A. Characteristics of Functions · Quadratic Functions

Families of Quadratic Functions with the Same Zeros

A2.4About 65 minutes

Learning goals

I can write the equation of the family of quadratic functions with given zeros.
I can describe how members of a family are related by vertical stretches, compressions and reflections.
I can find the member of a family that passes through a given point or has a given maximum or minimum.
I can explain which features all members of a family share.

Words to know

family of quadratic functions
All quadratic functions that share a property, such as the same zeros: f(x) = a(x − r)(x − s).

vertical stretch / compression
Multiplying every output by a factor |a| > 1 (stretch) or 0 < |a| < 1 (compression).

reflection in the x-axis
Multiplying every output by −1, which flips the graph over the x-axis.

Same zeros, different shapes

Two snow ramps at a terrain park can start and end at the same places on the ground and still have very different heights: one gentle, one steep. Both arches cross the ground at the same two points. In math language, they are quadratic functions with the same zeros.

All the quadratic functions that share a pair of zeros form a family. This lesson investigates how the members of a family are related (curriculum expectation A2.4) and how to pick out the one member you need.

The family equation

If a quadratic function has zeros r and s, then (x − r) and (x − s) are factors, so

f(x) = a(x − r)(x − s), a ≠ 0.

Every value of a gives a different member of the family. Each member has:

the same zeros, r and s;
the same axis of symmetry, x = (r + s)/2;
a different vertex height, and a different y-intercept;
opening up if a > 0 and down if a < 0.

Four members of the family f(x) = a(x − 1)(x − 5). All pass through (1, 0) and (5, 0) and share the axis of symmetry x = 3. Only the vertex height changes: −4, −8, −2 and 4.

The transformational relationship

Compare y = (x − 1)(x − 5) with y = 2(x − 1)(x − 5). For every x, the second output is exactly 2 times the first. So the second graph is a vertical stretch by a factor of 2 of the first.

The zeros do not move, because 2 × 0 = 0.
The vertex (3, −4) moves to (3, −8).
The y-intercept 5 moves to 10.

In general, every member of the family is a vertical stretch or compression of y = (x − r)(x − s) by a factor of |a|, with a reflection in the x-axis when a < 0. The points on the x-axis are the only points that stay fixed. (These are the same transformations, y = af(x), that you study in Functions and Equivalent Expressions.)

Solid curve: y = (x − 1)(x − 5). Dashed curve: y = 2(x − 1)(x − 5). Multiplying by 2 doubles every y-coordinate: (3, −4) becomes (3, −8) and (0, 5) becomes (0, 10). The zeros (1, 0) and (5, 0) do not move.

Worked example

Example 1: Write the family, then choose a member

(a) Write an equation for the family of quadratic functions with zeros −2 and 4. (b) Find the member that passes through (0, 8).

(a) The factors are (x + 2) and (x − 4), so the family is f(x) = a(x + 2)(x − 4).
(b) Substitute the point (0, 8): 8 = a(0 + 2)(0 − 4) = a(2)(−4) = −8a.
So a = −1.
The member is f(x) = −(x + 2)(x − 4). In standard form: −(x² − 2x − 8) = −x² + 2x + 8.

f(x) = −(x + 2)(x − 4), or −x² + 2x + 8.

Worked example

Example 2: Choose a member by its maximum

A member of the family with zeros 0 and 6 has a maximum value of 18. Find its equation.

Family: f(x) = a(x − 0)(x − 6) = ax(x − 6).
The axis is halfway between the zeros: x = 3. The maximum value happens there, so f(3) = 18.
f(3) = a(3)(3 − 6) = −9a. So −9a = 18 and a = −2.
a < 0, so the vertex really is a maximum ✓.

f(x) = −2x(x − 6).

Worked example

Example 3: Describe the transformation

Describe how the graph of g(x) = −½(x − 1)(x − 5) is related to y = (x − 1)(x − 5). Then give the vertex of g.

Each output of g is −½ times the output of y = (x − 1)(x − 5).
So g is a vertical compression by a factor of ½ and a reflection in the x-axis.
The vertex of y = (x − 1)(x − 5) is (3, −4). Multiply its y-coordinate by −½: (3, 2).
Check: g(3) = −½(2)(−2) = 2 ✓.

