Grade by Grade
C. Polynomial and Rational Functions · Polynomial Functions and Equations

Families of Polynomial Functions and Equations from Given Conditions

C1.8 C1.7 C1.5About 70 minutes
Learning goals
  • I can write the equation of the family of polynomial functions with given zeros.
  • I can find the member of a family that passes through a given point.
  • I can write the equation of a polynomial function from conditions such as degree, zeros and their orders, a point, or symmetry.
  • I can read the zeros, their orders and one other point from a graph and write its equation.

Words to know

family of functions
A set of functions that share a feature. Polynomial functions with the same zeros form a family y = k(x − r₁)(x − r₂)...(x − rₙ), k ≠ 0.
member of a family
One function from the family, chosen by giving k a particular value.
conjugate pair
Two irrational zeros such as 1 + √3 and 1 − √3. Their factors multiply to a quadratic with whole-number coefficients.

Same zeros, different stretches

Remember from Grade 11: in Quadratic Functions you wrote families of parabolas y = a(x − r)(x − s) with the same zeros. The same idea works for any degree.

All polynomial functions with zeros r₁, r₂, ..., rₙ (and no others) can be written as

y = k(x − r₁)(x − r₂)...(x − rₙ), k ≠ 0.

Each value of k gives one member of the family. Every member crosses the x-axis at the same places. The value of k stretches the graph vertically, flips it when k < 0, and changes the y-intercept.

To pick out one member, you need one more piece of information, usually a point on the graph. Substitute it and solve for k.

xy−2135−20−101020k = 1k = 2k = −1k = ½
Four members of the family y = k(x + 1)(x − 2)(x − 4). They all share the zeros −1, 2 and 4, but each value of k gives a different stretch, and k < 0 flips the graph.
Worked example

Example 1: A member through a point

(a) Write the family of cubic functions with zeros −1, 2 and 4. (b) Find the member that passes through (1, 12) and write it in standard form.

  1. (a) y = k(x + 1)(x − 2)(x − 4).
  2. (b) Substitute x = 1, y = 12: 12 = k(2)(−1)(−3) = 6k, so k = 2.
  3. Expand: (x + 1)(x − 2) = x² − x − 2. Times (x − 4): x³ − 5x² + 2x + 8. Times 2: 2x³ − 10x² + 4x + 16.

y = 2(x + 1)(x − 2)(x − 4) = 2x³ − 10x² + 4x + 16

Worked example

Example 2: Orders and a y-intercept

Find the quartic function with a zero of order 2 at x = −2, zeros at 1 and 3, and a y-intercept of −6.

  1. Family: y = k(x + 2)²(x − 1)(x − 3). The degree is 2 + 1 + 1 = 4 ✓.
  2. The y-intercept is the value at x = 0: k(4)(−1)(−3) = 12k = −6, so k = −½.

y = −½(x + 2)²(x − 1)(x − 3)

Worked example

Example 3: Irrational zeros

Find the cubic function with zeros 2, 1 + √3 and 1 − √3 that passes through (0, 8). Give integer coefficients in standard form.

  1. Multiply the conjugate factors first: (x − 1 − √3)(x − 1 + √3) = (x − 1)² − (√3)² = x² − 2x + 1 − 3 = x² − 2x − 2.
  2. Family: y = k(x − 2)(x² − 2x − 2).
  3. At x = 0: k(−2)(−2) = 4k = 8, so k = 2.
  4. Expand: (x − 2)(x² − 2x − 2) = x³ − 4x² + 2x + 4; times 2.

y = 2x³ − 8x² + 4x + 8

Worked example

Example 4: Reading an equation from a graph

The quartic graph below crosses the x-axis at −3 and 4, touches it at 1, and passes through (2, 10). Find its equation.

  1. Crossing at −3 and 4: factors (x + 3) and (x − 4). Touching at 1: factor (x − 1)². Degree 1 + 2 + 1 = 4 ✓.
  2. y = k(x + 3)(x − 1)²(x − 4).
  3. Use (2, 10): k(5)(1)(−2) = −10k = 10, so k = −1.
  4. Check the end behaviour: k < 0 and even degree, so both ends go down, as in the graph ✓. Check the y-intercept: −(3)(1)(−4) = 12 ✓.

y = −(x + 3)(x − 1)²(x − 4)

xy−4−2−1235−20204060−314(2, 10)(0, 12)
y = −(x + 3)(x − 1)²(x − 4): crosses at −3 and 4, touches at 1, passes through (2, 10) and (0, 12). Both ends go down because k = −1 and the degree is 4.
Worked example

Example 5: Using symmetry as a condition

An odd cubic function has a zero at x = 3 and passes through (1, −16). Find its equation.

