18.3 why a normal quadratic has −∞<x<∞

🟦 Concept

The domain of a graph is the set of all possible x-values that can be used in the function.

A quadratic equation in standard form is:$$y=ax^2+bx+c$$

where:$$a\neq0$$

For a normal quadratic function, x can be any real number.

Therefore, the domain is:$$-\infty<x<\infty$$

It can also be written in interval notation as:$$(-\infty,\infty)$$

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🟩 Why Can $x$ Be Any Real Number?

Consider:$$y=x^2-4x+3$$

We can substitute any real value for $x$.

For example, if:$$x=-2$$

then:$$y=(-2)^2-4(-2)+3$$$$y=15$$

If:$$x=0$$

then:$$y=3$$

If:$$x=5$$

then:$$y=8$$

We could also use decimal, fractional, very large positive, or very large negative values of $x$.

There is no value of xx that makes an ordinary quadratic expression undefined.

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🟪 What Does This Look Like on the Graph?

A quadratic curve continues indefinitely:

← to the left and to the right →

This means the graph contains x-values extending without limit in both directions.

Therefore:$$x\in\mathbb{R}$$

which means:

x belongs to the set of all real numbers.

So the domain is:$$(-\infty,\infty)$$

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🟨 Domain vs Range

Do not confuse domain with range.

Domain → possible x-values

Range → possible y-values

For:$$y=x^2-4x+3$$

the domain is:$$-\infty<x<\infty$$

The turning point is:$$(2,-1)$$

Since the parabola opens upwards, its range is:$$y\geq-1$$

So:

Domain: all real x-values

Range: y-values from $-1$ upwards

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💡 Exam Tip

For an ordinary quadratic function:$$y=ax^2+bx+c$$

the domain is always:$$-\infty<x<\infty$$

unless the question specifically restricts the values of $x$.

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⚠️ Common Mistake

Students sometimes use the turning point to find the domain.

The turning point determines the minimum or maximum y-value, so it is important when finding the range, not the domain.

For a normal quadratic curve, the graph continues indefinitely to both the left and right.

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⭐ Quick Summary

The domain tells us which x-values are allowed.

For a quadratic function:$$y=ax^2+bx+c$$

every real value of $x$ is allowed.

Therefore:$$-\infty<x<\infty$$

or:$$x\in\mathbb{R}$$

Remember

Quadratic graph → extends left and right forever → domain is all real numbers.