Question

On a coordinate plane, a cube root function goes through (negative 2.5, negative 3.5), crosses the y-axis at (0, negative 3), has an inflection point at (1, negative 2), and goes through (2, negative 1). Determine the values of h and k in the equation of g(x). G(x) = RootIndex 3 StartRoot x minus h EndRoot+ k h = k =

Answer 1

Answer:

$g(x) = \sqrt[3]{x-1} - 2$

Step-by-step explanation:

We want to find h and k in:

$g(x) = \sqrt[3]{x-h} + k$

At the inflection point, the second derivative is equal to zero, so:

$g'(x) = \frac{1}{3} (x-h)^{\frac{-2}{3}}$

$g''(x) = \frac{1}{3} \frac{-2}{3}(x-h)^{\frac{-5}{3}} = 0$

Then x - h = 0.

Inflection point is located at (1, -2), replacing this x value we get:

1 - h = 0

h = 1

We know that the point (-2.5, -3.5) belongs to the function, so:

$-3.5 = \sqrt[3]{-2.5-1} + k$

k ≈ -2

All data, used or not, are shown in the picture attached.