SAT Math Practice Question #396 (Hard (800)) | Test Citadel
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SAT Math Difficulty: Hard (800)

Digital SAT Math Practice Question #396

Test your problem-solving accuracy on this official-caliber item. Select an answer choice to check your reasoning instantly.

The quadratic function \( f \) is defined by \( f(x) = a(x - h)^2 + k \), where \( a \), \( h \), and \( k \) are constants. The graph of \( y = f(x) \) in the \(xy\)-plane has a vertex at \( (3, -4) \) and passes through the point \( (5, 8) \). If the graph of \( y = f(x) \) is translated 2 units to the left and 5 units down to create the graph of \( y = g(x) \), what is the \( y \)-intercept of the graph of \( y = g(x) \)?
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Socratic AI Engine Step-by-Step Derivation
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Tactical Insight (Hint 1)

Identify the quadratic form: consider converting to vertex form \( y = a(x-h)^2 + k \) or using \( x = - rac{b}{2a} \) to find the line of symmetry.

Elimination Framework (Hint 2)

Desmos Tactical Shortcut: Plot the left-hand side and right-hand side as separate equations (e.g. \( y_1 = f(x) \) and \( y_2 = g(x) \)). Click the intersection points directly to read exact coordinates.

Masterclass Solution & Distractor Trap Analysis

This question tests your ability to build a quadratic function from a vertex and a point, and then accurately apply geometric transformations. **Step 1: Find the equation of \( f(x) \).** We are given the vertex form: \( f(x) = a(x - h)^2 +...

Distractor Analysis: Trap choice eliminates careless test-takers who confuse roots with coordinates...

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