Example: Sketching a Tangent to a point on a line and estimating its slope

Below, we have the line graphed for y = f(x),function f(x) shown plotted on an x-y graph

a) Roughly sketch the tangent to the points x = 1 , x = 3 and x = 5

So, remember the tangent to a point on a line, is the straight line that intersects that point and runs at the slope that point is on,

Roughly sketching the line to the point x = 1 would give us,function f(x) shown plotted on an x-y graph and the tangent shown at x = 1

For x = 3, it is slightly different, roughly, x = 3 looks like the point where the graphed line turns and starts to go in the other direction, so if you imagine what the tangent would be up until the point x = 3 and then what the tangent would be right after the point x = 3, the turning point at x = 3 would be roughly have the tangent halfway between the tangents before and after the point,

This would look like a straight line running parallel to the x - axis,function f(x) shown plotted on an x-y graph and the tangent shown at x = 3

For x = 5, we have the line looking like it roughly runs in the opposite way it does at x = 1, draw the line for the tangent at the point x = 5function f(x) shown plotted on an x-y graph and the tangent shown at x = 5

b) Roughly guess what the slope of that tangent would be

So, roughly guessing what the slope of the tangent would be at each point,

At x = 1, it roughly looks like the line of the tangent goes down one y-value for every one x - value,

Remember our slope (or gradient, m) =  
change in y
change in x
So, the y-value change is -1, and the x-value change is 1, giving us a tangent slope of roughly,

-1
1
  = -1

At x = 0, it roughly looks like the y-value does not go up or down, so the change would be 0, this tells us that no matter what the change in the x-value is, our slope for the tangent will be 0, as if the change in y = 0, then,

The change in y
The change in x
  =
0
The change in x
  = 0 (as 0 divided by anything is = 0)

For our last point, x = 3, the tangents slope seems to be opposite to what x = 1 is,

The y-value roughly goes up 1 for every x-value it goes right,

So,  
the change in y
the change in x
  =
1
1
  = 1
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