Understanding Quadratic Functions

What Is a Quadratic Function?

A quadratic function is a polynomial function in which the highest power of the variable is 2.

A quadratic function is usually written in standard form:

f(x) =ax² + bx + c​

where:

• a, b, and c are constants.
• a ≠ 0.
• x is the variable.
• ax² is the quadratic term.
• bx is the linear term.
• c is the constant term.

The condition a ≠ 0 is important. If a = 0, the x² term disappears and the function is no longer quadratic.

Example

Consider:

f(x) = 2x² + 5x − 3

In this function:

a = 2, b = 5, c = −3

Because the highest power of x is 2, this is a quadratic function.

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Recognising Standard Form

Quadratic functions do not need to contain all three terms.

For example, all of these are quadratic:

y = x² + 4x + 7

y = 3x² − 5

y = −2x² + 6x

y = 5x²

Each function is quadratic because its highest power of x is 2.

The terms may also appear in a different order. For example:

y = 7 + 3x − 2x²

can be rearranged into standard form:

y = −2x² + 3x + 7

This makes it easier to identify:

a = −2, b = 3, c = 7

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The Degree of a Polynomial

The degree of a polynomial is the highest exponent of its variable.

For example:

A quadratic function therefore has degree 2.

Quick Check

What is the degree of:

f(x) = 7x² − 3x + 9

The highest exponent is 2.

Therefore:

Degree = 2​

The function is quadratic.

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Quadratic or Not?

One of the easiest ways to identify a quadratic function is to look for its highest exponent.

Consider:

y = 4x + 1

The highest exponent is 1, so this is linear, not quadratic.

Now consider:

y = 4x² + 1

The highest exponent is 2, so this is quadratic.

Finally:

y = 4x³ + x2 + 1

Although this equation contains an x2 term, its highest exponent is 3. It is therefore a cubic function, not a quadratic function.

Important

Simply seeing x² does not automatically mean that a function is quadratic.

The x² term must contain the highest power of x.

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Linear Functions vs Quadratic Functions

A linear function has the general form:

y = mx + b

Its graph is a straight line.

A quadratic function has the general form:

y = ax² + bx + c

Its graph is a curve called a parabola.

For example:

Linear:

y = 2x + 1

Quadratic:

y = x²

For y = x², a few values are:

Notice that the y-values do not increase at a constant rate. This changing rate produces a curved graph rather than a straight line.

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The Parabola

The graph of every quadratic function is a parabola.

A simple example is:

y = x²

Its graph is U-shaped and has a lowest point at (0, 0).

For a general quadratic,

y = ax² + bx + c

the value of a helps determine the direction of the parabola.

• If a > 0, the parabola opens upward.
• If a < 0, the parabola opens downward.

For example:

y = x²

opens upward, while

y = −x²

opens downward.

The turning point of a parabola is called its vertex.

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Why Do Quadratic Functions Form Parabolas?

In a linear function, equal changes in x produce a constant change in y. This produces a straight line.

In a quadratic function, the x² term causes the rate of change to change as x changes.

For y = x²:

0, 1, 4, 9, 16, 25,…

The first differences are:

1, 3, 5, 7, 9,…

These differences are changing.

However, the differences between those differences are constant:

2, 2, 2, 2,…

This pattern of constant second differences is a characteristic of quadratic functions and corresponds to the curved shape of a parabola.

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Quadratics in the Real World

Quadratic functions are useful because many situations involve quantities that change at a changing rate.

For example, the approximate path of a thrown ball can be modelled by a quadratic function.

Quadratic models can appear when studying:

• The path of a thrown or launched object.
• The height of an object over time.
• The shape of some arches and reflectors.
• Maximum-area and minimum-cost problems.
• Relationships involving acceleration.

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Worked Example

Identify whether each function is quadratic.

A.

f(x) = 3x² − 7x + 2

The highest exponent is 2.

Quadratic​

B.

f(x) = 5x − 8

The highest exponent is 1.

Linear​

C.

f(x) = x³ + 2x2 − 4

The highest exponent is 3.

Cubic​

D.

f(x) = −6x² + 9

The highest exponent is 2.

Quadratic​

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Key Vocabulary

Quadratic function – A polynomial function with degree 2.

Standard form – The form f(x) = ax² + bx + c where a ≠ 0.

Degree – The highest exponent of the variable in a polynomial.

Polynomial – An expression made from variables, constants, and non-negative whole-number exponents.

Parabola – The curved graph produced by a quadratic function.

Vertex – The turning point of a parabola.

Coefficient – A number multiplying a variable.

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Key Takeaways

• A quadratic function has degree 2.
• Its standard form is: f(x) = ax² + bx + c,a ≠ 0​
• The highest power of x must be 2.
• A linear function has degree 1, while a quadratic function has degree 2.
• Having an x² term does not make a function quadratic if a higher-power term is also present.
• The graph of a quadratic function is called a parabola.
• Parabolas open upward when a > 0 and downward when a < 0.
• Quadratic functions have constant second differences when x-values are equally spaced.