# Math Help - Help with parabola questions for project

1. ## Help with parabola questions for project

i am doing a project involving the Gateway Arch as a parabola. I need to do a series of questions most of which i can do but cant do without 4 questions that i have no idea how to do. for my coordinates i have (0,0) (315,630) (630,0). any help would be great since this isnt my strong topics. The questions i need help with are

Providing and equation that models my parabola

My x intercepts and what it means in the context of the situation

My y intercepts and what it means in the context of the situation

The practical domain/range

2. Hello Ryan7123

Welcome to Math Help Forum!
Originally Posted by Ryan7123
... equation that models my parabola
Any parabola with its axis parallel to the y-axis (like the one you've got here) will have an equation that looks like the standard quadratic:

$y = ax^2 + bx+c$

You know three points that lie on the parabola. So they tell you three pairs of values of $x$ and $y$. So plug these into this equation, and you get:

First, $(0,0): 0 = a.0 + b.0 + c \Rightarrow c = 0$

So the equation becomes $y = ax^2 + bx$.

Then the other two points:

$(315, 630): 630 = a.315^2 + b.315$ (1)

$(630,0): 0 = a.630^2 + b.630 \Rightarrow b = -630a$

Substitute into (1):

$630 = 315^2a - 630.315a = 315(315 - 630)a = 315(-315)a$

$\Rightarrow a = -\frac{630}{315.315}= -\frac{2}{315}$

$\Rightarrow b = -630a = 4$

So there's your equation: $y = -\frac{2}{315}x^2 + 4x$

My x intercepts and what it means in the context of the situation
The x intercepts are the values of x where the curve crosses the x-axis. They are 0 and 630. So if we assume that the x-axis represents the ground, and the units are measured in feet (which I don't know for sure - I'm only guessing!), this means that the arch is 630 feet wide at its base.

My y intercepts and what it means in the context of the situation
The y-intercept is the value of y at the point where the curve crosses the y-axis, which is zero. Again, if the x-axis represents the ground, this means that the height of the arch above the ground at any point is given by the value of y, and the highest point will be at (315, 630) - so it's 630 feet high.

The practical domain/range
The domain is the set of values of x that you will need to use in the problem, and the range is the set of values of y. In each case, then, these are numbers from 0 to 630.