Graphing Cubic Functions
A cubic function has the form y = ax^3 + bx^2 + cx + d. Its graph has an S-shaped curve with up to two turning points and passes through all y-values.
What You Need to Know
Key Concept Diagram
Basic cubic y = x^3 passes through (-1,-1), (0,0), (1,1) with an inflection at origin
If a > 0 the curve rises left-to-right; if a < 0 it falls left-to-right
Cubics can have 1, 2 or 3 x-intercepts (real roots)
The y-intercept occurs at x = 0: substitute to find d
Key Vocabulary
Cubic function
A polynomial function where the highest power of x is 3
Inflection point
Where the curve changes from concave up to concave down (or vice versa)
Turning point
A local maximum or minimum on the curve
Root
An x-value where the function equals zero (x-intercept)
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the y-intercept of y = 2x^3 - 3x^2 + x - 4?
Question 2
How many x-intercepts can a cubic function have at most?
Question 3
For y = -x^3, as x increases toward positive infinity, y:
Key Concepts Summary
- ●Basic cubic y = x^3 passes through (-1,-1), (0,0), (1,1) with an inflection at origin
- ●If a > 0 the curve rises left-to-right; if a < 0 it falls left-to-right
- ●Cubics can have 1, 2 or 3 x-intercepts (real roots)
- ●The y-intercept occurs at x = 0: substitute to find d