quadratic functions: how do you know how wide the parabola is supposed to be?

Question

my equation is:

f:x --> -1/2x^2

what i have so far:

a maximum
axis of symmetry = 0
vertex = 0

i thought that a (ax^2+bx+c) was supposed to determine the width...? the [sample] answers in the back of the book say it's +/- 3 wide.

go easy on me - it's my first time with these :P

yeah, i'm trying to graph it, but my textbook is practically chinese for all i can understand.

Answer 1

I presume you're trying to sketch it and not sure how wide or narrow to go.

So for you only have the point (0,0).
Just calculate an extra few points.
eg. when x = +/-1, y = -1/2
So now you have the points (-1, -1/2) and (1,-1/2)

Now it becomes a lot easier to sketch.

Answer 2

Your a value determines your width of the parabola most of the times, but b and c also has some effect on it. Overall, after plotting all points, you can see that they are distanced apart in either a short distance or long distance. If it is distanced very short, it is not wide and if it is distanced long it is wide. Quite obvious isn't it? Now you said the lower the a value, you wider it is. Well, if the a value were negative, it would have the same width as it would as its absolute values, depending on whether or not it has a b or c value. If it has either b or c or both, they wouldn't be the same. Now why would I say that? This would be only a little related to the quadratic equation, but when you graph the y value by plugging in a x value, you can see that you grow 2 curves that meet at the vertex. a and b are coefficients and c is a constant. If a were larger, the y value would go crazy high even though the x value is very low. This pattern shows that it is not wide. What if a was low? The x value would be very high although the y value is very low. This makes it wide. By this is affected when you have a b or c value because if you had an a value as negative and another a value as the absolute value of it, and you b or c value is positive. The negative a value would sort of cancel each other out, while the absolute value one wouldn't. This can seem pretty similar to slope, but is completely different - defining width is kind of little y per unit of x. The higher the value you get, the less the width.

Answer 3

width of a parabola depends on where you measure it, since it keeps getting wider the farther you go from the vertex.

parent function just squares x to get y, so plotting from the vertex you go over 1 (left AND right), up 1, over 2 (again from vertex), up 4, over 3, up 9. whatever the "a" value in the equation, that multiplies the "up" from the parent function. yours has vertex at origin, so when you go over 1 (±), you go down (negative up) 1/2 of 1², and when you go over 2, you go down 1/2 of 2².