Determine the domain and range of quadratic functions and represent them using inequalities, and describe representations of quadratic functions in relation to their solutions and the real-world situations they model (TEKS A.6A, A.6B).

What this topic is asking

TEKS A.6A and A.6B sit in the Quadratic Functions category and ask two connected things: find the domain and range of a quadratic (with inequalities), and describe its representations in relation to its solutions and the real-world situation it models. The defining feature is that a parabola's range is bounded by its vertex, unlike a line.

Domain: all real numbers (unless restricted)

For a bare quadratic $f(x) = ax^2 + bx + c$, you can input any real number, so the domain is all real numbers. The parabola never stops extending sideways. This is the same as a line, and it is why the domain is rarely the interesting part of a quadratic question, except in a context.

Range: bounded by the vertex

The range is where quadratics differ from lines. Because a parabola turns around at its vertex, the outputs reach a single extreme and then reverse.

• Opens up ($a > 0$). The vertex is the lowest point. The range is $y \ge k$, all outputs at or above the minimum.
• Opens down ($a < 0$). The vertex is the highest point. The range is $y \le k$, all outputs at or below the maximum.

So $f(x) = x^2 - 4$ (vertex $(0, -4)$, opening up) has range $y \ge -4$, and $g(x) = -(x - 1)^2 + 5$ (vertex $(1, 5)$, opening down) has range $y \le 5$.

Representations and real-world models

A quadratic can be shown as an equation, a graph, or a table, and each highlights different features: the graph shows the vertex and intercepts; the factored equation shows the zeros; vertex form shows the maximum or minimum. A.6B asks you to connect these to a situation. For a projectile, the vertex is the peak height, the zeros are launch and landing times, and the range is the set of achievable heights, restricted to non-negative values. Matching the algebra to the physical meaning is the assessed skill.

How STAAR examines this topic

• Multiple choice. Choose the range (the vertex-bounded inequality), with the domain offered as a distractor.
• Inline choice. Identify where a maximum occurs (the vertex), and the restricted range in context.
• Drag and drop. Match a representation to a feature or a situation.

A clarifying idea is that the vertex is the gatekeeper of the range: find the vertex and the opening direction, and the range follows immediately as "at or above" or "at or below" the vertex's height.

Finding the range when only standard form is given

If a quadratic arrives in standard form rather than vertex form, you still need the vertex's $y$-coordinate to state the range. Compute the axis of symmetry $x = \frac{-b}{2a}$, substitute it back to get the minimum or maximum value $k$, then write $y \ge k$ or $y \le k$ depending on the opening direction. For $f(x) = x^2 - 6x + 5$, the axis is $x = 3$, and $f(3) = 9 - 18 + 5 = -4$, so the range is $y \ge -4$. Skipping the substitution and guessing the range from the constant term $c$ is a common error, because $c$ is the $y$-intercept, not the vertex value, unless the vertex happens to lie on the $y$-axis.

Why the domain and range describe different axes

Keeping domain and range straight is easier when you remember they answer different questions. The domain asks which horizontal positions the curve occupies, and a parabola spans the whole $x$-axis, so the answer is all real numbers. The range asks which heights the curve reaches, and a parabola only reaches heights on one side of its vertex, so the answer is a half-line in $y$. Because both can be written with inequalities that look similar, the discriminator on a multiple-choice item is usually whether the variable is $x$ (domain) or $y$ (range) and whether the bound is the vertex value.

Try this

Q1. State the range of $f(x) = -(x + 2)^2 + 3$. [1 point]

• Cue. Vertex $(-2, 3)$, opens down: $y \le 3$.

Q2. What is the domain of $f(x) = x^2 + 5x - 1$ with no restriction? [1 point]

• Cue. All real numbers.