Comparing Different Forms

Why Quadratics Have Different Forms

A quadratic function can often be written in several equivalent forms. The three most common are:

Standard form

y = ax² + bx + c

Vertex form

y = a(x − h)² + k

Factored form

y = a(x − r₁)(x − r₂)

These forms can represent exactly the same parabola, but each form makes different graph features easier to identify.

The important skill is not simply knowing all three forms. It is knowing which form is most useful for the information you need.

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The Same Quadratic in Three Forms

Consider these equations:

Standard form:

y = x² − 6x + 5

Factored form:

y = (x − 1)(x − 5)

Vertex form:

y = (x − 3)² − 4

All three equations describe the same quadratic function.

They therefore have the same:

• graph
• vertex
• roots
• x-intercepts
• y-intercept
• axis of symmetry
• maximum or minimum
• domain and range

Only the representation has changed.

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What Standard Form Reveals

Standard form is:

y = ax² + bx + c

The easiest graph feature to identify from standard form is the y-intercept.

At the y-intercept:

x = 0

Therefore:

y = a(0)² + b(0) + c

y = c

So the y-intercept is:

(0, c)

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Example: Reading Standard Form

Consider:

y = 2x² − 8x + 6

Immediately we know:

a = 2

b = −8

c = 6

Therefore:

• y-intercept = (0, 6)
• a > 0, so the parabola opens upward
• |a| > 1, so the parabola is vertically stretched compared with y = x²

The vertex and roots are not immediately obvious.

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What the Coefficient a Reveals

The coefficient a appears in all three forms.

It tells us about the direction and shape of the parabola.

If:

a > 0

the parabola opens upward.

If:

a < 0

the parabola opens downward.

If:

|a| > 1

the parabola is vertically stretched and appears narrower.

If:

0 < |a| < 1

the parabola is vertically compressed and appears wider.

So the value of a provides useful information regardless of the form being used.

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Finding the Vertex from Standard Form

The vertex is not immediately visible in standard form.

For:

y = ax² + bx + c

the x-coordinate of the vertex is:

x = −b/(2a)

Then substitute this x-value into the equation to find the y-coordinate.

For example:

y = x² − 6x + 5

a = 1

b = −6

So:

x = −(−6)/(2 × 1)

x = 3

Substitute x = 3:

y = 3² − 6(3) + 5

y = 9 − 18 + 5

y = −4

Therefore:

vertex = (3, −4)

This works, but vertex form would reveal the vertex much more quickly.

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What Vertex Form Reveals

Vertex form is:

y = a(x − h)² + k

The vertex is immediately visible:

vertex = (h, k)

The axis of symmetry is also immediately visible:

x = h

This makes vertex form the best form for identifying:

• vertex
• axis of symmetry
• maximum or minimum
• horizontal translation
• vertical translation

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Example: Reading Vertex Form

Consider:

y = −2(x − 4)² + 7

Immediately we can identify:

• vertex = (4, 7)
• axis of symmetry = x = 4
• a = −2
• parabola opens downward
• vertical stretch factor = 2
• maximum value = 7

We can also describe the transformations from y = x²:

• shift 4 units right
• stretch vertically by factor 2
• reflect across the x-axis
• shift 7 units upward

Vertex form makes all of this information relatively easy to see.

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Vertex Form and Maximum or Minimum Values

If:

a > 0

the parabola opens upward.

Therefore, the vertex represents a minimum.

If:

a < 0

the parabola opens downward.

Therefore, the vertex represents a maximum.

Consider:

y = 3(x + 2)² − 5

Vertex:

(−2, −5)

Since a = 3 > 0:

minimum value = −5

This occurs at:

x = −2

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What Factored Form Reveals

Factored form is:

y = a(x − r₁)(x − r₂)

The roots are immediately visible:

x = r₁

and:

x = r₂

Therefore, the x-intercepts are:

(r₁, 0)

and:

(r₂, 0)

Factored form is therefore the best form for identifying roots and x-intercepts.