Compression by ½ and reflection in the x-axis; vertex (3, 2).

One point picks one member

The zeros fix the family. Any one more point that is not a zero (a y-intercept, a vertex, any point on the graph) fixes the value of a, and so picks exactly one member.

Hands-on activity

A family of ramps

You need: graph paper, a pencil, a ruler and three coloured pencils (any colours).

Draw axes and mark the zeros (2, 0) and (8, 0).
Fill in a table of values for y = (x − 2)(x − 8) at x = 2, 3, 4, 5, 6, 7, 8. Plot and join the points.
In two new columns, multiply each y-value by −1 and by −½. Plot those two members in different colours.
Measure the vertical distance from the x-axis to each vertex. How are the three distances related?

What to look for: the y-values of y = (x − 2)(x − 8) are 0, −5, −8, −9, −8, −5, 0. Multiplying by −1 gives an arch with maximum 9 at x = 5; multiplying by −½ gives an arch with maximum 4.5. All three curves pass through (2, 0) and (8, 0) and are symmetric about x = 5. The vertex distances are in the ratio 1 : 1 : ½.

Just for fun

Family reunion for the quadratics with zeros 1 and 5: everyone shows up at the same two points, stands on the same axis of symmetry, and then argues for hours about who is the most stretched.

Your turn

Write the family of quadratic functions with zeros 2 and 6.

Find the member of that family whose vertex is (4, −12).

Find the member of the family with zeros −3 and 1 that passes through (−1, −2).

A member of the family with zeros −1 and 7 has a maximum value of 16. Find it.

Find the member of the family with zeros 3 and −3 and y-intercept −18.

Describe how y = 4(x + 1)(x − 3) is related to y = (x + 1)(x − 3).

More practice: warm-up

Name the zeros of every member of f(x) = a(x − 5)(x + 2).

What is the axis of symmetry of every member of f(x) = a(x − 5)(x + 2)?

Does y = −3(x − 1)(x − 4) open up or down?

y = (x − 1)(x − 5) has vertex (3, −4). What is the vertex of y = 3(x − 1)(x − 5)?

Which transformation changes y = (x − 2)(x + 6) into y = −(x − 2)(x + 6)?

More practice: on your own

Find the member of the family with zeros −2 and 2 that has y-intercept 6.

Find the member of the family with zeros −1 and 3 that passes through (2, −12).

A member of the family with zeros 1 and 7 has a maximum value of 18. Find it and state its y-intercept.

Members of the family f(x) = a(x + 4)(x − 4) have a vertex on the y-axis. Find the member with vertex (0, 4).

An arch is 120 m wide at ground level and 36 m high in the middle. Place it with zeros at x = 0 and x = 120 and find its equation.

Challenge

Two members of the family with zeros 0 and 4 are f(x) = 5x(x − 4) and g(x) = −x(x − 4). How far apart are their vertices?

Show that every member of the family f(x) = a(x − r)(x − s) has its vertex at ((r + s)/2, −a(r − s)²/4).

Find the member of the family with zeros −5 and 1 that passes through the vertex of g(x) = (x + 2)² + 9. Is the vertex of g also the vertex of your function?

Check your understanding

Quiz

Which equation describes the family of quadratic functions with zeros 3 and −4?

What do all members of f(x) = a(x − 2)(x − 8) have in common?

The member of f(x) = a(x − 1)(x − 3) that passes through (0, 6) has a =

How is y = −3(x − 1)(x − 6) related to y = (x − 1)(x − 6)?

A member of the family with zeros 0 and 10 has a maximum value of 50. What is a?

Which points stay fixed when y = (x − r)(x − s) is stretched vertically?

For parents and teachers

Big idea: Expectation A2.4 (first part): investigate the transformational relationship among quadratic functions with the same zeros. Every member is a vertical stretch, compression or reflection of y = (x − r)(x − s).

How to help: Ask your teen to explain why changing a cannot move the zeros. Ask "How many points do you need to pin down one member?" (The zeros plus one more point.)

Common difficulties:

Sign errors in the factors: a zero of −2 gives (x + 2).
Substituting a zero as the extra point. It always gives 0 = 0 and cannot find a.
Confusing vertical stretches (multiply y) with horizontal ones.