Find the equation of the hyperbola given its graph, vertical asymptote at x=0, and passing through points…

Question

Find the equation of the hyperbola given its graph, vertical asymptote at x=0, and passing through points (−4, 0) and (2, 3).

ScanSolve AI Tutor · Accepted Answer

To determine the equation of the hyperbolic function shown in the graph, we use the general form $y = \frac{a}{x+p} + q$ .

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 $y = \frac{a}{x} + q$ .
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:
$0 = \frac{a}{-4} + q$ $q = \frac{a}{4} \quad \text{(Equation 1)}$
Substitute point B $(2, 3)$ into the simplified equation:
$3 = \frac{a}{2} + q \quad \text{(Equation 2)}$
Solve the system of equations for $a$ and $q$ .
Substitute Equation 1 into Equation 2:
$3 = \frac{a}{2} + \frac{a}{4}$
Find a common denominator for the fractions:
$3 = \frac{2a}{4} + \frac{a}{4}$ $3 = \frac{3a}{4}$
Multiply both sides by 4:
$12 = 3a$
Divide by 3:
$a = 4$
Substitute the value of $a$ back into Equation 1 to find $q$ .
$q = \frac{a}{4} = \frac{4}{4}$ $q = 1$
Write the final equation of the hyperbola.
Substitute the values of $a=4$ and $q=1$ into $y = \frac{a}{x} + q$ :
$y = \frac{4}{x} + 1$

Find the equation of the hyperbola given its graph, vertical asymptote at x=0, and passing through points (-4, 0) and (2, 3).