SAT Question de91a6fc: Nonlinear Functions answer and explanation with Desmos solution

It’s given that the number of bowls sold can be modeled by a quadratic function based on the price, where a maximum of $9$ bowls were sold for $$10$ each and $5$ bowls were sold for $$12$ each. This quadratic function can be defined by an equation of the form $N\left(p\right)=a(p-h{)}^2+k$, where $N(p)$ is the number of bowls sold when the price is $$p$ each, and $a$, $h$, and $k$ are constants such that the function reaches its maximum value, $k$, when $p=h$. Since the model indicates the maximum number of bowls sold is $9$ when the price is $$10$ each, $N(p)$ reaches its maximum value, $9$, when $p=10$. Therefore, $k=9$ and $h=10$. Substituting $10$ for $h$ and $9$ for $k$ in the function $N\left(p\right)=a(p-h{)}^2+k$ yields $N\left(p\right)=a(p-10{)}^2+9$. Since the model indicates the number of bowls sold is $5$ when the price is $$12$ each, the value of $N(p)$ is $5$ when $p=12$. Substituting $12$ for $p$ and $5$ for $N(p)$ in the equation $N\left(p\right)=a(p-10{)}^2+9$ yields $5=a(12-10{)}^2+9$, or $5=4a+9$. Subtracting $9$ from both sides of this equation yields $-4=4a$. Dividing both sides of this equation by $4$ yields $-1=a$. Therefore, the model can be represented by the equation $N\left(p\right)=-(p-10{)}^2+9$. Substituting $8$ for $p$ in this equation yields $N\left(8\right)=-(8-10{)}^2+9$, or $N(8)=-4+9$, or $N(8)=5$. Therefore, according to the model, the number of bowls sold for $$8$ each was $5$.