The given functions can be generalized using the form f(x) = y
Given the following functions;
[tex]\begin{gathered} f(-2)=2 \\ f(8)=-3 \end{gathered}[/tex]These functions can be written as coordinates points (-2, 2) and (8, -3)
The equation of the linear function in slope-intercept form is expressed as y = mx + b
m is the slope:
b is the y-intercept
Get the slope of the line passing through the points:
[tex]\begin{gathered} m=\frac{y_2-y_1}{x_2-x_1} \\ m=\frac{-3-2}{8-(-2)} \\ m=\frac{-5}{8+2} \\ m=-\frac{5}{10} \\ m=-\frac{1}{2} \end{gathered}[/tex]Get the y-intercept using the point(-2, 2) and m = -0.5
[tex]\begin{gathered} y=mx+b \\ 2=-0.5(-2)+b \\ 2=1+b \\ b=2-1 \\ b=1 \end{gathered}[/tex]Write the equation in slope-intercept form where m = -0.5 and b = 1;
[tex]y=-\frac{1}{2}x+1[/tex]Write in point-slope form;
The point-slope form of the equation is expressed as;
[tex]y-y_1=m(x-x_1)[/tex]Using the following parameters;
[tex]\begin{gathered} m=-\frac{1}{2} \\ (x_1,y_1)=(-2,2) \end{gathered}[/tex]Substitute the given parameters into the point-slope form of the equation;
[tex]\begin{gathered} y-2=-\frac{1}{2}(x-(-2)_{}) \\ y-2=-\frac{1}{2}(x+2) \end{gathered}[/tex]This gives the point-slope form of the equation.
For the standard form:
The standard form of the linear equation is expressed as:
[tex]Ax+By=C[/tex]Recall that;
[tex]y=-\frac{1}{2}x+1[/tex]Rearrange in standard form as shown:
[tex]\begin{gathered} 2y=-x+2 \\ \end{gathered}[/tex]Add "x" to both sides of the equation:
[tex]\begin{gathered} 2y+x=-x+x+2 \\ 2y+x=2 \\ x+2y=2 \end{gathered}[/tex]This gives the required linear equation in standard form.
