Question

The table represents the function f(x).

Answer 1

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

$m=\frac{y2-y1}{x2-x1}$

Let

$A(-6,0)\\B(0,4)$

substitute

$m=\frac{4-0}{0+6}$

$m=\frac{4}{6}$

$m=\frac{2}{3}$

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 $B(0,4)$

so

the y-intercept is equal to $4$

Step 3

Verify each case  

we know that

the equation of the line into slope-intercept form is equal to

$y=mx+b$

where

m is the slope

b is the y-intercept

case A) $f(t)=t+4$

In this case we have

$m=1\\y-intercept=4$

therefore

the function of case A) does not have the same slope as the function f(x)

case B) $f(t)=t-6$

In this case we have

$m=1\\y-intercept=-6$

therefore

the function of case B) does not have the same slope and y-intercept as the function f(x)

case C) $f(t)=\frac{2}{3}t+4$

In this case we have

$m=\frac{2}{3}\\y-intercept=4$

therefore

the function of case C) does have the same slope and y-intercept as the function f(x)

case D) $f(t)=\frac{2}{3}t+6$

In this case we have

$m=\frac{2}{3}\\y-intercept=6$

therefore

the function of case D) does not have the same  y-intercept as the function f(x)

therefore

the answer is

$f(t)=\frac{2}{3}t+4$

Answer 2

Answer:

f(t) = 2/3 t + 4