Question

drag each tile to the correct box. not all tiles will be used.

find the increasing functions. then arrange the increasing functions in order from least to greatest rate of change.

\\y = \frac{5}{2}x + 10\\
\\y = -\frac{1}{2}x + \frac{1}{2}\\
\\y = \frac{3}{2}x - \frac{11}{2}\\
\\y = \frac{1}{2}x - 2\\
\\y = \frac{4}{3}x - \frac{7}{3}\\
\\y = \frac{3}{4}x - 10\\

Explanation:

Identify increasing functions

Increasing linear functions have a positive slope (rate of change):

• \(y = \frac{5}{2}x + 10\) (Slope: \(\frac{5}{2} = 2.5\)) -> Increasing
• \(y = \frac{3}{4}x - 10\) (Slope: \(\frac{3}{4} = 0.75\)) -> Increasing
• \(y = -\frac{1}{2}x + \frac{1}{2}\) (Slope: \(-\frac{1}{2} = -0.5\)) -> Decreasing
• \(y = \frac{3}{2}x - \frac{11}{2}\) (Slope: \(\frac{3}{2} = 1.5\)) -> Increasing
• \(y = \frac{1}{2}x - 2\) (Slope: \(\frac{1}{2} = 0.5\)) -> Increasing
• \(y = \frac{4}{3}x - \frac{7}{3}\) (Slope: \(\frac{4}{3} \approx 1.33\)) -> Increasing

Order slopes from least to greatest

Compare the positive slopes of the five increasing functions:

$$ 0.5 < 0.75 < 1.33 < 1.5 < 2.5 $$

$$ \frac{1}{2} < \frac{3}{4} < \frac{4}{3} < \frac{3}{2} < \frac{5}{2} $$

Match functions to ordered slots

Arrange the corresponding functions in order:

1. \(y = \frac{1}{2}x - 2\)
2. \(y = \frac{3}{4}x - 10\)
3. \(y = \frac{4}{3}x - \frac{7}{3}\)
4. \(y = \frac{3}{2}x - \frac{11}{2}\)
5. \(y = \frac{5}{2}x + 10\)

Answer:

Find the increasing functions. Then arrange the increasing functions in order from least to greatest rate of change.

<blank>\(y = \frac{1}{2}x - 2\)</blank> \(<\) <blank>\(y = \frac{3}{4}x - 10\)</blank> \(<\) <blank>\(y = \frac{4}{3}x - \frac{7}{3}\)</blank> \(<\) <blank>\(y = \frac{3}{2}x - \frac{11}{2}\)</blank> \(<\) <blank>\(y = \frac{5}{2}x + 10\)</blank>