1)Which of the following results in the graph of f(x) = x2 being expanded vertically by a factor of 4? the answer is

f(x) = 4x2. can you explain how.

2)Which is true for the graph of y=(x^2-9)/(x^2-4). the answer is vertical asymptotes at x = ±2 . please explain.

1) To expand a function vertically by a factor of 4, we need to multiply the function by 4. In this case, we have the function f(x) = x^2 and we want to expand it vertically by a factor of 4. So, we can multiply it by 4 to get f(x) = 4x^2.

To understand why this results in a vertical expansion, let's compare the graphs of f(x) = x^2 and f(x) = 4x^2. When you plug in different values of x into both functions, you will notice that for any given x, f(x) = 4x^2 will always be greater than f(x) = x^2. This means that the graph of f(x) = 4x^2 will be vertically stretched along the y-axis, making it 4 times taller than the original graph of f(x) = x^2.

2) The given function is y = (x^2 - 9) / (x^2 - 4). To find the vertical asymptotes of this function, we need to determine the values of x for which the denominator of the fraction becomes zero.

Setting the denominator (x^2 - 4) equal to zero, we get:
x^2 - 4 = 0

This equation can be factored as a difference of squares:
(x - 2)(x + 2) = 0

Solving for x, we find two possible values: x = 2 and x = -2. These values represent vertical asymptotes of the function.

To understand why these values result in vertical asymptotes, let's analyze the behavior of the function as x approaches these values. As x approaches 2 or -2, the denominator of the fraction becomes extremely close to zero, but it never actually reaches zero. However, the numerator (x^2 - 9) does not become zero. As a result, the function becomes extremely large or small, and it approaches infinity or negative infinity as x gets closer to 2 or -2.

Therefore, we conclude that the graph of y = (x^2 - 9) / (x^2 - 4) has vertical asymptotes at x = ±2.