A. Quadratic Functions · Quadratic Functions

Comparing the Three Forms of a Quadratic Function

A2.10 A2.7 A2.8 A2.9About 65 minutes

Learning goals

I can describe what the standard, vertex and factored forms each show directly.
I can choose the best form to answer a question.
I can convert between the three forms.
I can explain why some quadratic functions have no real factored form.

Words to know

standard form
f(x) = ax² + bx + c; shows a and the y-intercept c.

vertex form
f(x) = a(x − h)² + k; shows a, the vertex (h, k) and the axis x = h.

factored form
f(x) = a(x − r)(x − s); shows a and the zeros r and s.

leading coefficient
The value a; it is the same in every form of the same quadratic function.

Three outfits, one function

You now know three ways to write a quadratic function. They are not three different functions: they are one function written three ways, and each way puts different information on display.

Standard form: f(x) = ax² + bx + c
Vertex form: f(x) = a(x − h)² + k
Factored form: f(x) = a(x − r)(x − s)

The value of a is the same in all three forms. It always tells you the direction of opening and the vertical stretch or compression.

One parabola, three equations. Standard form shows the y-intercept −10, vertex form shows the vertex (2, −18), and factored form shows the zeros −1 and 5. All three have a = 2.

form	shows directly	takes a little work
standard f(x) = ax² + bx + c	a (direction, shape) and the y-intercept c	vertex: h = −b ÷ (2a), or complete the square; zeros: factor or use the quadratic formula
vertex f(x) = a(x − h)² + k	a, the vertex (h, k), the axis x = h, the max or min value k, the range	y-intercept: ah² + k; zeros: solve a(x − h)² + k = 0
factored f(x) = a(x − r)(x − s)	a, the zeros r and s	axis x = (r + s) ÷ 2; vertex: substitute; y-intercept: a·r·s

What each form tells you at a glance.

Worked example

Example 1: Read every form

f(x) = 2x² − 8x − 10 = 2(x − 2)² − 18 = 2(x + 1)(x − 5). Use the best form for each feature, and verify that the forms agree.

Direction and shape: a = 2 in every form: opens up, vertical stretch by 2.
y-intercept (standard form): −10.
Vertex, axis, minimum, range (vertex form): vertex (2, −18), axis x = 2, minimum −18, R = {y ∈ ℝ | y ≥ −18}.
Zeros (factored form): −1 and 5.
Check: the zeros are symmetric about x = 2 (−1 is 3 left of 2 and 5 is 3 right) ✓. Expanding 2(x + 1)(x − 5) = 2(x² − 4x − 5) = 2x² − 8x − 10 ✓.

a = 2, y-intercept −10, vertex (2, −18), zeros −1 and 5.

Pick the form that answers your question directly. If the function is in a different form, convert it first.

Worked example

Example 2: Choosing a form in context

A ball's height in metres after t seconds is h(t) = −5t² + 10t + 15. (a) Write it in vertex form and in factored form. (b) Use the best form to find the starting height, the maximum height, and when it lands.

(a) Complete the square: −5(t² − 2t) + 15 = −5(t² − 2t + 1) + 5 + 15 = −5(t − 1)² + 20.
Factor: −5(t² − 2t − 3) = −5(t − 3)(t + 1).
(b) Starting height (standard form, c): 15 m.
Maximum height (vertex form): 20 m, at t = 1 s.
Landing (factored form): t = 3 or t = −1. Reject the negative time. It lands at 3 s.

Starts at 15 m, reaches 20 m at 1 s, lands at 3 s.

Worked example

Example 3: Matching equations to a graph

A parabola opens up, has zeros −2 and 4 and y-intercept −8. Which equations match it? A: f(x) = (x + 2)(x − 4) B: f(x) = −(x + 2)(x − 4) C: f(x) = (x − 2)(x + 4) D: f(x) = (x − 1)² − 9

A: zeros −2 and 4; opens up; f(0) = (2)(−4) = −8. Matches.
B: same zeros, but a = −1, so it opens down (and f(0) = 8). No.
C: zeros 2 and −4. No.
D: vertex (1, −9). Zeros: (x − 1)² = 9, x − 1 = ±3, x = 4 or −2. f(0) = 1 − 9 = −8. Opens up. Matches.
A and D are the same function in two forms: (x + 2)(x − 4) = x² − 2x − 8 = (x − 1)² − 9.

A and D both match; they are equivalent.

