M
Math Coach Amy
Amy Ferguson Moncure

H Algebra 2/Trig · Unit 3

Turning Points

Find how many turning points are possible, read them from a graph, and sketch a polynomial that fits.

Jump to section:

Key Ideas Easy Medium Hard

Degree n: at most n − 1

R turning points: degree at least R + 1

Odd degree: 0, 2, 4, …

Even degree: 1, 3, 5, …

What to Remember

How many

A turning point is where the graph changes direction.

Degree n has at most n − 1 turning points.

R turning points means the degree is at least R + 1.

Odd and even degree

An odd degree has an even number of turning points. The fewest is 0.

An even degree has an odd number of turning points. The fewest is 1.

Two turning points on a positive odd graph give two increasing intervals and one decreasing interval.

Real zeros

Real zeros are the x-intercepts.

Imaginary zeros can add turning points. A graph can turn when it has fewer real zeros than the degree.

Level 1

Easy

Use the degree, or the number of turning points.

Example
A polynomial has degree 5. At most, how many turning points can it have? Sketch a graph with that many turning points.

Degree 5

At most n − 1

5 − 1

4

Answer: 4

1
A polynomial has degree 8. At most, how many turning points can it have? Sketch a graph with that many turning points.
Show Answer

Degree 8

At most n − 1

8 − 1

7

Answer: 7

2
A polynomial has degree 6. At most, how many turning points can it have? Sketch a graph with that many turning points.
Show Answer

Degree 6

At most n − 1

6 − 1

5

Answer: 5

3
A polynomial has degree 7. At most, how many turning points can it have? Sketch a graph with that many turning points.
Show Answer

Degree 7

At most n − 1

7 − 1

6

Answer: 6

4
A polynomial has 4 turning points. What is the least degree it can have? Sketch a graph with that degree and that many turning points.
Show Answer

4 turning points

At least R + 1

4 + 1

5

Answer: 5

5
A polynomial has 5 turning points. What is the least degree it can have? Sketch a graph with that degree and that many turning points.
Show Answer

5 turning points

At least R + 1

5 + 1

6

Answer: 6

Level 2

Medium

Count the turns, then the x-intercepts.

Example
How many turning points? What is the least degree? How many real zeros?
−2−1123456−4−22468101214

2 turning points

Least degree

2 + 1

3

Real zeros

1

Imaginary zeros can add turning points.

−2−1123456−4−22468101214

Answer: 2 turning points
Least degree 3
1 real zero

1
How many turning points? What is the least degree? How many real zeros?
−4−3−2−11234567−6−4−2246810121416
Show Answer

1 turning point

Least degree

1 + 1

2

Real zeros

2

−4−3−2−11234567−6−4−2246810121416

Answer: 1 turning point
Least degree 2
2 real zeros

2
How many turning points? What is the least degree? How many real zeros?
−112345−3−2−112345
Show Answer

0 turning points

Least degree

0 + 1

1

Real zeros

1

−112345−3−2−112345

Answer: 0 turning points
Least degree 1
1 real zero

3
How many turning points? What is the least degree? How many real zeros?
−6−5−4−3−2−1123456−7−6−5−4−3−2−1123
Show Answer

3 turning points

Least degree

3 + 1

4

Real zeros

2

Imaginary zeros can add turning points.

−6−5−4−3−2−1123456−7−6−5−4−3−2−1123

Answer: 3 turning points
Least degree 4
2 real zeros

4
How many turning points? What is the least degree? How many real zeros?
−6−4−2246−8−6−4−22468101214161820
Show Answer

4 turning points

Least degree

4 + 1

5

Real zeros

5

−6−4−2246−8−6−4−22468101214161820

Answer: 4 turning points
Least degree 5
5 real zeros

5
How many turning points? What is the least degree? How many real zeros?
−6−5−4−3−2−1123456−4−3−2−1123456
Show Answer

3 turning points

Least degree

3 + 1

4

Real zeros

4

−6−5−4−3−2−1123456−4−3−2−1123456

Answer: 3 turning points
Least degree 4
4 real zeros

Level 3

Hard

Match the degree, the turns, and the zeros.

Example
Sketch a graph that fits. If the conditions are not possible, explain why.
Degree 5, the leading coefficient is negative, there is one real zero, and the graph decreases on (−∞, ∞).

Degree 5, negative

At most n − 1

5 − 1

4

Odd degree: 0, 2, or 4

Decreasing on (−∞, ∞) means 0 turning points

0 is allowed

Answer: Possible

1
Sketch a graph that fits. If the conditions are not possible, explain why.
Degree 8 and no real zeros.
Show Answer

Degree 8

At most n − 1

8 − 1

7

Even degree: 1, 3, 5, or 7

Both ends go up

The graph stays above the x-axis

3 turning points is allowed

Answer: Possible

2
Sketch a graph that fits. If the conditions are not possible, explain why.
Degree 4, only two real zeros, at x = −3 and x = 4, and no repeated zeros.
Show Answer

Degree 4

At most n − 1

4 − 1

3

Even degree: 1 or 3

Cross at x = −3 and x = 4

The graph does not cross again

3 turning points is allowed

−34

Answer: Possible

3
Sketch a graph that fits. If the conditions are not possible, explain why.
Degree 4, positive leading coefficient, a relative maximum at x = 1, and absolute minimums at x = −4 and x = 5.
Show Answer

Degree 4, positive

At most n − 1

4 − 1

3

Even degree: 1 or 3

Minimum at x = −4

Maximum at x = 1

Minimum at x = 5

−415

Answer: Possible

4
Sketch a graph that fits. If the conditions are not possible, explain why.
Degree 6 and no turning points.
Show Answer

Degree 6

At most n − 1

6 − 1

5

Even degree: 1, 3, or 5

0 is not allowed

Answer: Not possible

5
Sketch a graph that fits. If the conditions are not possible, explain why.
Degree 7 and only 5 turning points.
Show Answer

Degree 7

At most n − 1

7 − 1

6

Odd degree: 0, 2, 4, or 6

5 is not allowed

Answer: Not possible

Need help with turning points?

Book a Session with Amy