Respuesta :
[tex]\bf \qquad \textit{parabola vertex form}
\\\\\\
y=a(x-{{ h}})^2+{{ k}}\\
x=a(y-{{ k}})^2+{{ h}}\qquad\qquad vertex\ ({{ h}},{{ k}})[/tex]
so.. based on the provided points, we can more or less make that is a vertical parabola, so "x" is the squared variable, so we'll use the 1st of the above ones
[tex]\bf y=a(x-{{ h}})^2+{{ k}}\qquad \begin{cases} vertex\ (-4,6)\to \begin{cases} h=-4\\ k=6 \end{cases}\\\\\\ \textit{another point}\\ (-3,14)\to \begin{cases} x=-3\\ y=14 \end{cases} \end{cases} \\\\\\ y=a(x-(-4))^2+6\implies y=a(x+4)^2+6\impliedby \textit{what's "a"?} \\\\\\ \underline{14}=a(\underline{-3}+4)^2+6[/tex]
solve for "a"
so.. based on the provided points, we can more or less make that is a vertical parabola, so "x" is the squared variable, so we'll use the 1st of the above ones
[tex]\bf y=a(x-{{ h}})^2+{{ k}}\qquad \begin{cases} vertex\ (-4,6)\to \begin{cases} h=-4\\ k=6 \end{cases}\\\\\\ \textit{another point}\\ (-3,14)\to \begin{cases} x=-3\\ y=14 \end{cases} \end{cases} \\\\\\ y=a(x-(-4))^2+6\implies y=a(x+4)^2+6\impliedby \textit{what's "a"?} \\\\\\ \underline{14}=a(\underline{-3}+4)^2+6[/tex]
solve for "a"
Answer: The coefficient of squared expression is,
[tex]8\text{ or } -\frac{1}{36}[/tex]
Step-by-step explanation:
There can be two cases,
Case 1 : If the parabola is along y-axis,
Since, the standard equation of parabola along y-axis is,
[tex]y=a(x-h)^2+k[/tex]
Where, (h,k) is the vertex of the parabola,
Here, (h,k) = (-4,6),
By substituting the values,
[tex]y=a(x-(-4))^2+6[/tex]
[tex]y=a(x+4)^2+6[/tex]
Given, another point of the parabola is (-3, 14),
⇒ (-3, 14) must satisfy the above equation,
[tex]14=a(-3+4)^2+6[/tex]
[tex]14=a+6[/tex]
[tex]\implies a=8[/tex]
Hence, the coefficient of squared expression is 8.
Case 2 : If the parabola is along x-axis,
The standard equation of parabola along x-axis is,
[tex]x=a(y-h)^2+k[/tex]
Where, (h,k) is the vertex of the parabola,
Here, (h,k) = (-4,6),
By substituting the values,
[tex]x=a(y-(-4))^2+6[/tex]
[tex]x=a(y+4)^2+6[/tex]
Given, another point of the parabola is (-3, 14),
⇒ (-3, 14) must satisfy the above equation,
[tex]-3=a(14+4)^2+6[/tex]
[tex]-3-6=a(18)^2[/tex]
[tex]-9=324 a[/tex]
[tex]\implies a=-\frac{9}{324}=-\frac{1}{36}[/tex]
Hence, the coefficient of squared expression is -1/36.
