Sketching Polynomials

y-axis Intercept: This is the point where the curve crosses the vertical axis, found by setting $x = 0$. For any polynomial $f(x) = a_n x^n + ... + a_0$, the $y$-intercept is always at the coordinate $(0, a_0)$.

x-axis Intercepts (Roots): These are the solutions to $f(x) = 0$. They represent where the graph meets the horizontal axis. A polynomial of degree $n$ can have a maximum of $n$ real roots.

Multiplicity and Root Behavior: If a root $(x - k)$ is repeated twice ($n=2$), the graph touches the axis at $(k, 0)$ and turns back (tangency). If it is repeated three times ($n=3$), the graph forms a stationary point of inflection and crosses the axis.

Leading Term Dominance: As $x \to \pm \infty$, the behavior of the polynomial is dictated entirely by the highest power term $a_n x^n$. All other terms become negligible in comparison at large values.

Degree Parity: Even-degree polynomials ($x^2, x^4$) have the same end behavior at both ends (both up or both down). Odd-degree polynomials ($x^3, x^5$) have opposite end behaviors (one up, one down).

Leading Coefficient Sign: For a positive leading coefficient, the graph always ends by going 'up' as $x \to \infty$. Conversely, a negative leading coefficient forces the graph 'down' as $x \to \infty$.

Turning Points: These are local maxima or minima where the curve changes direction. A polynomial of degree $n$ has at most $n-1$ turning points.

Using Differentiation: Turning points are found by solving the derivative equation $f'(x) = 0$. This identifies the $x$-coordinates where the gradient is zero.

Verification: By calculating the coordinates of these turning points, you can ensure your sketch correctly portrays the peaks and valleys of the function between the identified roots.

Checklist for Sketching: Before drawing, explicitly list the $y$-intercept, the roots (with their behavior), and the quadrants where the graph begins and ends.

Consistency Check: Ensure your turning points sit logically between your roots. If a turning point is found at $y=5$ but the graph must cross the $x$-axis between those points, there must be an error in calculation.

Coordinate Labeling: In a sketch, you must mark all axial intercepts. Use coordinates like $(k, 0)$ or simply mark the value on the axis to demonstrate precision.