To solve the problem of finding the value of \( p \) in terms of \( q \) for the graphs \( y = px^2 + q \) and \( y = 2x^2 - 1 \) that intersect at \( x = 2 \), we will follow these steps: ### Step 1: Set the equations equal to each other at the point of intersection Since the two graphs intersect at \( x = 2 \), we can set their equations equal to each other at this point: \[ px^2 + q = 2x^2 - 1 \] ### Step 2: Substitute \( x = 2 \) into the equations Now, we substitute \( x = 2 \) into both equations: 1. For the first equation \( y = px^2 + q \): \[ y = p(2^2) + q = 4p + q \] 2. For the second equation \( y = 2x^2 - 1 \): \[ y = 2(2^2) - 1 = 2(4) - 1 = 8 - 1 = 7 \] ### Step 3: Set the two expressions for \( y \) equal to each other Now we have: \[ 4p + q = 7 \] ### Step 4: Solve for \( p \) in terms of \( q \) To isolate \( p \), we can rearrange the equation: \[ 4p = 7 - q \] \[ p = \frac{7 - q}{4} \] ### Conclusion: Identify the correct option From our calculation, we find that: \[ p = \frac{7 - q}{4} \] This matches option **B**. ### Step 5: Analyze the other options Now, let's briefly analyze the other options to understand why they are incorrect: - **Option A: \(\frac{q - 8}{7}\)** This option suggests a different relationship between \( p \) and \( q \) that does not satisfy the equation we derived. - **Option C: \(\frac{8 - q}{2}\)** This option also does not align with our derived equation. If we substitute values, it does not yield the correct intersection point. - **Option D: \(\frac{7 + q}{8}\)** This option suggests a positive correlation between \( p \) and \( q \) that contradicts our derived relationship. ### Summary of Key Points - We set the two equations equal at the intersection point \( x = 2 \). - We substituted \( x = 2 \) into both equations to find their \( y \)-values. - We derived the equation \( 4p + q = 7 \) and solved for \( p \). - The correct answer is \( p = \frac{7 - q}{4} \), which corresponds to option **B**. ### Revision Summary - To find intersections, set the equations equal at the given \( x \)-value. - Substitute the \( x \)-value into both equations to find corresponding \( y \)-values. - Rearrange the resulting equation to isolate the variable of interest. - Always check other options to confirm the correctness of your answer.