The graph of the function f(x)=x^2 - 2x. draw the line tangent to the point (0,0). then estimate the slope at the point?

1 Answer
Apr 10, 2018

m=-2m=2

Explanation:

.

y=x^2-2xy=x22x

We can find the slope of the tangent to the curve by taking its derivative and evaluating it at the point of tangency:

dy/dx=2x-2=2(0)-2=0-2=-2dydx=2x2=2(0)2=02=2

m=-2m=2

We can now write the equation of the tangent line:

y=mx+by=mx+b

y=-2x+by=2x+b

We then use the coordinates of the point of tangency to solve foe bb which is the yy-intercept of the line:

0=-2(0)+b0=2(0)+b

0=0+b0=0+b

b=0b=0

Therefore, the equation of the tangent line at (0,0)(0,0) is:

y=-2xy=2x

The graph below shows the function of the parabola in purple and the tangent line in red:

enter image source here

If you wanted to estimate the slope you could visually inspect the graph and see that the slope is:

("Rise")/("Run")RiseRun

Then pick a point on the xx-axis, x=-1x=1 and draw a vertical line up to the tangent line. This line is shown in green.

Now, you can estimate the length of the green line which is the ("Rise")(Rise). It appears to be about 22. The ("Run")(Run) is -11. Therefore,

("Rise")/("Run")=2/(-1)=-2RiseRun=21=2

If we guess at two values for the ("Rise")(Rise), one slightly larger and one slightly smaller then 22, such as 2.12.1 and 1.91.9 and write the equation of the tangent line with each slope value we will have:

x=2.1, :. y=-2.1x

x=1.9, :. y=-1.9x

We can test each one by setting the equation of the tangent line equal to the equation of the curve and solving for the x-coordinates of the intersection points.

x^2-2x=-2.1x, :. x^2+0.1x=0, :. x(x+0.1)=0, :. x=0 and -0.1

Plugging these values into the equation of either function will give us the y-coordinates of these points.

x=0, y=0, and x=-0.1, y=0.2

This means the line intersects the parabola at two points:

(0,0) and (-0.1,0.2)

which means the line is not a tangent line.

Let's test the other value for ("Rise"):

x^2-2x=-1.9x, :. x^2-0.1x=0, :. x(x-0.1)=0, :. x=0 and 0.1

x=0, y=0, and x=0.1, y=-0.2

This mean th eline intersects the parabola at two points:

(0,0) and (0.1,-0.2)

which again means the line is not a tangent line.

This trial and error process will result in ("Rise")=2, m=-2, and the equation of the tangent line to be:

y=-2x