  1. Odd means f(−x) = −f(x). So if 3 is a zero, then f(−3) = −f(3) = 0: −3 is also a zero. And f(0) = 0 for every odd function.
  2. So the zeros are −3, 0, 3: y = kx(x − 3)(x + 3) = k(x³ − 9x).
  3. At x = 1: k(1 − 9) = −8k = −16, so k = 2.

y = 2x³ − 18x

A checklist for "find the equation" questions

  1. List the zeros and their orders (from the conditions or the graph). Add zeros forced by symmetry.
  2. Check that the orders add up to the degree.
  3. Write the family with k.
  4. Use one extra point (often the y-intercept) to find k.
  5. Check: end behaviour, y-intercept, and any other given point.
Just for fun

At the polynomial family reunion, everyone has exactly the same zeros, but nobody can agree on how tall to stand. The relative with k = −1 insists on standing on their head.

Hands-on activity

Design a toy-car hill

You need: grid paper, a pencil and a calculator.

You are drawing the side view of a hill for a toy-car track. The x-axis is the level table top (1 unit = 1 cm). The hill must leave the table smoothly (touching, not crossing) at x = 0, come back down to the table at x = 6, and pass through the point (2, 4).

  1. Explain why y = kx²(x − 6) has the right zeros and orders.
  2. Use (2, 4) to find k.
  3. Make a table for x = 0 to 6 and plot the hill.
  4. From your table, where is the hill highest, and how high is it?
  5. Your little cousin wants a hill only 4 cm high at its highest point. Which number in your equation would you change, and to what?

What to look for: a zero of order 2 at 0 makes the curve leave the table smoothly; a zero of order 1 at 6 makes it cross the x-axis there. k(4)(−4) = −16k = 4 gives k = −¼, so y = −¼x²(x − 6). Values for x = 0 to 6: 0, 1.25, 4, 6.75, 8, 6.25, 0. The highest point is about 8 cm at x = 4. Halving k to −⅛ halves every height, giving a 4 cm hill with the same zeros (a different member of the same family).

Your turn

1

Write the family of cubic functions with zeros −3, 0 and 5.

2

Find the member of y = k(x + 1)(x − 2)(x − 4) that passes through (3, 24).

3

A quartic has zeros ±1 and ±2 and a y-intercept of 8. Find its equation.

4

A cubic has a zero of order 2 at x = 2, a zero at x = −1, and passes through (1, 4). Find its equation.

5

Write the family of cubic functions with zeros 3, 1 + √2 and 1 − √2, using integer coefficients inside the brackets.

6

Are y = 2(x − 1)(x − 2)(x − 3) and y = (2x − 2)(x − 2)(x − 3) the same function?

7

Write the quartic with a zero of order 3 at x = −1, a zero at x = 2, and leading coefficient 3.

More practice: warm-up

1

What do all members of a family y = k(x − 1)(x − 4)(x + 2) have in common?

2

What does the value of k change?

3

Can k = 0?

4

Give the y-intercept of y = k(x − 1)(x − 2)(x − 3) in terms of k.

5

Write two members of the family with zeros −2 and 5 (degree 2).

More practice: on your own

1

A cubic has zeros −2, ½ and 3 and passes through (2, −15). Find its equation.

2

A quartic has a zero of order 2 at x = 0 and zeros at ±3, and passes through (1, −16). Find its equation.

3

A cubic graph touches the x-axis at the origin, crosses at x = 3, and passes through (2, −8). Find its equation.

4

An even quartic has zeros at 1 and 3 and a y-intercept of 18. Find its equation.

5

Find the member of y = k(x − 2 − √5)(x − 2 + √5)(x + 1) that passes through (0, 2).

Challenge

1

A cubic function with leading coefficient 1 has zeros 1 and 3 and a y-intercept of 6. Find the third zero.

2

Find the quartic function with zeros ±√2 and a zero of order 2 at x = 3 that passes through (0, −36). Use integer coefficients.

3

Explain why no member of y = k(x − 1)(x − 2)(x − 3) passes through both (0, 6) and (5, 10).

Check your understanding

Quiz

1.

Which equation describes the family of cubic functions with zeros 0, 2 and −5?

2.

The family y = k(x − 1)(x + 2)(x − 3) has a member through (0, 12). What is k?

3.

A quartic touches the x-axis at x = 2 and crosses it at x = −1 and x = 4. Which form fits?

4.

Which quadratic factor has the zeros 3 + √2 and 3 − √2?

5.

An odd cubic function has a zero at x = 2 and passes through (1, 6). What is its equation?

6.

How many conditions beyond the zeros are needed to pick out one member of a family y = k(x − r₁)(x − r₂)(x − r₃)?

For parents and teachers

Big idea (C1.7, C1.8): polynomials with the same zeros form a family y = k(x − r₁)...(x − rₙ). One extra condition, usually a point, fixes k. Orders of zeros (touch or cross) and symmetry (even or odd) are also conditions you can build in.

How to help: Ask your teen to explain each factor in their answer ("Why is (x + 3) squared?") and to check the final equation with the given point. Graphing the answer is a great check.

Common difficulties:

  • Sign errors: the zero −3 gives (x + 3).
  • Forgetting to square (or cube) factors for repeated zeros.
  • Arithmetic slips when substituting the point; substitute carefully, one factor at a time.
  • With irrational zeros, not multiplying the conjugate pair first, which makes expanding much harder.