Question 1118375
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You are given the form for the answer: {{{y = a(x-h)^2+k}}}<br>
In that form, the vertex of the parabola is (h,k).  Since you are given that the vertex is (-3,35), the equation is of the form<br>
{{{y = a(x+3)^2+35}}}<br>
The usual way of finding the value of the coefficient a is to plug in the x and y values of the other known point on the parabola:<br>
{{{23 = a(-5+3)^2+35 = a(-2)^2+35}}}
{{{23 = 4a+35}}}
{{{-12 = 4a }}}
{{{a = -12/4 = -3}}}<br>
Then the complete equation is<br>
{{{y = -3(x+3)^2+35}}}<br>
If you understand a bit about parabolas and quadratic functions, you can find the value of a without plugging in the x and y values of the other point.  Here is the shortcut; you can see where the shortcut comes from by looking at the detailed steps used above to find the value of a.<br>
(1) The given point that is not the vertex is 2 units away in the x direction from the vertex (right or left doesn't matter because of the symmetry of the parabola).
(2) In the equation, the x term is squared; so square the distance from (1): 2^2 = 4.
(3) The y value, from the vertex to the other point, changed by -12, from 35 to 23.
(4) The coefficient a is -12/4 = 3.<br>
This is similar to what you know about slopes of linear equations.<br>
In the linear equation y = ax+b, the coefficient a is the slope; it is the change in y divided by the change in x.<br>
For a quadratic equation, in which the x term is SQUARED, the coefficient a is the change in y divided by the SQUARE of the change in x.<br>
To go through the calculation again for your example, we have:
change in x from vertex to other point: 2
change in y from vertex to other point: -12
coefficient a: -12/(2^2) = -12/4 = -3