Two horizontal hyperbolas: algebra and calculator checks

$\dfrac{(x-0)^2}{2^2}-\dfrac{(y-0)^2}{1^2}=1$, so the centre is $(0,0)$ and the vertices are $(-2,0),(2,0)$. Set the right side to zero for asymptotes: $x^2/4-y^2=0$, hence $y^2=x^2/4$ and $y=\pm x/2$. The lines of symmetry are $x=0,y=0$. Draw two horizontal branches approaching these lines. This working fills the Student copy’s blanks using the compared Teacher copy.

$\dfrac{[x-(-4)]^2}{3^2}-\dfrac{(y-2)^2}{4^2}=1$, giving centre $(-4,2)$ and vertices $(-7,2),(-1,2)$. The homogeneous equation is $(x+4)^2/9-(y-2)^2/16=0$, or $16(x+4)^2=9(y-2)^2$. Thus $y-2=\pm\tfrac43(x+4)$ and the asymptotes are $y=\tfrac43x+\tfrac{22}{3}$ and $y=-\tfrac43x-\tfrac{10}{3}$. Symmetry lines: $x=-4,y=2$.

Follow the source table: apps → [2:Conics] → [3:HYPERBOLA]. Press mode for CONIC SETTINGS, choose FUNC (functions) and MAN (manual window settings), then ESC to return. Select [1:], the horizontal form $(X-H)^2/A^2-(Y-K)^2/B^2=1$. Enter A=2, B=1, H=0, K=0. To SOLVE, press alpha, enter. The feature screen shows centre $(0,0)$, vertices $(-2,0),(2,0)$, focus readouts $(-2.236,0)$ and $(2.2361,0)$, and slope $\pm0.5$. The focus readouts are displayed decimals; the exact focus coordinates are $(\pm\sqrt5,0)$ (an authored check). The centre/vertices give symmetry $x=0,y=0$, and the slope gives asymptotes $y=\pm x/2$. Press graph; trace coordinates may be inaccurate. Press zoom for the displayed choices 1:Zoom Conic, 2:Zoom In, 3:Zoom Out, 4:ZBox, 5:ZSquare, 6:ZStandard. Without MAN, the source says a Zoom Conic command is performed by default. In MAN, press window for manual controls; the screenshot has Xmin=−10, Xmax=10, Xscl=1, Ymin=−10, Ymax=10, Yscl=1. These are source display settings, not part of the equation.

Select horizontal form [1:] and enter A=3, B=4, H=−4, K=2. Press alpha, enter to SOLVE. The feature screen shows centre $(-4,2)$, vertices $(-7,2),(-1,2)$, foci $(-9,2),(1,2)$ and slope readout $\pm1.3333$. The exact slopes are $\pm4/3$. The centre/vertices give symmetry $x=-4,y=2$; combine the slopes with the centre to obtain $y-2=\pm\tfrac43(x+4)$, not lines through the origin. Press graph and trace if needed, retaining the source warning about numerical trace accuracy. The exact foci also follow from $\sqrt{3^2+4^2}=5$ (an authored verification).