Ranges from Linear and Vertex-Form Quadratic Rules

What Range Can a Rule Produce?

A function’s range is the set of all output values it actually produces from inputs in its domain. Finding a range means answering two questions: which outputs are possible, and are there any outputs the rule can never produce?

Here we consider rules whose inputs can be any real number. This is important: a formula may have a different range if its domain is restricted. For example, a rule considered only for nonnegative inputs might produce fewer outputs than the same rule considered for all real inputs.

Two familiar forms make range-finding especially direct. A nonconstant linear rule can produce every real output. A quadratic in vertex form has a lowest or highest output, determined by its vertex, and extends without bound in the other direction.

Ranges of Linear Rules

A linear rule has the form \(f(x)=mx+b\), where \(m\) and \(b\) are real constants. If \(m\ne 0\), then every real number is an output. To see why, let \(y\) be any real number and solve \(y=mx+b\) for the input:

Because \(m\ne 0\), this is a valid real input for every real \(y\). Substituting it into the rule gives \(f(x)=y\). Thus no real output is missed, and the range is \(\mathbb{R}\).

If \(m=0\), the rule is instead \(f(x)=b\) for every input. It produces just one output, so its range is the singleton set \(\{b\}\). A constant rule is sometimes included as a special case of a linear rule; it must not be confused with a nonconstant one when finding the range.

Vertex Form and the Turning Value

A quadratic in vertex form is written as

where \(a\), \(h\), and \(k\) are real numbers and \(a\ne 0\). The condition \(a\ne 0\) ensures the rule is genuinely quadratic. The input \(x=h\) makes the squared term zero, so the output there is \(k\). The point \((h,k)\) is called the vertex.

The square \((x-h)^2\) is always nonnegative. If \(a>0\), multiplying it by \(a\) leaves it nonnegative, so no output can be less than \(k\). If \(a<0\), the product is nonpositive, so no output can be greater than \(k\). This gives the bound on the outputs. To establish the full range, we must also show that every value on the allowed side of the bound is attained.

Why the Quadratic Range Formula Works

First suppose \(a>0\). For every real \(x\), \((x-h)^2\ge 0\), so \(a(x-h)^2\ge 0\), and therefore \(f(x)\ge k\). Thus no output below \(k\) is possible. Also, for any \(y\ge k\), the number \((y-k)/a\) is nonnegative. It has a real square root, and the input

produces output \(y\), since

So every value at least \(k\) occurs, and none below it does. The range is \([k,\infty)\).

Now suppose \(a<0\). Since the square is nonnegative, \(a(x-h)^2\le 0\), hence \(f(x)\le k\). For any \(y\le k\), \((y-k)/a\ge 0\): both numerator and denominator are nonpositive, with a zero numerator allowed. Define \(x=h+\sqrt{(y-k)/a}\). Then the same substitution gives \(f(x)=y\). Thus every value at most \(k\) occurs, and none above it does. The range is \((-\infty,k]\).

Worked Examples with Vertex Form

A Constant Rule Is a Special Case

Although the quadratic range theorem assumes \(a\ne 0\), it is useful to notice what happens if the coefficient of the squared term is zero. The expression \(0(x-h)^2+k\) is just the constant rule \(f(x)=k\). Every real input then gives the same output, so its range is \(\{k\}\), not an unbounded interval.

A Common Mistake: Including or Excluding the Vertex

A frequent error is to use a parenthesis rather than a bracket at the vertex value, as if the quadratic only approached that value. But the vertex input \(x=h\) is allowed, and direct substitution gives \(f(h)=k\). The endpoint must therefore be included.

For example, writing the range of \(2(x-3)^2-5\) as \((-5,\infty)\) is incorrect. That interval excludes \(-5\), even though

The correction is \([-5,\infty)\). The opposite mistake can occur if the coefficient is negative: the maximum is included as well, so the correct form for \(-\frac12(x+2)^2+4\) is \((-\infty,4]\), not \((-\infty,4)\).

Another useful check is to keep the direction of the bound straight. A positive coefficient gives a minimum and outputs at or above it; a negative coefficient gives a maximum and outputs at or below it. The vertex value itself is always attained.

A Reliable Range-Finding Process

For these forms, a short sequence of checks prevents most errors:

Why These Rules Matter

Range gives information about what outputs a model can produce, without requiring a list of individual input-output pairs. For a nonconstant linear rule with unrestricted real inputs, any real output is possible. For a quadratic in vertex form, the coefficient’s sign and the vertex value immediately tell us which outputs are ruled out.

These conclusions depend on the domain. If the input is restricted, the reasoning that every permitted output can be reached may no longer apply. Always read the domain before determining the range; do not assume that a formula is being used for every real input unless that is stated or established.