For the linear function h , the graph of y = h x in the xy

Choice B is correct. It’s given that the graph of the linear function $h$, where $y=h\left(x\right)$, passes through the points $\left(7,21\right)$ and $\left(9,25\right)$ in the xy-plane. An equation defining $h$ can be written in the form $y=mx+b$, where $y=h\left(x\right)$, $m$ represents the slope of the graph in the xy-plane, and $b$ represents the y-coordinate of the y-intercept of the graph. The slope can be found using any two points, $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, and the formula $m=\frac{\left(y_2-y_1\right)}{\left(x_2-x_1\right)}$. Substituting $\left(7,21\right)$ and $\left(9,25\right)$ for $\left(x_1,y_1\right)$ and $\left(x_2,y_2\right)$, respectively, in the slope formula yields $m=\frac{25-21}{9-7}$, which is equivalent to $m=\frac{4}{2}$, or $m=2$. Substituting $2$ for $m$ and $\left(7,21\right)$ for $\left(x,y\right)$ in the equation $y=mx+b$ yields $21=\left(2\right)\left(7\right)+b$, or $21=14+b$. Subtracting $14$ from each side of this equation yields $7=b$. Substituting $2$ for $m$ and $7$ for $b$ in the equation $y=mx+b$ yields $y=2x+7$. Since $y=h\left(x\right)$, it follows that the equation that defines $h$ is $h\left(x\right)=2x+7$.

Choice A is incorrect. For this function, the graph of $y=h\left(x\right)$ in the xy-plane would pass through $\left(7,0\right)$, not $\left(7,21\right)$, and $\left(9,1\right)$, not $\left(9,25\right)$.

Choice C is incorrect. For this function, the graph of $y=h\left(x\right)$ in the xy-plane would pass through $\left(7,70\right)$, not $\left(7,21\right)$, and $\left(9,84\right)$, not $\left(9,25\right)$.

Choice D is incorrect. For this function, the graph of $y=h\left(x\right)$ in the xy-plane would pass through $\left(7,88\right)$, not $\left(7,21\right)$, and $\left(9,106\right)$, not $\left(9,25\right)$.