Step 1
Find the slope of the function f(x)
we know that
The formula to calculate the slope between two points is equal to
[tex]m=\frac{y2-y1}{x2-x1}[/tex]
Let
[tex]A(-6,0)\\B(0,4)[/tex]
substitute
[tex]m=\frac{4-0}{0+6}[/tex]
[tex]m=\frac{4}{6}[/tex]
[tex]m=\frac{2}{3}[/tex]
Step 2
Find the y-intercept of the function f(x)
The y-intercept is the value of the function when the value of x is equal to zero
in this problem the y-intercept of the function is the point [tex]B(0,4)[/tex]
so
the y-intercept is equal to [tex]4[/tex]
Step 3
Verify each case
we know that
the equation of the line into slope-intercept form is equal to
[tex]y=mx+b[/tex]
where
m is the slope
b is the y-intercept
case A) [tex]f(t)=t+4[/tex]
In this case we have
[tex]m=1\\y-intercept=4[/tex]
therefore
the function of case A) does not have the same slope as the function f(x)
case B) [tex]f(t)=t-6[/tex]
In this case we have
[tex]m=1\\y-intercept=-6[/tex]
therefore
the function of case B) does not have the same slope and y-intercept as the function f(x)
case C) [tex]f(t)=\frac{2}{3}t+4[/tex]
In this case we have
[tex]m=\frac{2}{3}\\y-intercept=4[/tex]
therefore
the function of case C) does have the same slope and y-intercept as the function f(x)
case D) [tex]f(t)=\frac{2}{3}t+6[/tex]
In this case we have
[tex]m=\frac{2}{3}\\y-intercept=6[/tex]
therefore
the function of case D) does not have the same y-intercept as the function f(x)
therefore
the answer is
[tex]f(t)=\frac{2}{3}t+4[/tex]
