the standard form of a quadratic function. The vertex form is an alternative representation that provides the vertex of the parabola explicitly.