Question

1. use the equation $y = 2(4 + x)$ to answer each question.

a. complete the table for each value of $x$.
b. plot the points from the table on a coordinate grid. what do you notice about the points?
c. make a conjecture about the relationship between the change in $x$ and the change in $y$.

$x$	$y$
2
3
4
5
6

Explanation:

Step1: Substitute x=1 into the equation

Given the equation \( y = 2(4 + x) \), when \( x = 1 \), we have \( y = 2(4 + 1)=2\times5 = 10 \).

Step2: Substitute x=2 into the equation

When \( x = 2 \), \( y = 2(4 + 2)=2\times6 = 12 \).

Step3: Substitute x=3 into the equation

When \( x = 3 \), \( y = 2(4 + 3)=2\times7 = 14 \).

Step4: Substitute x=4 into the equation

When \( x = 4 \), \( y = 2(4 + 4)=2\times8 = 16 \).

Step5: Substitute x=5 into the equation

When \( x = 5 \), \( y = 2(4 + 5)=2\times9 = 18 \).

Step6: Substitute x=6 into the equation

When \( x = 6 \), \( y = 2(4 + 6)=2\times10 = 20 \).

For part b, when we plot the points \((1,10)\), \((2,12)\), \((3,14)\), \((4,16)\), \((5,18)\), \((6,20)\) on a coordinate grid, we will notice that all the points lie on a straight line. This is because the equation \( y = 2(4 + x) \) can be rewritten as \( y=2x + 8 \), which is a linear equation in the form \( y=mx + b \) (where \( m = 2 \) is the slope and \( b = 8 \) is the y - intercept), and the graph of a linear equation is a straight line.

For part c, let's find the change in \( x \) (\(\Delta x\)) and the change in \( y \) (\(\Delta y\)) between consecutive points. For example, between \( x = 1 \) and \( x = 2 \), \(\Delta x=2 - 1=1\), and \(\Delta y = 12-10 = 2\). Between \( x = 2 \) and \( x = 3 \), \(\Delta x=3 - 2 = 1\), and \(\Delta y=14 - 12=2\). In general, for a linear equation of the form \( y=mx + b \), the slope \( m=\frac{\Delta y}{\Delta x}\). Here, \( m = 2 \), so when \( \Delta x = 1 \) (the change in \( x \) is 1 unit), \( \Delta y=2 \) (the change in \( y \) is 2 units). So the conjecture is that for every 1 - unit increase in \( x \), \( y \) increases by 2 units.

Answer:

Part a:

The completed table is:

\( x \)	\( y \)
2	12
3	14
4	16
5	18
6	20

Part b:

The points lie on a straight line.

Part c:

For every 1 - unit increase in \( x \), \( y \) increases by 2 units.