What can be inferred about the roots of a cubic function based on its graph?
Practice Questions
Q1
What can be inferred about the roots of a cubic function based on its graph?
It can have at most two real roots.
It can have at most three real roots.
It can have no real roots.
It must have at least one real root.
Questions & Step-by-Step Solutions
What can be inferred about the roots of a cubic function based on its graph?
Step 1: Understand what a cubic function is. A cubic function is a mathematical expression of the form f(x) = ax^3 + bx^2 + cx + d, where a, b, c, and d are constants and a is not zero.
Step 2: Know that the graph of a cubic function is a continuous curve. This means there are no breaks or gaps in the graph.
Step 3: Learn about the Intermediate Value Theorem. This theorem states that if a function is continuous on an interval and takes on different values at the endpoints of that interval, then it must cross the x-axis at least once within that interval.
Step 4: Apply the Intermediate Value Theorem to cubic functions. Since a cubic function is continuous, it must cross the x-axis at least once, which means it has at least one real root.
Step 5: Recognize that a cubic function can have up to three real roots. The graph can touch or cross the x-axis at one, two, or three points, depending on the specific function.
