QUESTION STEM:A linear function h has slope 3/2.The point \((1, 3)\) lies on the graph of h.For what value of...

1. TRANSLATE the problem information

• Given information:
  • Linear function h has \(\mathrm{slope} = \frac{3}{2}\)
  • Point \((1, 3)\) lies on the graph
  • Need x-value where \(\mathrm{h(x)} = 6\)

2. INFER the approach

• Since we have a linear function with known slope, we can use slope-intercept form
• Strategy: Find the complete function first, then solve for x

3. Set up the linear function

• Linear function form: \(\mathrm{h(x)} = \mathrm{mx} + \mathrm{b}\)
• With slope \(\mathrm{m} = \frac{3}{2}\): \(\mathrm{h(x)} = \frac{3}{2}\mathrm{x} + \mathrm{b}\)
• Still need to find b (the y-intercept)

4. SIMPLIFY to find the y-intercept

• Use the given point \((1, 3)\):
• \(\mathrm{h(1)} = 3\), so: \(\frac{3}{2}(1) + \mathrm{b} = 3\)
• \(\frac{3}{2} + \mathrm{b} = 3\)
• \(\mathrm{b} = 3 - \frac{3}{2} = \frac{6}{2} - \frac{3}{2} = \frac{3}{2}\)

5. Write the complete function

• \(\mathrm{h(x)} = \frac{3}{2}\mathrm{x} + \frac{3}{2}\)

6. SIMPLIFY to solve for x when h(x) = 6

• Set up equation: \(\frac{3}{2}\mathrm{x} + \frac{3}{2} = 6\)
• Subtract \(\frac{3}{2}\): \(\frac{3}{2}\mathrm{x} = 6 - \frac{3}{2} = \frac{9}{2}\)
• Divide by \(\frac{3}{2}\): \(\mathrm{x} = \frac{9}{2} \div \frac{3}{2} = \frac{9}{2} \times \frac{2}{3} = 3\)

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students try to solve for x immediately without first finding the complete linear function. They might attempt to use just the slope and the condition \(\mathrm{h(x)} = 6\), not realizing they need the y-intercept first. This leads to confusion and guessing since they can't set up a proper equation.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify the need to find b but make arithmetic errors with fractions. Common mistakes include:

• Calculating \(\mathrm{b} = 3 - \frac{3}{2}\) as \(\frac{3}{2}\) instead of \(\frac{3}{2}\)
• Making sign errors when isolating x
• Converting division by a fraction incorrectly

This may lead them to get an incorrect numerical answer.

The Bottom Line:

This problem tests whether students understand that finding a specific input value requires knowing the complete function first - you can't skip the step of finding the y-intercept just because you have the slope.