y = ax^2 - cIn the equation above, a and c are positive constants. How many times does the graph...

1. TRANSLATE the problem information

• Given equations:
  • \(\mathrm{y = ax^2 - c}\) (where \(\mathrm{a \gt 0}\), \(\mathrm{c \gt 0}\))
  • \(\mathrm{y = a + c}\)
• Find: Number of intersection points

2. INFER the approach

• To find where graphs intersect, set the y-values equal to each other
• This gives us: \(\mathrm{ax^2 - c = a + c}\)
• We need to solve for x to find intersection points

3. SIMPLIFY the intersection equation

• Start with: \(\mathrm{ax^2 - c = a + c}\)
• Add c to both sides: \(\mathrm{ax^2 = a + 2c}\)
• Divide by a: \(\mathrm{x^2 = \frac{a + 2c}{a} = 1 + \frac{2c}{a}}\)

4. INFER the number of solutions

• Since \(\mathrm{a \gt 0}\) and \(\mathrm{c \gt 0}\), we know that \(\mathrm{\frac{2c}{a} \gt 0}\)
• Therefore: \(\mathrm{x^2 = 1 + \frac{2c}{a} \gt 1 \gt 0}\)
• A positive value under the square root gives us \(\mathrm{x = \pm\sqrt{1 + \frac{2c}{a}}}\)
• This means exactly two real solutions

5. VISUALIZE to confirm

• \(\mathrm{y = ax^2 - c}\): upward-opening parabola with vertex at \(\mathrm{(0, -c)}\)
• Since \(\mathrm{c \gt 0}\), the vertex is below the x-axis
• \(\mathrm{y = a + c}\): horizontal line above the x-axis (since \(\mathrm{a + c \gt 0}\))
• An upward parabola below the x-axis intersects a horizontal line above it at exactly two points

Answer: C. Two

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that finding intersections requires setting the equations equal to each other. Instead, they might try to analyze each equation separately or look for some other relationship between the constants a and c.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set \(\mathrm{ax^2 - c = a + c}\) but make algebraic errors when solving, such as forgetting to move all constants to one side or incorrectly manipulating the equation to isolate \(\mathrm{x^2}\).

This may lead them to get an incorrect expression that doesn't clearly show two solutions, causing them to select Choice B (One) or abandon the algebraic approach and guess.

The Bottom Line:

This problem requires students to bridge algebraic manipulation with graphical understanding. The key insight is recognizing that intersection problems are fundamentally about solving equations, and that the resulting quadratic equation will have two solutions when the discriminant (or in this case, the value under the square root) is positive.