https://file.aiquickdraw.com/mathbot/file/1724360882266i8r2fhdn.jpg

To find the values of \( p \), \( a \), and \( q \) in the given equation, we need to analyze the hyperbola equation provided:

\[
y = \frac{p}{(x - a)^2} + q
\]

Given that the hyperbola passes through the point \( A(1.5, -3.677) \), we can substitute \( x = 1.5 \) and \( y = -3.677 \) into the equation to find \( p \), \( a \), and \( q \).

### Step 1: Substitute the Point \( A(1.5, -3.677) \)
\[
-3.677 = \frac{p}{(1.5 - a)^2} + q
\]

### Step 2: Analyze the Graph
From the image, we observe that the graph shows a hyperbola centered along a certain vertical line. The asymptote's intersection with the y-axis gives us clues about \( q \), the vertical shift. Similarly, the horizontal shift \( a \) can be estimated by the x-value where the hyperbola changes direction (often near the center of the graph).

#### a. Estimate \( q \)
From the graph, the horizontal asymptote seems to be approaching a specific y-value, likely \( q \). We can estimate \( q \) visually from the graph.

#### b. Estimate \( a \)
\( a \) is the x-value at the center of the hyperbola. By examining the graph, we can estimate this value.

### Step 3: Solve for \( p \)
Once \( a \) and \( q \) are estimated, we can solve for \( p \) using the equation derived above.

Would you like me to estimate these values based on the visual information, or do you have any particular details or zoomed-in sections of the graph that could provide more precise measurements?

Learn how to solve the hyperbola equation y = p / (x - a)^2 + q using a given point. This problem involves analyzing the graph of a hyperbola to determine the constants p, a, and q. Suitable for high school students studying conic sections and graphical analysis.

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