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Example: Reading Factored Form

Consider:

y = −2(x + 3)(x − 5)

The roots are:

x = −3

and:

x = 5

Therefore, the x-intercepts are:

(−3, 0) and (5, 0)

Since:

a = −2

the parabola opens downward.

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Axis of Symmetry from Factored Form

Factored form also makes the axis of symmetry relatively easy to find.

The axis lies halfway between the roots.

Use:

x = (r₁ + r₂)/2

For:

y = (x − 2)(x − 8)

the roots are:

2 and 8

Therefore:

x = (2 + 8)/2

x = 5

The axis of symmetry is:

x = 5

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Finding the Vertex from Factored Form

Once the axis of symmetry is known, substitute its x-value into the equation.

For:

y = (x − 2)(x − 8)

we found:

x = 5

Now:

y = (5 − 2)(5 − 8)

y = (3)(−3)

y = −9

Therefore:

vertex = (5, −9)

Factored form does not show the vertex directly, but the roots make it relatively easy to calculate.

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Comparing What Each Form Reveals

There is no single "best" quadratic form.

The best form depends on the question.

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One Function, Different Information

Consider:

y = 2x² − 12x + 10

Factor:

y = 2(x − 1)(x − 5)

Convert to vertex form:

y = 2(x − 3)² − 8

All three equations represent the same quadratic.

Standard Form

y = 2x² − 12x + 10

Immediately reveals:

y-intercept = (0, 10)

Factored Form

y = 2(x − 1)(x − 5)

Immediately reveals:

x-intercepts = (1, 0) and (5, 0)

Vertex Form

y = 2(x − 3)² − 8

Immediately reveals:

vertex = (3, −8)

and:

axis of symmetry = x = 3

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Comparing Equivalent Equations

Equivalent quadratic equations produce the same y-value for every x-value.

For example:

y = x² − 8x + 12

y = (x − 2)(x − 6)

y = (x − 4)² − 4

These equations look very different, but they are equivalent.

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Verifying Factored and Standard Forms

Start with:

y = (x − 2)(x − 6)

Expand:

y = x² − 6x − 2x + 12

Combine:

y = x² − 8x + 12

Therefore:

(x − 2)(x − 6) = x² − 8x + 12

The two forms are equivalent.

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Verifying Vertex and Standard Forms

Start with:

y = (x − 4)² − 4

Expand:

y = x² − 8x + 16 − 4

Therefore:

y = x² − 8x + 12

So all three equations are equivalent:

y = x² − 8x + 12

y = (x − 2)(x − 6)

y = (x − 4)² − 4

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Checking Graph Features

We can also compare the features revealed by each form.

Standard

y = x² − 8x + 12

y-intercept:

(0, 12)

Factored

y = (x − 2)(x − 6)

roots:

2 and 6

x-intercepts:

(2, 0) and (6, 0)

Axis:

x = (2 + 6)/2

x = 4

Vertex

y = (x − 4)² − 4

vertex:

(4, −4)

axis:

x = 4

All of these features belong to the same parabola.

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Choosing a Form to Find the Vertex

Suppose you are given a choice between:

y = x² − 10x + 21

y = (x − 3)(x − 7)

y = (x − 5)² − 4

and asked:

What is the vertex?

The third equation is the most useful.

From:

y = (x − 5)² − 4

we immediately know:

vertex = (5, −4)

No additional calculation is required.

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Choosing a Form to Find the Roots

Now suppose the question asks:

What are the roots?

Use:

y = (x − 3)(x − 7)

The roots are immediately visible:

x = 3 and x = 7

Factored form is therefore the most useful.

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Choosing a Form to Find the y-Intercept

Suppose the question asks:

What is the y-intercept?

Use:

y = x² − 10x + 21

The constant term is 21.

Therefore:

y-intercept = (0, 21)

Standard form is the most useful.

