Question

The table represents the linear function f(x), and the equation represents the linear function g(x).

Compare the y-intercepts and slopes of the linear functions f(x) and g(x) and choose the answer that best describes them.


x f(x)
0 1
2 11
4 21

g(x) = 4x + 1
The slope of f(x) is less than the slope of g(x). The y-intercept of f(x) is equal to the y-intercept of g(x).
The slope of f(x) is less than the slope of g(x). The y-intercept of f(x) is greater than the y-intercept of g(x).
The slope of f(x) is greater than the slope of g(x). The y-intercept of f(x) is equal to the y-intercept of g(x).
The slope of f(x) is less than the slope of g(x). The y-intercept of f(x) is greater than the y-intercept of g(x).

Answer 1

Answer: The correct option is

(C) The slope of f(x) is greater than the slope of g(x). The y-intercept of f(x) is equal to the y-intercept of g(x).

Step-by-step explanation: Given that the following table represents the linear function f(x), and the equation represents the linear function g(x) :

x 0 2 4

f(x) 1 11 21

and

`g(x)=4x+1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)`

We are to compare the y-intercepts and slopes of the linear functions f(x) and g(x) and choose the answer that best describes them.

From the graph, we note that (0, 1), (2, 11) and (4, 21) are three points on the line graph of function f(x).

We know that

the slope of any line having two points (a, b) and (c, d) on it is given by

`m=(d-b)/(c-a).`

So, the slope of the line of f(x) is

`m=(11-1)/(2-0)=(10)/(2)=5.`

Since the line of f(x) passes through the point (0, 1), so its equation will be

`f(x)-1=m(x-0)\\\\\Rightarrow f(x)-1=5x\\\\\Rightarrow f(x)=5x+1~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)`

The slope-intercept form of the equation of a line is

`y=mx+c,` where m is the slope and c is the y-intercept of the line.

Comparing equation (ii) and (i) with the slope-intercept form, we get

slope of f(x) = 5 and y-intercept of f(x) = 1

and

slope of g(x) = 4 and y-intercept of g(x) = 1.

Thus, the slope of f(x) is greater than the slope of g(x). The y-intercept of f(x) is equal to the y-intercept of g(x).

Option (C) is CORRECT.

Answer 2

The slope of f(x) is greater than the slope of g(x). The y-intercept of f(x) is equal to the y-intercept of g(x).

Explanation:

Both have y-intercepts at (0,1) but g(x) has a slope of 4 while f(x) has a slope of 5.