4.2 Quartic Graphs
Loading...
...
Learning Objectives
- Understand the general shape of quartic functions
- Sketch graphs of basic quartic functions from their equations
- Recognise the effect of transformations on quartic functions
- Identify turning points and roots of quartic graphs
- Solve equations and inequalities involving quartic functions
Key Concepts
What Is a Quartic Function?
A **quartic function** is a polynomial of degree 4, typically written as _y = ax⁴ + bx³ + cx² + dx + e_.
The highest power of _x_ is 4, and the graph of a quartic function can have up to **3 turning points** and **4 real roots**.
The general shape of the graph depends on the sign and value of the leading coefficient _a_.
Shapes of Quartic Graphs
If _a > 0_, the ends of the graph rise (like a 'W' shape or flattened 'U').
If _a < 0_, the ends fall (like an 'M' or upside-down 'U').
Quartic graphs can have **0, 1, 2, 3 or 4 real roots**, depending on the discriminant and shape.
Example 1
Sketch the graph of y = x⁴ − 5x²
Solution:
Factor: y = x²(x² − 5) = x²(x − √5)(x + √5). Roots at x = 0, ±√5. The graph is symmetrical about the y-axis and has a 'W' shape since the leading coefficient is positive.
Turning Points and Symmetry
Quartic graphs may have **up to 3 turning points** (local maxima or minima).
If a graph is even-powered and has no odd powers of _x_, it is **symmetric about the y-axis**.
Turning points can be estimated by sketching or found using calculus.
Transformations of Quartic Graphs
Transformations follow similar rules as with quadratics and cubics:
- _y = (x − a)⁴_ translates the graph **right** by _a_ units.
- _y = x⁴ + b_ shifts it **up** by _b_ units.
- _y = −x⁴_ reflects it in the **x-axis**.
- _y = (kx)⁴_ compresses or stretches it **horizontally**.
Solving Quartic Equations and Inequalities
To solve equations like _x⁴ − 5x² = 0_, factorise where possible.
To solve inequalities, sketch the graph and determine where it lies above or below the x-axis.
Solutions may be given as intervals depending on where the graph is positive or negative.
Example 1
Solve the inequality x⁴ − 4x² < 0
Solution:
Factor: x²(x² − 4) < 0 → x²(x − 2)(x + 2) < 0. Sketch the graph: it crosses at x = −2, 0, and 2. Inequality is satisfied in the interval (−2, 0) ∪ (0, 2).
Test Your Knowledge
Ready to test your understanding of 4.2 Quartic Graphs?