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Choosing a Form for Transformations

Suppose the question asks:

Describe the transformations from y = x².

Consider:

y = −3(x − 2)² + 5

Vertex form immediately shows:

• shift right 2
• vertical stretch by factor 3
• reflection across the x-axis
• shift up 5

Vertex form is therefore usually the best choice for transformation questions.

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Choosing a Form for Sketching

All three forms can help sketch a parabola, but they provide different starting points.

Standard Form

Provides:

• y-intercept
• direction of opening
• stretch/compression

More work is required to find the vertex and roots.

Vertex Form

Provides:

• vertex
• axis of symmetry
• direction
• stretch/compression

This is often excellent for sketching.

Factored Form

Provides:

• x-intercepts
• axis of symmetry from the roots
• direction
• stretch/compression

This is also very useful for sketching when real roots exist.

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Comparing Vertex and Factored Forms for Sketching

Consider:

y = (x − 4)² − 9

Vertex form immediately gives:

vertex = (4, −9)

and:

axis = x = 4

But:

y = (x − 1)(x − 7)

immediately gives:

x-intercepts = (1, 0) and (7, 0)

The roots also show that the axis lies halfway between 1 and 7:

x = 4

Both forms are useful.

The better choice depends on which graph features you need first.

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Using Different Forms in Real Problems

Different forms can make different parts of the same real-world problem easier.

Suppose the height of a ball is modeled by:

h = −5t² + 30t

Standard form immediately gives the initial height:

at t = 0,

h = 0

Factor:

h = −5t(t − 6)

Now we immediately see that:

h = 0 at t = 0 and t = 6

So the ball returns to ground level after:

6 seconds

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Using Vertex Form in the Same Problem

Convert:

h = −5t² + 30t

to:

h = −5(t − 3)² + 45

Now we immediately see:

vertex = (3, 45)

Therefore:

• maximum height = 45 m
• maximum occurs at 3 s

The factored and vertex forms reveal different important information about the same motion.

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Selecting the Best Form

A useful decision guide is:

This does not mean other forms cannot answer the question.

It means one form usually requires less work.

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Information Hidden vs Revealed

An important mathematical idea is that changing the form does not change the function.

Instead, it changes which information is visible.

For example:

y = x² − 6x + 5

hides the roots and vertex.

But:

y = (x − 1)(x − 5)

reveals the roots.

And:

y = (x − 3)² − 4

reveals the vertex.

The mathematics has not changed.

Only the representation has changed.

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Why Mathematicians Use Multiple Representations

Different representations allow mathematicians to focus on different properties.

This idea appears throughout mathematics.

A useful representation can:

• make patterns easier to see
• simplify calculations
• reveal important features
• make predictions easier
• make a solution more efficient
• help communicate mathematical information

Choosing an appropriate representation is therefore an important problem-solving skill.

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Worked Comparison

Consider the following three equations:

A: y = −2x² + 16x − 24

B: y = −2(x − 2)(x − 6)

C: y = −2(x − 4)² + 8

From A

Standard form gives:

a = −2

b = 16

c = −24

Therefore:

y-intercept = (0, −24)

From B

Factored form gives:

roots:

x = 2 and x = 6

Therefore:

x-intercepts = (2, 0) and (6, 0)

Axis:

x = (2 + 6)/2

x = 4

From C

Vertex form gives:

vertex = (4, 8)

Axis:

x = 4

Since a = −2:

• parabola opens downward
• maximum value = 8

All three equations describe the same parabola.

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Verifying the Worked Comparison

Expand B:

y = −2(x − 2)(x − 6)

y = −2(x² − 8x + 12)

y = −2x² + 16x − 24

This gives equation A.

Now expand C:

y = −2(x − 4)² + 8

y = −2(x² − 8x + 16) + 8

y = −2x² + 16x − 32 + 8

y = −2x² + 16x − 24

Again, we obtain equation A.

