Question

Write an inequality for the shaded region shown in the figure.​

Answer 1

Answer:

y ≥  x^2 - 1

Step-by-step explanation:

First, we can see that the shaded region is above what seems to be a parabola, and we also can see that the lines of the parabola are solid lines (which means that the points on the curve itself are solutions, so the symbol ≥ is used)

Then:

y ≥ a*x^2 + b*x + c

where a*x^2 + b*x + c is the general quadratic equation.

Now let's find the equation for the parabola:

f(x) = a*x^2 + b*x + c

We also can see that the vertex of the parabola is at the point (0, -1)

This means that:

f(0) = -1 = a*0^2 + b*0 + c

     = -1 = c

then we have that c = -1

Then:

f(x) = a*x^2 + b*x - 1

Now we can look at the graph again, to see that the zeros of the parabola are at +1 and -1

Which means that:

f(1) = 0 = a*1^2 + b*1 - 1 = a + b - 1

f(-1) = 0 = a*(-1)^2 + b*(-1) - 1 = a - b - 1

Then we got two equations:

a + b - 1 = 0

a - b - 1 = 0

from this we can conclude that b must be zero.

Then:

b = 0

and these equations become:

a - 1 = 0

a - 1 = 0

solving for a, we get:

a = 1

Then the quadratic equation is:

f(x) = 1*x^2 + 0*x - 1

f(x) = x^2 - 1

And the inequality is:

y ≥  x^2 - 1