Do I have this right?
A first degree polynomial crosses the x axis
A second degree polynomial touches the y axis without crosisng
A third degree polynomial flattens against the y axis.
Not quite. Let me explain the concepts of polynomial functions and crossing/touching the x/y axis.
A polynomial function is an algebraic expression that consists of terms comprising variables raised to non-negative integer exponents, multiplied by coefficients. The highest power of the variable in a polynomial determines the degree of the polynomial.
- A first-degree polynomial is called a linear function. It has a degree of 1 and represents a straight line. It can both cross or touch the x-axis, depending on the slope of the line. If the slope is non-zero, the line will cross the x-axis. If the slope is zero, the line will touch the x-axis at a single point.
- A second-degree polynomial is known as a quadratic function. It has a degree of 2 and can take the form of a parabola. A quadratic function can cross the x-axis at most twice or touch the x-axis at a single point. Whether it crosses or touches the x-axis depends on the vertex (the highest or lowest point) of the parabola and the direction in which it opens. If the vertex is above the x-axis, the parabola will cross the x-axis at two points. If the vertex lies on the x-axis, the parabola will touch the x-axis at that point. If the vertex is below the x-axis, the parabola will not intersect the x-axis at all.
- A third-degree polynomial is a cubic function with a degree of 3. Cubic functions can have various shapes and behaviors. It is not accurate to state that a cubic function "flattens against the y-axis." A cubic function can cross the y-axis, touch the y-axis, or not interact with the y-axis at all. The behavior of the cubic function is determined by various factors, including the coefficients and the number of real roots it possesses.
In summary, the crossing or touching of the x-axis and y-axis depends on the specific characteristics of the polynomial function, such as its degree and coefficients. The behavior and shape of a polynomial function can be determined by analyzing its equation or graphing it.