SAT Question 16988f9c: Two-variable Data Models And Scatterplots answer and explanation with Desmos solution

Choice D is correct. A linear model can be written in the form $y=mx+b$, where $m$ is the slope of the graph of the model in the xy-plane and $(0,b)$ is the y-intercept. The graph of an appropriate linear model for this relationship passes near the points $(1,3)$ and $(9,10)$ in the xy-plane. Two points on a line, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, can be used to find the slope of the line using the slope formula, $m=\frac{y_2-y_1}{x_2-x_1}$. Substituting the points $(1,3)$ and $(9,10)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{10-3}{9-1}$, or $m=0.875$. Therefore, the value of $m$ for an appropriate linear model is approximately $0.875$. Substituting $0.875$ for $m$ in $y=mx+b$ yields $y=0.875x+b$. Since an appropriate linear model passes near the point $(1,3)$, the approximate value of $b$ can be found by substituting $1$ for $x$ and $3$ for $y$ in the equation $y=0.875x+b$, which yields $3=(0.875)(1)+b$, or $3=0.875+b$. Subtracting $0.875$ from both sides of this equation yields $2.125=b$. Therefore, the value of $b$ for an appropriate linear model is approximately $2.125$. Thus, of the given choices, $y=0.9x+2.2$ is the most appropriate linear model for this relationship.

Alternate approach: A linear model can be written in the form $y=mx+b$, where $m$ is the slope of the graph of the model in the xy-plane and $(0,b)$ is the y-intercept. The scatterplot shows that as the x-values of the data points increase, the y-values of the data points increase, which means the graph of an appropriate linear model has a positive slope. Of the given choices, $y=0.9x+2.2$ is the only linear model whose graph has a positive slope.