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) Ah, the graph of f(x) = x^2 being expanded vertically by a factor of 4? Well, imagine if the graph of f(x) = x^2 went to the gym and decided to bulk up! So, by multiplying the function by 4, we give it some protein shakes and weights, making it f(x) = 4x^2. Now it's like the graph has been doing squats and lunges, hitting the gym hard and stretching itself vertically. That's how the graph gets expanded vertically by a factor of 4. You go, graph!
2) Ah, the graph of y=(x^2-9)/(x^2-4) having vertical asymptotes at x = ±2? It's like the graph has run into a pair of vertical walls. It can't cross these walls, no matter how hard it tries. Those walls are actually vertical asymptotes at x = ±2.
Think of it this way: when x gets really close to 2 or -2, the denominator (x^2-4) becomes very, very close to zero. But dividing by zero? That's a big no-no in the land of mathematics. So as x approaches 2 or -2, the graph of the function goes wacky and shoots off to positive or negative infinity, like a rocket trying to escape the gravitational pull of a black hole. Hence, we get vertical asymptotes at x = ±2. Watch out for those black holes, graph!