At how many points do the graphs of the equations y = x^2 and y = 2x - 1 intersect...

1. INFER the solution strategy

• Key insight: Intersection points occur where the two graphs have the same x and y coordinates
• This means we need to find where \(\mathrm{y = x²}\) and \(\mathrm{y = 2x - 1}\) are simultaneously true
• Strategy: Set the right sides of both equations equal to each other

2. TRANSLATE the intersection condition into algebra

• Since both expressions equal y, we can set them equal:

\(\mathrm{x² = 2x - 1}\)

3. SIMPLIFY to standard quadratic form

• Move all terms to one side: \(\mathrm{x² - 2x + 1 = 0}\)
• This is now in the standard form \(\mathrm{ax² + bx + c = 0}\)

4. SIMPLIFY by factoring

• Notice that \(\mathrm{x² - 2x + 1}\) is a perfect square trinomial
• Factor: \(\mathrm{(x - 1)² = 0}\)
• Solve: \(\mathrm{x = 1}\) (this is a double root)

5. SIMPLIFY to find the y-coordinate

• Substitute \(\mathrm{x = 1}\) into either original equation
• Using \(\mathrm{y = 2x - 1}\): \(\mathrm{y = 2(1) - 1 = 1}\)
• The intersection point is \(\mathrm{(1, 1)}\)

6. INFER the final answer

• Since we found exactly one value of x, there is exactly one intersection point
• The parabola and line are tangent to each other at \(\mathrm{(1, 1)}\)

Answer: B. 1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may think that since quadratic equations typically have two solutions, there must be two intersection points. They see the quadratic \(\mathrm{x² - 2x + 1 = 0}\) and expect two different x-values, not recognizing that \(\mathrm{(x - 1)² = 0}\) gives a double root representing a single point of tangency.

This leads to confusion about whether the answer should be 1 or 2, often causing them to guess Choice C (2).

Second Most Common Error:

Poor SIMPLIFY execution: Students make algebraic errors when rearranging \(\mathrm{x² = 2x - 1}\) to standard form. Common mistakes include getting the signs wrong (like \(\mathrm{x² + 2x - 1 = 0}\)) or making calculation errors during factoring.

These algebraic errors lead to incorrect quadratic equations that don't factor properly, causing students to get stuck and randomly select an answer.

The Bottom Line:

The key challenge is recognizing that a double root in a quadratic equation corresponds to exactly one intersection point where the graphs are tangent to each other, not two separate intersection points.