Question

The table below shows the relationship between x and y

Answer 1

Solution:

Given the table showing the relationship between x and y, below:

We can represent the relationship between x and y using the equation of a line expressed as

$\begin{gathered} y=mx+c\text{ ----- equation 1} \\ where \\ m\Rightarrow slope\text{ of the line} \\ c\Rightarrow y-intercept\text{ of the line} \end{gathered}$

step 1: Evaluate the slope m of the line.

The slope m of the line is expressed as

$m=\frac{rise}{run}=\frac{y_2-y_1}{x_2-x_1}\text{ ----- equation 2}$

To find the equation of the line, we select two points (x, y) from the table of values.

Thus, selecting the points (-1,-1) and (0,1), this implies that

$\begin{gathered} x_1=-1 \\ y_1=-1 \\ x_2=0 \\ y_2=1 \end{gathered}$

Substituting these values into equation 2, we have

$\begin{gathered} m=\frac{1-(-1)}{0-(-1)} \\ =\frac{1+1}{0+1} \\ =\frac{2}{1} \\ \Rightarrow m=2 \end{gathered}$

step 2: Evaluate the y-intercept of the line.

The y-intercept of the line is the value of y when the line cuts the y-axis. In other words, it's the value of y when the value of x equals zero.

In the table of values, when the value of x equals zero, the value of y is 1.

Thus, the y-intercept of the line is

$c=1$

step 3: Substitute the values of m and c into equation 1.

From equation 1,

$\begin{gathered} y=mx+c \\ where \\ m=2,\text{ c=1} \end{gathered}$

Thus, the function that represents the relationship between the quantities in the table is

$y=2x+1$

The correct option is F.