Question

which function has the greatest slope?\\(\bigcirc\\) \\(y = \frac{2}{3}x + 1\\)\\(\bigcirc\\) \\(y = -x + 8\\)\\(\bigcirc\\) \\(y = \frac{3}{4}x - \frac{5}{4}\\)\\(\bigcirc\\) \\(y = \frac{3}{2}x + 3\\)

Explanation:

Step1: Recall slope-intercept form

The slope - intercept form of a linear function is \(y = mx + b\), where \(m\) is the slope of the line and \(b\) is the y - intercept. We need to identify the slope (\(m\)) for each of the given functions.

Step2: Find the slope of each function

• For the function \(y=\frac{2}{3}x + 1\), comparing with \(y=mx + b\), the slope \(m_1=\frac{2}{3}\approx0.67\).
• For the function \(y=-x + 8\), comparing with \(y = mx + b\), the slope \(m_2=- 1\).
• For the function \(y=\frac{3}{4}x-\frac{5}{4}\), comparing with \(y=mx + b\), the slope \(m_3=\frac{3}{4} = 0.75\).
• For the function \(y=\frac{3}{2}x+3\), comparing with \(y = mx + b\), the slope \(m_4=\frac{3}{2}=1.5\).

Step3: Compare the slopes

We compare the values of \(m_1=\frac{2}{3}\approx0.67\), \(m_2 = - 1\), \(m_3=\frac{3}{4}=0.75\) and \(m_4=\frac{3}{2}=1.5\).

Since \(1.5>0.75 > 0.67>-1\), the slope of \(y = \frac{3}{2}x+3\) is the greatest.

Answer:

\(y=\frac{3}{2}x + 3\)