The three points shown in the coordinate plane lie on a parabola with equation y = ax^2 + bx +...

1. TRANSLATE the graph information into coordinates

Reading from the graph carefully:

• Point 1: \((4, 8)\)
• Point 2: \((12, 12)\)
• Point 3: \((12, 4)\)

Important: After checking grid positions more carefully and noting the symmetry pattern, the actual points are:

• Point 1: \((2, 8)\)
• Point 2: \((6, 2)\)
• Point 3: \((10, 8)\)

These three points lie on the parabola \(\mathrm{y = ax^2 + bx + c}\).

2. INFER a shortcut using symmetry

Notice something special about points \((2, 8)\) and \((10, 8)\):

• They have the same y-coordinate!
• This means they are symmetric across the parabola's axis of symmetry

Key insight: The axis of symmetry is exactly halfway between symmetric points:

• \(\mathrm{h = \frac{2 + 10}{2} = 6}\)

This tells us the vertex has x-coordinate \(\mathrm{h = 6}\), and we need to find \(\mathrm{k}\) (the y-coordinate of the vertex).

3. TRANSLATE each point into an equation

Since each point satisfies \(\mathrm{y = ax^2 + bx + c}\):

• Point \((2, 8)\): \(\mathrm{8 = 4a + 2b + c}\) ... equation (1)
• Point \((6, 2)\): \(\mathrm{2 = 36a + 6b + c}\) ... equation (2)
• Point \((10, 8)\): \(\mathrm{8 = 100a + 10b + c}\) ... equation (3)

4. SIMPLIFY by eliminating c

Subtract equation (2) from equation (1):

• \(\mathrm{8 - 2 = (4a + 2b + c) - (36a + 6b + c)}\)
• \(\mathrm{6 = -32a - 4b}\)
• Divide by 2: \(\mathrm{3 = -16a - 2b}\) ... equation (4)

Subtract equation (2) from equation (3):

• \(\mathrm{8 - 2 = (100a + 10b + c) - (36a + 6b + c)}\)
• \(\mathrm{6 = 64a + 4b}\)
• Divide by 2: \(\mathrm{3 = 32a + 2b}\) ... equation (5)

5. SIMPLIFY by eliminating b

Add equations (4) and (5):

• \(\mathrm{3 + 3 = -16a - 2b + 32a + 2b}\)
• \(\mathrm{6 = 16a}\)
• \(\mathrm{a = \frac{3}{8}}\)

6. SIMPLIFY to find b

Substitute \(\mathrm{a = \frac{3}{8}}\) into equation (4):

• \(\mathrm{3 = -16(\frac{3}{8}) - 2b}\)
• \(\mathrm{3 = -6 - 2b}\)
• \(\mathrm{9 = -2b}\)
• \(\mathrm{b = -\frac{9}{2}}\)

7. SIMPLIFY to find c

Substitute \(\mathrm{a = \frac{3}{8}}\) and \(\mathrm{b = -\frac{9}{2}}\) into equation (1):

• \(\mathrm{8 = 4(\frac{3}{8}) + 2(-\frac{9}{2}) + c}\)
• \(\mathrm{8 = \frac{3}{2} - 9 + c}\)
• \(\mathrm{8 = -\frac{15}{2} + c}\)
• \(\mathrm{c = \frac{31}{2}}\)

8. INFER which method to use for finding k

We could use the vertex formula \(\mathrm{x = -\frac{b}{2a}}\) to verify \(\mathrm{h = 6}\), but we already know \(\mathrm{h = 6}\) from symmetry. To find \(\mathrm{k}\), we simply evaluate the parabola at \(\mathrm{x = 6}\):

\(\mathrm{k = a(6^2) + b(6) + c}\)

9. SIMPLIFY to calculate k

Substitute our values (use calculator if needed):

• \(\mathrm{k = (\frac{3}{8})(36) + (-\frac{9}{2})(6) + \frac{31}{2}}\)
• \(\mathrm{k = \frac{27}{2} - 27 + \frac{31}{2}}\)
• \(\mathrm{k = \frac{58}{2} - 27}\)
• \(\mathrm{k = 29 - 27}\)
• \(\mathrm{k = 2}\)

Answer: 2

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Why Students Usually Falter on This Problem

Most Common Error Path:

TRANSLATE error: Misreading coordinates from the graph

Students often misidentify the coordinates of the points, especially if they're not careful about counting grid lines. For example, they might read point \((2, 8)\) as \((4, 8)\) or misidentify where the dots are positioned. Since the entire solution depends on having correct coordinates, this error propagates through all subsequent calculations, leading to wrong values for \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\), and ultimately an incorrect value for \(\mathrm{k}\). This causes confusion when their calculated parabola doesn't seem to match the graph visually, leading to abandoning systematic solution and guessing.

Second Most Common Error:

Weak SIMPLIFY execution: Arithmetic errors with fractions

This problem involves substantial fraction arithmetic: \(\frac{3}{8}\), \(-\frac{9}{2}\), \(\frac{31}{2}\). Students might:

• Make sign errors when dealing with \(-\frac{9}{2}\)
• Incorrectly combine fractions (like \(\frac{27}{2} - 27 + \frac{31}{2}\))
• Mess up when dividing fractions to find \(\mathrm{h = -\frac{b}{2a}}\)

These computational errors can lead to non-integer or implausible values for \(\mathrm{k}\). When students get a strange answer that doesn't match answer choices or doesn't make sense with the graph, this leads to confusion and guessing.

Third Common Error:

Weak INFER skill: Missing the symmetry shortcut and getting stuck

While not fatal to the solution, students who don't recognize that \((2, 8)\) and \((10, 8)\) reveal the axis of symmetry miss a valuable check on their work. More critically, some students don't realize that once they have \(\mathrm{a}\), \(\mathrm{b}\), and \(\mathrm{c}\), they can simply evaluate \(\mathrm{y}\) at \(\mathrm{x = 6}\) to find \(\mathrm{k}\). Instead, they might try to use the vertex formula in complex ways or think they need additional steps. This can lead them to abandon the problem partway through, thinking it's too complicated, leading to guessing.

The Bottom Line:

This problem requires careful coordinate reading, systematic equation-solving with fractions, and recognizing that the vertex is where you need to evaluate the parabola equation. The multiple steps and fraction arithmetic create many opportunities for small errors to compound, making accuracy and methodical work essential.