Question

select the correct answer.

considering only functions with a greater rate of change than that of the function represented on the graph, which function has the lowest rate of change?

\\(y = \frac{13}{7}x - \frac{7}{3}\\)

\\(y = \frac{7}{5}x + \frac{2}{5}\\)

\\(y = \frac{11}{8}x + \frac{3}{8}\\)

\\(y = \frac{5}{4}x + \frac{13}{4}\\)

\\(y = \frac{5}{3}x - \frac{8}{3}\\)

Explanation:

Find the rate of change of the graphed function

Using the Rate of Change and Linear Functions knowledge points, we identify two points on the line: \((1, 0)\) and \((5, 5)\).
The rate of change (slope \(m\)) is:

$$ m = \frac{5 - 0}{5 - 1} = \frac{5}{4} = 1.25 $$

Determine the rates of change of the options

We find the slope \(m\) for each linear equation \(y = mx + b\):

• Option 1: \(y = \frac{13}{7}x - \frac{7}{3} \implies m = \frac{13}{7} \approx 1.857\)
• Option 2: \(y = \frac{7}{6}x + \frac{2}{9} \implies m = \frac{7}{6} \approx 1.167\)
• Option 3: \(y = \frac{11}{8}x + \frac{3}{8} \implies m = \frac{11}{8} = 1.375\)
• Option 4: \(y = \frac{5}{4}x + \frac{13}{4} \implies m = \frac{5}{4} = 1.25\)
• Option 5: \(y = \frac{5}{3}x - \frac{8}{3} \implies m = \frac{5}{3} \approx 1.667\)

Filter functions with a greater rate of change

We select functions where \(m > 1.25\):

• \(y = \frac{13}{7}x - \frac{7}{3}\) (\(m \approx 1.857 > 1.25\))
• \(y = \frac{11}{8}x + \frac{3}{8}\) (\(m = 1.375 > 1.25\))
• \(y = \frac{5}{3}x - \frac{8}{3}\) (\(m \approx 1.667 > 1.25\))

Identify the lowest rate of change among the filtered functions

Comparing the slopes of the filtered functions:

$$ 1.375 < 1.667 < 1.857 \implies \frac{11}{8} < \frac{5}{3} < \frac{13}{7} $$

The function with the lowest rate of change greater than \(1.25\) is:

$$ y = \frac{11}{8}x + \frac{3}{8} $$

Answer:

• (A) \(y = \frac{13}{7}x - \frac{7}{3}\)
• (B) \(y = \frac{7}{6}x + \frac{2}{9}\)</mcq-correct>
• (C) \(y = \frac{11}{8}x + \frac{3}{8}\) (Correct answer)

<mcq-option>(D) \(y = \frac{5}{4}x + \frac{13}{4}\)

• (E) \(y = \frac{5}{3}x - \frac{8}{3}\)