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: -6, 3, 8 and y: 45, 225, 6 (note: likely a typo in original table, correct y should be -4.5, 2.25, 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 = (change in y)/(change in x) = 39/14 (note: likely incorrect in original, correct calculation should use -4.5, 6 and -6, 8) 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: Match with y - values

For \(x=-6\), \(y = - 4.5\); for \(x = 3\), \(y=2.25\); for \(x = 8\), \(y = 6\).

\(x\)	\(y\)
3	2.25
8	6

Step1: Recall slope formula

The slope formula is \(m=\frac{y_2 - y_1}{x_2 - x_1}\), where \((x_1,y_1)\) and \((x_2,y_2)\) are two points on the line.

Step2: Identify points

Using row 1 \((x_1=-6,y_1 = - 4.5)\) and row 3 \((x_2 = 8,y_2=6)\).

Step3: Calculate change in y and x

Change in \(y=y_2 - y_1=6-(-4.5)=6 + 4.5 = 10.5=\frac{21}{2}\)
Change in \(x=x_2 - x_1=8-(-6)=8 + 6=14\)

Step4: Calculate slope

\(m=\frac{\frac{21}{2}}{14}=\frac{21}{2}\times\frac{1}{14}=\frac{3}{4}=0.75\) (The given \(\frac{39}{14}\) is incorrect. Let's correct it.)

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 \(m=\frac{3}{4}\), and we can use one of the points, say \((3,2.25)\) (or \((-6,-4.5)\) or \((8,6)\)).

Step2: Substitute into equation

Using the point \((x = 3,y=2.25)\) and \(m=\frac{3}{4}\) in \(y=mx + b\).
\(2.25=\frac{3}{4}\times3 + b\)
\(2.25 = 2.25+b\)

Step3: Solve for b

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

Answer:

\(x\)	\(y\)
3	2.25
8	6

4.1.2)