Question

given the following three ordered pairs, construct the linear proportional function and represent it as a graph, equation, table of values, and mapping. then, answer the questions.\\{(-6, -4.5), (3, 2.25), (8, 6)\\}
4.1.1) write the set of ordered pairs as a table, with the x-values listed from least to greatest.
(table with x and y columns: x=-6, y=45; x=3, y=225; x=8, y=6)
4.1.2) calculate the slope of the function using row 1 and row 3 from the table in the previous question. show or explain your work.
$m = \frac{\text{change in } y}{\text{change in } x} = \frac{39}{14}$
4.1.3) calculate the y-intercept of the function. show or explain your work.
$b = $
4.1.4) write the equation of the linear function.

Explanation:

4.1.1)

Step1: Order x-values

The x - values are - 6, 3, 8. They are already in order from least to greatest.

Step2: Get correct y - values

The ordered pairs are \((-6,-4.5)\), \((3,2.25)\), \((8,6)\). So the correct table is:

x	y
3	2.25
8	6

Step1: Recall slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\). For row 1 (\(x_1=-6,y_1 = - 4.5\)) and row 3 (\(x_2 = 8,y_2=6\)).

Step2: Calculate change in y and x

Change in \(y=y_2 - y_1=6-(-4.5)=6 + 4.5 = 10.5\)
Change in \(x=x_2 - x_1=8-(-6)=8 + 6=14\)

Step3: Calculate slope

\(m=\frac{10.5}{14}=\frac{105}{140}=\frac{3}{4}=0.75\)

Step1: Recall linear equation form

The linear equation is \(y=mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. We know \(m = 0.75\) (or \(\frac{3}{4}\)) and we can use one of the points, say \((3,2.25)\).

Step2: Substitute into equation

Substitute \(x = 3\), \(y=2.25\) and \(m = 0.75\) into \(y=mx + b\):
\(2.25=0.75\times3 + b\)

Step3: Solve for b

\(2.25 = 2.25+b\)
Subtract \(2.25\) from both sides: \(b=2.25 - 2.25=0\)

Answer:

x	y
3	2.25
8	6

4.1.2)