Sketching Polynomials (AQA A-Level Mathematics): Revision Notes

Example: Sketch the following quartic $y = (x + 1)(x + 2)(x - 3)(x - 1)$

1. Find the roots (x-intercepts) where $y = 0$:

• Set $y = 0$: $(x + 1)(x + 2)(x - 3)(x - 1) = 0$

• The roots are $x = -2, -1, 1, 3$.

2. Find the y-intercept (where $x = 0$):

• Substitute $x = 0$: $y = (0 + 1)(0 + 2)(0 - 3)(0 - 1)$

$y = (1)(2)(-3)(-1) = 6$

3. Draw the graph:

• Plot the x-intercepts $x = -2, -1, 1, 3$.
• Plot the y-intercept $(0, 6)$.
• Draw a smooth curve passing through these points.

Important Note

• Positive Quartic:
• Starts from the top left and ends at the top right.
• Example: $y = x^4$.
• Negative Quartic:
• Starts from the bottom left and ends at the bottom right.
• Example: $y = -x^4$.

Summary Steps for Sketching Quartic Graphs

4. Find roots ($x$-intercepts):

• Set $y = 0$ and solve for $x$.

5. Find the y-intercept:

• Set $x = 0$ and solve for $y$.

6. Determine the end behaviour:

• If the coefficient of $x^4$ is positive, the graph starts and ends at the top.
• If the coefficient of $x^4$ is negative, the graph starts and ends at the bottom.

7. Sketch the graph:

• Plot the intercepts.
• Ensure the curve smoothly passes through these points and follows the correct end behaviour.

Example: Sketch $y = (x + 2)(x - 3)(x - 6)$ 8. Find the roots (x-intercepts) where $y = 0$:

• Set $y = 0:$
• $(x + 2)(x - 3)(x - 6) = 0$
• The roots are $x = -2, 3, 6$.

9. Find the y-intercept (where $x = 0$):

• Substitute $x = 0$:
• $y = (0 + 2)(0 - 3)(0 - 6)$
• $y = 2 \cdot (-3) \cdot (-6)$
• $y = 36$

10. Sketch the graph:

• Plot the x-intercepts $x = -2, 3, 6$.
• Plot the y-intercept $(0, 36)$.
• Draw a smooth curve passing through these points, starting from the bottom left (since the coefficient of $x^3$ is positive) and ending at the top right.

Points to Note

• Label all intercepts clearly on the graph.
• Ensure the curve does not go back on itself and smoothly passes through the intercepts.
• For the best example:
• Clearly mark the intercepts.
• Ensure the curve has the correct shape for the given cubic equation.

Example 1: Sketch $y = (6 - x)(x + 3)(x - 5)$

11. Find the roots ($x$-intercepts) where $y = 0$:

• Set $y = 0$: $(6 - x)(x + 3)(x - 5) = 0$

• The roots are $x = 6, -3, 5$.

12. Find the y-intercept (where $x = 0$):

• Substitute $x = 0:$ $y = (6 - 0)(0 + 3)(0 - 5)$

$y = (6)(3)(-5)$

$y = -90$

1. Sketch the graph:

• Plot the $x$-intercepts $x = 6, -3, 5$.
• Plot the y-intercept $(0, -90)$.
• Draw a smooth curve passing through these points.

Example 2: Sketch $y = (x + 2)(x - 3)(x + 2)$

13. Notice the double root:

• The equation has a double root at $x = -2$.

14. Rewrite the equation: $y = (x + 2)^2(x - 3)$

15. Find the roots (x-intercepts) where $y = 0$:

• Set $y = 0$: $(x + 2)^2(x - 3) = 0$

• The roots are $x = -2$ (double root), $x = 3$.

15. Find the y-intercept (where $x = 0$):

• Substitute $x = 0$: $y = (0 + 2)^2(0 - 3)$

$y = (2)^2(-3)$

$y = -12$

2. Sketch the graph:

• Plot the $x$-intercepts $x = -2, 3$.
• Plot the $y$-intercept $(0, -12)$.
• Draw a smooth curve passing through these points, noting the double root behaviour at $x = -2$.

Example 3: Sketch $y = (x - 3)^3$ 16. Identify the triple root:

• The equation has a triple root at $x = 3$.

17. Find the root ($x$-intercept) where $y = 0$:

• Set $y = 0$: $(x - 3)^3 = 0$

• The root is $x = 3$.

18. Find the y-intercept (where $x = 0$):

• Substitute $x = 0$: $y = (0 - 3)^3$

$y = -27$

3. Sketch the graph:

• Plot the $x$-intercept $x = 3$.
• Plot the y-intercept $(0, -27)$.
• Draw a smooth curve passing through these points, noting the triple root behaviour at $x = 3$.

Summary

• For single roots, the curve crosses the $x$-axis.
• For double roots, the curve touches the $x$-axis and turns around.
• For triple roots, the curve flattens at the $x$-axis.