graph the following equation and discuss briefly the steps. 1. 9x²-16y²-144=0
Answer (1)
Step-by-step explanation:To graph the equation \( 9x^2 - 16y^2 - 144 = 0 \), we first need to rewrite it in a standard form. Here are the steps involved:### Step 1: Rearranging the EquationStart by moving the constant term to the other side:\[9x^2 - 16y^2 = 144\]### Step 2: Dividing by 144Next, divide every term by 144 to get the equation in the standard form of a hyperbola:\[\frac{9x^2}{144} - \frac{16y^2}{144} = 1\]This simplifies to:\[\frac{x^2}{16} - \frac{y^2}{9} = 1\]### Step 3: Identify the Standard FormNow we have the equation in the standard form of a hyperbola:\[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\]where \( a^2 = 16 \) and \( b^2 = 9 \). Thus, \( a = 4 \) and \( b = 3 \).### Step 4: Determine the Center and AxesThe center of the hyperbola is at the origin (0,0) because there are no \( h \) or \( k \) terms in the equation. - The transverse axis (which is horizontal since the \( x^2 \) term is positive) has a length of \( 2a = 8 \).- The conjugate axis (vertical) has a length of \( 2b = 6 \).### Step 5: Plotting the AsymptotesThe equations for the asymptotes of this hyperbola are given by:\[y = \pm \frac{b}{a} x = \pm \frac{3}{4} x\]### Step 6: Sketch the Graph1. Begin by plotting the center at (0,0).2. Next, plot points for the vertices along the transverse axis at (4,0) and (-4,0).3. Plot the points for the vertices along the conjugate axis at (0, 3) and (0, -3).4. Draw dashed lines for the asymptotes through the origin, ensuring they extend in both directions.5. Finally, sketch the branches of the hyperbola that open left and right from the center.### ConclusionThe resulting graph represents a hyperbola centered at the origin (0,0) with horizontally opening branches. The vertices are at (4,0) and (-4,0), while the asymptotes guide the shape of the graph. This process allows for a clear understanding of how the hyperbola is oriented and defined by the given equation. If you would like to delve into any specific part or need illustrations, let me know!