Therefore:

A = B = C

as functions.

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Common Mistakes

Mistake: "Different-looking equations must produce different graphs."
Equivalent equations can look very different while representing exactly the same function.

Mistake: "Standard form is always the best form."
The best form depends on the information required.

Mistake: "The vertex is easiest to find from factored form."
It can be calculated from factored form, but it is immediately visible in vertex form.

Mistake: "The roots are easiest to find from vertex form."
They may be calculated from vertex form, but they are immediately visible in factored form.

Mistake: "Factored form gives the y-intercept immediately."
You must usually substitute x = 0. Standard form reveals it directly.

Mistake: "Changing form changes the function."
Correct algebraic conversion preserves the function.

Mistake: "One form contains more information than another."
Equivalent forms contain the same mathematical information, but they make different features easier to see.

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Key Terms

Quadratic function — function whose graph is a parabola and whose highest power of the variable is 2.

Standard form — quadratic representation y = ax² + bx + c.

Vertex form — quadratic representation y = a(x − h)² + k.

Factored form — quadratic representation y = a(x − r₁)(x − r₂).

Equivalent equations — equations that produce the same output for every permitted input.

Vertex — turning point of a parabola.

Root — x-value that makes the quadratic equal to zero.

Zero — another term for a root.

x-intercept — point where the graph meets the x-axis.

y-intercept — point where the graph meets the y-axis.

Axis of symmetry — vertical line dividing a parabola into two mirror-image halves.

Maximum — greatest y-value of a downward-opening parabola.

Minimum — smallest y-value of an upward-opening parabola.

Representation — a particular way of expressing the same mathematical object or relationship.

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Key Takeaways

• Quadratic functions can be represented in standard, vertex and factored forms.
• Equivalent forms represent exactly the same parabola.
• Standard form is y = ax² + bx + c.
• Vertex form is y = a(x − h)² + k.
• Factored form is y = a(x − r₁)(x − r₂).
• Vertex form reveals the vertex most easily.
• Vertex form also reveals the axis of symmetry and maximum or minimum quickly.
• Factored form reveals the roots and x-intercepts most easily.
• The axis of symmetry can be found from factored form by averaging the roots.
• Standard form reveals the y-intercept most easily.
• The coefficient a reveals the direction and vertical stretch/compression in all three forms.
• Vertex form is particularly useful for transformations and optimization problems.
• Factored form is particularly useful when solving for where y = 0.
• Standard form is useful for identifying coefficients and performing algebraic calculations.
• Changing the form of an equation does not change its graph if the equations remain equivalent.
• Different forms do not contain different mathematical information; they simply make different features easier to see.
• Choosing the most useful form can make a problem much faster and easier to solve.

Check Your Understanding

1. Which quadratic form reveals the vertex most easily?
2. Which quadratic form reveals the roots most easily?
3. Which quadratic form reveals the y-intercept most easily?
4. Identify the vertex of y = 3(x − 4)² − 7.
5. Identify the roots of y = 2(x − 1)(x + 5).
6. Identify the y-intercept of y = −2x² + 8x + 12.
7. Find the axis of symmetry of y = (x − 3)(x − 9).
8. Which form would you choose to find the maximum height of a projectile? Explain.
9. Which form would you choose to determine when a projectile reaches the ground? Explain.
10. Show that y = x² − 8x + 12 and y = (x − 2)(x − 6) are equivalent.
11. Show that y = x² − 8x + 12 and y = (x − 4)² − 4 are equivalent.
12. For y = −2(x − 3)² + 8, identify the vertex, axis of symmetry, direction of opening and maximum or minimum.
13. For y = −2(x − 1)(x − 5), identify the roots, x-intercepts, axis of symmetry and direction of opening.
14. Explain why two quadratic equations can look very different but still produce exactly the same graph.
15. A quadratic is available in all three forms. Explain which form you would choose to find the vertex, roots and y-intercept, and why.