MHB -4.3.1 find quadratic eq given 3 pts

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To find the quadratic equation given the points (-3,0), (3,0), and (0,-4), the general form y = ax^2 + bx + c can be used. By substituting (0,-4) into the equation, it is determined that c = -4. The roots at x = -3 and x = 3 lead to the equation y = k(x-3)(x+3), and substituting x = 0 gives -9k = -4, resulting in k = 4/9. Thus, the quadratic equation is y = (4/9)(x-3)(x+3). This method effectively combines substitution and root analysis to derive the equation.
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how do you find the quadratic equation given (-3,0) (3,0) and (0,-4)

thus y=k(x-3)(x+3) for the zero's but how do you find k or any other better method of finding the equation

thanks much(Cool)
 
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This can be done by looking at the general form of a quadratic equation: [math]y=ax^2+bx+c[/math]. We need to solve for a,b and c in order to write our equation and we have three points so we can do this through substitution. First use (0,-4) for (x,y) and you get [math]-4=a(0)^2+b(0)+c[/math] which means that c=-4. Now use the other two points the same way and you will have to solve a two variable system of equations for a and b. Once you have a,b and c you have your answer.
 
karush said:
how do you find the quadratic equation given (-3,0) (3,0) and (0,-4)

thus y=k(x-3)(x+3) for the zero's but how do you find k or any other better method of finding the equation
Starting from y=k(x-3)(x+3) (which gives the value 0 when x = 3 or –3), all you need to do is to put x=0 to see that y = –9k when x=0. But you want y to be –4 when x=0. Therefore –9k = –4. So k = 4/9, and $y = \frac49(x-3)(x+3).$
 
karush said:
how do you find the quadratic equation given (-3,0) (3,0) and (0,-4)

Another way (if you have covered the Lagrange Interpolation Polynomial):

$y=0L_1+0L_2-4L_3=-4\dfrac{(x+3)(x-3)}{(0+3)(0-3)}=\dfrac{4}{9}(x+3)(x-3)$
 
Fernando Revilla said:
Another way (if you have covered the Lagrange Interpolation Polynomial):

$y=0L_1+0L_2-4L_3=-4\dfrac{(x+3)(x-3)}{(0+3)(0-3)}=\dfrac{4}{9}(x+3)(x-3)$

no have not heard of it. looks valuable tho so will look it up thanks
 
$y = \dfrac{4}{9}(x-3)(x+3)$

ok I can't seem to write tikx to plot this

$\begin{tikzpicture}[scale=0.50]
%preamble \usepackage{pgfplots}
\begin{axis}[xmin=-3.5, xmax=3.5, ymin=-5, ymax=5, axis lines=middle, ticks=none]
\addplot[
draw = black, smooth, ultra thick,
domain=-4:4,
] {exp((4/9)*(x^2-9)}
foreach \x in {-3,3} { (axis cs:{\x},0) node[below left] {\x} };
\end{axis}
\end{tikzpicture}$
$$y = \dfrac{4}{9}(x-3)(x+3)$$
 
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