Linear Function

Question

Write a rule for a linear function y=f (x), given that f (0) =-5 and f (-5) =-6. Write answers using fractions or integers. 

 

The function rule is f(x) = 

Answer 1

The graph of a linear function is a straight line.  

 

f(0) = -5 means that (0,-5) is a point on the line.  Similarly, (-5,-6) is also a point on the line.  

 

Slope of line = m = [-6-(-5)]/[-5-0] = -1/(-5) = 1/5

 

Using the point-slope form, an equation of the line is:

 

y - (-5) = (1/5)(x - 0)

 

So, y = (1/5)x - 5

 

 

Answer 2

Since f(0) = -5, -5 is the y-intercept

 

y = mx - 5

 

Δy / Δx = -1/-5

 

y = x/5 - 5

Answer 3

The problem gives two points (by the way, that's enough to define a line):

   (0,-5)

   (-5,-6)

 

Conveniently, there is a formula for a line given two points:

   y-y₁ =   m (x - x₁)

   y-y₁ = ( (y₂-y₁)/(x₂-x₁) ) (x - x₁)

 

So, plug in the values:

   y - (-5) = ( (-6-(-5))/(-5-0) ) (x - 0)

   y + 5 = ( (-1/-5) ) x       

   y + 5 = (1/5)x                          

   y = (1/5)x - 5

 

Check that the two points are on that line:

   Is  -5 = (1/5)(0) - 5  ?

        -5 = -5   ? yes

   Is  -6 = (1/5)(-5) - 5   ?

        -6 =  -1  -5   ?

        -6 = -6   ? yes

Answer 4

Hi Moin,

 

The questions is asking for you to write a linear function based on the two results. It is sometimes easier to think of these linear functions graphically - and a two variable linear function simply defines the path of a line on a 2-D Graph, with an x-axis and y-axis. The two variables in the linear function are x and y. Furthermore, linear functions have the form of f(x) = y=mx+b. x and y are your variables, m = slope or rate, and b= y-intercept. In this problem they have given you two points or (x,y) coordinates on the line that is defined by this linear function (0,-5) where x=0 and y=-5 and (-5,-6) where x=-5 and y=-6. If you have graph paper, you can draw a line between these two points on the graph. From that line you can calculate the slope "m" by counting the "rise" which is the change in the y-axis coordinate (the vertical "height"of the line between the two points) over the "run" which is the change in the x-axis (the horizontal distance between the two points).You can do this algebraically as well, where you calculate the change in the y-coordinates divided by the change in the x-coordinates, or (y2-y1) / (x2-x1). If you plug in the numbers [(-6)-(-5)]/[-5-0] = m. Once you have "m" you can re-arrange the f(x) equation to be y-mx = b, and you can plug in one of the coordinates such as (0,-5) to solve for b. Or you can use a trick, since "b" is the y-intercept, the y-coordinate where the line crosses the y-axis, you can see what the y-value is when the x=0 and they already gave you the value of the function when x=0 for f(0). Remember the question wants fractions and integers, so write down "m" and "b" as such, and not as decimals. Read through this and see if you can get the answer - if I am still on I may be able to check it for you.