Question

mrs. jackson gives the table below to her students.

x	y
5	23
6	a

in order for the function to be linear, what must a be and why?

• $a = 22$ because the rate of change is 1.
• $a = 20$ because the rate of change is 3.
• $a = 22$ because the rate of change is $-1$.
• $a = 20$ because the rate of change is $-3$.

Explanation:

Step1: Calculate rate of change between x=4 and x=5

The rate of change (slope) between two points \((x_1,y_1)\) and \((x_2,y_2)\) is \(\frac{y_2 - y_1}{x_2 - x_1}\). For \(x_1 = 4,y_1 = 26\) and \(x_2=5,y_2 = 23\), the rate of change is \(\frac{23 - 26}{5 - 4}=\frac{- 3}{1}=-3\)? Wait, no, wait. Wait, \(23-26=-3\), \(5 - 4 = 1\), so slope is \(- 3\)? Wait, no, wait let's recalculate. Wait, when \(x\) increases from 4 to 5 (an increase of 1), \(y\) decreases from 26 to 23 (a decrease of 3). So the rate of change (slope) \(m=\frac{23 - 26}{5 - 4}=\frac{-3}{1}=-3\). Now, for the function to be linear, the rate of change between \(x = 5\) and \(x=6\) should also be \(-3\). So when \(x\) increases from 5 to 6 (increase of 1), \(y\) should decrease by 3. So \(y\) at \(x = 6\) (which is \(a\)) should be \(23-3=20\). Let's check the options. The option that says \(a = 20\) because the rate of change is \(- 3\) is the correct one. Wait, let's check the options again. The options are:

1. \(a = 22\) because rate of change is 1: Incorrect, rate of change is negative.

1. \(a = 20\) because rate of change is 3: No, rate of change is - 3.

1. \(a = 22\) because rate of change is - 1: Incorrect.

1. \(a = 20\) because rate of change is - 3: Correct.

Wait, let's recalculate the rate of change again. From \(x = 4\) to \(x = 5\): \(y\) goes from 26 to 23. So \(\Delta y=23 - 26=-3\), \(\Delta x=5 - 4 = 1\). So slope \(m=\frac{\Delta y}{\Delta x}=\frac{-3}{1}=-3\). Then from \(x = 5\) to \(x = 6\), \(\Delta x = 1\), so \(\Delta y=m\times\Delta x=-3\times1=-3\). So \(y\) at \(x = 6\) is \(23+(-3)=20\). So the correct option is \(a = 20\) because the rate of change is \(-3\).

Answer:

D. \(a = 20\) because the rate of change is \(-3\) (assuming the last option is D, but in the given options, the last option is " \(a = 20\) because the rate of change is \(-3\)")