Question

directions: for each problem, circle the representation with the greatest rate of change. put a star by the representation with the greatest y - intercept. assume all representations have a constant rate of change.

1. $y = x$
2. $y = \frac{7}{4}x + 2$
3. $y = - 0.5x$

Explanation:

Step1: Find rate of change (slope) and y - intercept for each representation in problem 8

• For \(y = x\):
• The equation is in \(y=mx + b\) form (\(m\) is slope, \(b\) is y - intercept). Here \(m = 1\), \(b = 0\).
• **For the table \(

$$\begin{array}{|c|c|}\hline x&y\\\hline1&1.5\\\hline2&2\\\hline3&2.5\\\hline\end{array}$$

\)**:

• Slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Using \((x_1,y_1)=(1,1.5)\) and \((x_2,y_2)=(2,2)\), \(m=\frac{2 - 1.5}{2 - 1}=0.5\).
• Using \(y=mx + b\) and \((x = 1,y = 1.5)\), \(1.5=0.5\times1 + b\), so \(b = 1\).
• For the graph:
• Pick two points, say \((- 4,0)\) and \((0,0.5)\). Slope \(m=\frac{0.5-0}{0 + 4}=0.125\), \(b = 0.5\).
• The greatest rate of change is for \(y = x\) (circle \(y = x\)), the greatest y - intercept is for the table (put a star by the table).

Step2: Find rate of change (slope) and y - intercept for each representation in problem 9

• For \(y=\frac{7}{4}x + 2\):
• Slope \(m=\frac{7}{4}=1.75\), \(b = 2\).
• For the points \((1,1.5)\) and \((2,3)\):
• Slope \(m=\frac{3 - 1.5}{2 - 1}=1.5\). Using \(y=mx + b\) and \((x = 1,y = 1.5)\), \(1.5=1.5\times1+b\), \(b = 0\).
• For the graph (which is \(y = 2x\)):
• Slope \(m = 2\), \(b = 0\).
• The greatest rate of change is for the graph (circle the graph), the greatest y - intercept is for \(y=\frac{7}{4}x + 2\) (put a star by \(y=\frac{7}{4}x + 2\)).

Step3: Find rate of change (slope) and y - intercept for each representation in problem 10

• For \(y=-0.5x\):
• Slope \(m=-0.5\), \(b = 0\).
• **For the table \(

$$\begin{array}{|c|c|}\hline x&y\\\hline0&0\\\hline2&-2\\\hline5&-5\\\hline\end{array}$$

\)**:

• Slope \(m=\frac{-2-0}{2 - 0}=-1\).
• For the graph:
• Pick two points, say \((0,1)\) and \((2,-2)\). Slope \(m=\frac{-2 - 1}{2-0}=-1.5\), \(b = 1\).
• The greatest rate of change (closest to \(0\) among negative slopes) is for \(y=-0.5x\) (circle \(y=-0.5x\)), the greatest y - intercept is for the graph (put a star by the graph).

Answer:

• Problem 8: Circle \(y = x\), star by the table.
• Problem 9: Circle the graph (\(y = 2x\)), star by \(y=\frac{7}{4}x + 2\).
• Problem 10: Circle \(y=-0.5x\), star by the graph.