Find the type of graph of a function with the same zero(s) as the provided functions: a) x² + 5x + 8, b) x³ - 6x² + 11x - 6
Quadratic functions are those functions that have almost two zeroes and the highest order is two. They are used in various fields of engineering and science to find the values of different parameters. Cubic functions are those which have almost three zeroes. They are also used in many engineering applications.
Answer: For the equation, x² + 5x + 8, the graph is parabola which doesn't touch the x-axis, and for the equation, x³ - 6x² + 11x - 6, the graph cuts the x-axis at 3 points.
Let's understand the solution.
a) For the equation x² + 5x + 8:
⇒ Discriminant D = 5² - 4 × 8 = -7 < 0
Hence, the equation has imaginary roots, and it doesn't touch the x-axis. Since the equation is quadratic, the graph is a parabola.
b) For the equation x³ - 6x² + 11x - 6:
After we simplify the equation, we get:
⇒ x³ - 6x² + 11x - 6 = (x - 1) (x - 2) (x - 3)
Hence, the equation has three real roots, hence it cuts the x-axis three times.
For the equation x² + 5x + 8, the graph is parabola which doesn't touch the x-axis and, for the equation x³ - 6x² + 11x - 6, the graph cuts the x-axis at 3 points.