This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To determine the equation of the hyperbolic function shown in the graph, we use the general form y = (a)/(x+p) + q. Step 1: Identify the vertical asymptote from the graph. The graph shows a vertical asymptote at x=0 (the y-axis). In the general form, the vertical asymptote is x=-p. Therefore, -p=0, which means p=0. The equation simplifies to y = (a)/(x) + q. Step 2: Use the given points to form a system of equations. The graph provides two points on the hyperbola: A (-4, 0) and B (2, 3). Substitute point A (-4, 0) into the simplified equation: 0 = (a)/(-4) + q q = (a)/(4) (Equation 1) Substitute point B (2, 3) into the simplified equation: 3 = (a)/(2) + q (Equation 2) Step 3: Solve the system of equations for a and q. Substitute Equation 1 into Equation 2: 3 = (a)/(2) + (a)/(4) Find a common denominator for the fractions: 3 = (2a)/(4) + (a)/(4) 3 = (3a)/(4) Multiply both sides by 4: 12 = 3a Divide by 3: a = 4 Step 4: Substitute the value of a back into Equation 1 to find q. q = (a)/(4) = (4)/(4) q = 1 Step 5: Write the final equation of the hyperbola. Substitute the values of a=4 and q=1 into y = (a)/(x) + q: y = (4)/(x) + 1 The equation of the hyperbolic function is: y = (4)/(x) + 1 That's 2 down. 3 left today — send the next one.