When we look at cubic functions on a graph, we see some interesting patterns that can help us understand how they work. A simple cubic function can be written like this:
f(x) = ax^3 + bx^2 + cx + d
Shape: The graph of a cubic function looks like an "S." This means that it can move up and down smoothly. Unlike quadratic functions, which only change direction once, cubic functions can change direction two times.
Intercepts: A cubic function can have up to three real roots. Roots are the points where the graph touches or crosses the x-axis. For example, in the function f(x) = x^3 - 3x^2 + 2x, the roots are at x = 0, x = 1, and x = 2.
Turning Points: There can be up to two turning points. These points are where the graph makes a peak (maxima) or a dip (minima). They help shape the curve of the graph.
Cubic functions often have a special kind of symmetry around their turning points. This makes the graph look nice and balanced.
By understanding these features, students can better predict and draw cubic functions!
When we look at cubic functions on a graph, we see some interesting patterns that can help us understand how they work. A simple cubic function can be written like this:
f(x) = ax^3 + bx^2 + cx + d
Shape: The graph of a cubic function looks like an "S." This means that it can move up and down smoothly. Unlike quadratic functions, which only change direction once, cubic functions can change direction two times.
Intercepts: A cubic function can have up to three real roots. Roots are the points where the graph touches or crosses the x-axis. For example, in the function f(x) = x^3 - 3x^2 + 2x, the roots are at x = 0, x = 1, and x = 2.
Turning Points: There can be up to two turning points. These points are where the graph makes a peak (maxima) or a dip (minima). They help shape the curve of the graph.
Cubic functions often have a special kind of symmetry around their turning points. This makes the graph look nice and balanced.
By understanding these features, students can better predict and draw cubic functions!