Question 34·SAT Desmos Regression Practice·Advanced Math

Since $p$ is a quadratic polynomial, write $p(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are its coefficients, the numbers multiplying its terms. Substitute that into the given definition: $q(x)=\frac{ax^2+bx+c-5}{x+1}$. The table gives values of $q$, so fitting $ax^2+bx+c$ directly to those values would fit the wrong function.

Enter the table’s inputs in $x_1$ and its $q(x)$ values in $y_1$. Then type $y_1\sim\frac{ax_1^2+bx_1+c-5}{x_1+1}$. The tilde tells Desmos to find coefficients that fit the rows. Under PARAMETERS, Desmos shows $a=1.5$, $b=4$, and $c=0$, so the hidden quadratic is $p(x)=\frac32x^2+4x$.

The vertex is the turning point of the parabola. For $ax^2+bx+c$, it lies on the symmetry line $x=-\frac{b}{2a}$, which splits the parabola into mirror halves. Type $-\frac{b}{2a}$ using the fitted letters. Desmos shows $-1.333\ldots$; its fraction button gives $-\frac43$.

Define $p(x)=ax^2+bx+c$, then type $p\left(-\frac{b}{2a}\right)$. Desmos shows $-2.666\ldots$; its fraction button gives $-\frac83$. Use $p$, not $q$, for this output because the question asks for the vertex of $y=p(x)$. Its coordinates are $\left(-\frac43,-\frac83\right)$. Choice C.