Polynomial Functions & Their Graphs — Challenge
The degree and the leading coefficient decide the shape before you plot a single point.
What these problems cover
These questions use 8 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Exponent — How many times the base is used as a factor. In 53 the exponent is 3.
- Factor — A number that divides evenly into another.
- y-intercept — The output when x = 0 — where the line crosses the vertical axis.
- Polynomial — A sum of terms, each a number times a whole-number power of the variable.
- Roots — The x values where y = 0.
- Leading coefficient — The number in front of the highest power. Its sign decides which way the right end goes.
- Multiplicity — How many times a root repeats. It decides whether the graph crosses or just touches.
- Turning point — Where the graph changes from rising to falling or back. At most n - 1 of them.
Problem 1
What is the degree of a cubic polynomial?
- 1
- 2
- 3
- 4
- 5
Worked answer
3
Cubic refers to the third power, as in a cube.
So the highest power of the variable is 3, and the degree is 3.
Problem 2
As x grows large and positive, what happens to y=x3?
- It increases without bound
- It decreases without bound
- It approaches 0
- It approaches 1
- It oscillates
Worked answer
It increases without bound
Cubing a large positive number gives a much larger positive number.
There is no ceiling: for any target, a large enough x exceeds it.
So y increases without bound. For large negative x the cube is large and negative instead.
Problem 3
At most how many real roots can a quadratic have?
- 1
- 2
- 3
- 4
- 0
Worked answer
2
A polynomial of degree n has at most n real roots.
A quadratic has degree 2, so it has at most two.
It may have one repeated root, or none at all, as with x2+1.
Problem 4
What is the y-intercept of y=x3-2x+5?
- -2
- 0
- 3
- 5
- 1
Worked answer
5
The y-intercept is the value when x=0.
Every term with an x vanishes, leaving just the constant: y = 5.
For any polynomial written this way, the constant term is the y-intercept.
Problem 5
How many x-intercepts does y=(x-1)(x+2)(x-3) have?
- 1
- 2
- 3
- 4
- 0
Worked answer
3
The graph meets the x-axis where y=0, and a product is zero when any factor is zero.
That happens at x = 1, x = -2 and x = 3.
All three are different, so there are three distinct intercepts.
Problem 6
At most how many x-intercepts can a polynomial of degree 4 have?
- 2
- 3
- 4
- 5
- 8
Worked answer
4
Each x-intercept corresponds to a real root.
A polynomial of degree n has at most n roots, so a quartic has at most four.
It may have fewer: x4+1 has none at all.
Problem 7
The graph of y=x4 is symmetric about what?
- The y-axis
- The x-axis
- The origin
- The line y=x
- It has no symmetry
Worked answer
The y-axis
Replacing x by -x gives (-x)4 = x4, the same value, because the power is even.
So the point at -x has the same height as the point at x, which is mirror symmetry in the y-axis.
An odd power such as x3 would instead be symmetric about the origin.
Problem 8
What is the leading coefficient of -2x5+x?
- -2
- 2
- 5
- 1
- -5
Worked answer
-2
The leading term is the one of highest degree, which is -2x5.
Its coefficient, sign included, is -2.
The 5 is the degree, and the leading coefficient's sign decides which way the graph ends.
Problem 9
As x becomes large and negative, what happens to y=x3?
- It increases without bound
- It decreases without bound
- It approaches 0
- It approaches 1
- It oscillates
Worked answer
It decreases without bound
An odd power keeps the sign of its base, so cubing a negative number gives a negative result.
Cubing a large negative number gives a much larger negative number, such as (-10)3 = -1000.
So y decreases without bound. An even power such as x4 would rise at both ends instead.