Question

plot a graph for y = x^2 + 4x -1​

Answer 1

y = x² + 4x -1,

This curve meets the the x-axis when y=0

i.e x²+4x-1=0

⇒x=-4±√(4²-4(1)(-1)) /2(1) ( Quadratic formula)

⇒x = -4 ±√20 / 2

⇒x= -4 ±2√5 / 2

⇒x = -2±√5.

This curve meets y-axis when x=0,

i.e y=(0)²+4(0)-1

y = -1.

Differentiate the given function so that we can know where it is increasing and decreasing.

dy/dx= 2x+4        (Math Help: d/dx (x^n) = n *(x)^n-1)

Let's find where the first derivative is positive(Increasing) and negative(decreasing),

Whenever x>-2 the curve is increasing.

and when x<-2 the curve is decreasing.

Now let's find the maxima or minima of the function, in it's process equate dy/dx = 0

i.e 2x+4=0

⇒x= -2. To know whether this is maximum or minimum of that curve differentiate the function once again i.e second derivative,

d²y/dx² = 2 which is positive. Hence it is also positive when x= -2.

Therefore the curve has minimum value at x= -2.

At x = -2,

y=(-2)²+4(-2)-1

y= -5

To find the concavity or convexity of that curve you need to find second derivative, which we found above already i.e d²y/dx² = 2, which is positive.

This means that the curve always has its concave side facing up or convex side facing down.

Now you have everything required to plot the graph.

The graph looks like this.