When factored form does not exist

Every quadratic has a standard form and a vertex form. Completing the square always works. But factored form with real numbers needs real zeros.

f(x) = (x − 3)² + 2 opens up and its lowest point (3, 2) is above the x-axis. It never crosses the x-axis, so it has no real zeros and cannot be written as a(x − r)(x − s) with real r and s. The discriminant you met in Quadratic Expressions and Equations gives the same verdict: x² − 6x + 11 has b² − 4ac = 36 − 44 < 0.

Just for fun

Why does a quadratic function own three outfits?

Standard form for school photos (it shows off the intercept), vertex form for mountaintop selfies, and factored form for when it really needs to get down to zero.

Hands-on activity

Form-matching card game

You need: 12 small cards or slips of paper and a pen. A partner is optional.

Write each of these 12 expressions on its own card:

(x − 1)(x − 3), x² − 4x + 3, (x − 2)² − 1, −(x + 2)(x − 4), −x² + 2x + 8, −(x − 1)² + 9, 2(x − 1)(x + 3), 2x² + 4x − 6, 2(x + 1)² − 8, (x − 5)², x² − 10x + 25, (x − 5)(x − 5).

Shuffle the cards and lay them face up.
Sort them into four groups of three equivalent expressions. Justify each match by expanding, factoring or completing the square.
For each group, write down the vertex, the zeros and the y-intercept, and say which card shows each one directly.
Challenge your partner to a timed rematch, or make a new set of your own.

What to look for: the groups are {(x − 1)(x − 3), x² − 4x + 3, (x − 2)² − 1}: vertex (2, −1), zeros 1 and 3, y-intercept 3; {−(x + 2)(x − 4), −x² + 2x + 8, −(x − 1)² + 9}: vertex (1, 9), zeros −2 and 4, y-intercept 8; {2(x − 1)(x + 3), 2x² + 4x − 6, 2(x + 1)² − 8}: vertex (−1, −8), zeros 1 and −3, y-intercept −6; and {(x − 5)², x² − 10x + 25, (x − 5)(x − 5)}: vertex (5, 0), one zero 5, y-intercept 25.

Your turn

From f(x) = −3(x − 4)² + 7, state the vertex, axis, maximum or minimum value, and direction.

From f(x) = 2(x + 1)(x − 7), state the zeros, direction and axis.

From f(x) = x² − 5x + 6, state the y-intercept and direction.

Which form answers each question directly? (a) the maximum height (b) when it lands (c) the starting height

Write f(x) = (x − 3)(x + 1) in standard form and in vertex form.

Explain why f(x) = x² + 4 cannot be written in factored form with real numbers.

More practice: warm-up

Name the form: f(x) = 5(x − 2)(x + 3).

Name the form: f(x) = −(x + 6)² − 1.

Is the value of a the same in all three forms of one function?

State the y-intercept of f(x) = 3x² + x − 4.

State the zeros of f(x) = (x − 9)(x + 2).

More practice: on your own

Write f(x) = −2x² + 4x + 6 in factored form and vertex form. State the zeros, vertex and y-intercept.

f(x) = (x + 2)² − 9. Find the zeros, then write f in factored and standard form.

Which of f(x) = 2(x − 1)² + 3, g(x) = −2(x − 1)² + 3 and k(x) = 2(x − 1)² − 3 have real zeros? Explain.

A ball's height is h(t) = −5t² + 20t + 25 metres. Write h in vertex and factored form, then state the starting height, maximum height and landing time.

A parabola opens down, has vertex (3, 8) and zeros 1 and 5. Write its equation in all three forms.

Challenge

A quadratic function has zeros 2 and 6 and y-intercept 12. Write it in all three forms.

Explain why every quadratic function has a vertex form, but not every one has a factored form with real numbers.

Compare f(x) = (x − 2)(x − 8) and g(x) = 0.5(x − 2)(x − 8): zeros, axis, vertex and y-intercept.

Check your understanding

Quiz

Which form shows the vertex directly?

What are the zeros of f(x) = 4(x − 3)(x + 5)?

Which value is the same in all three forms of one quadratic function?

How many real zeros does f(x) = 2(x − 1)² + 5 have?

A ball's height is h(t) = −5(t − 4)(t + 1). Which question does this form answer most directly?

What is the y-intercept of f(x) = 3(x − 2)(x − 1)?

For parents and teachers

Big idea: expectation A2.10 asks students to describe the information that can be obtained by inspecting the standard form, the vertex form and the factored form of a quadratic function.

How to help: the card game in the activity is a good family challenge. When your teen solves a word problem, ask "Which form would answer that question fastest?"

Common difficulties:

Treating the three forms as three different functions.
Reading the vertex from standard form (c is the y-intercept, not k).
Sign errors when reading h, r and s.
Expecting every quadratic to factor.