What does the STAAR reference sheet give for quadratics?

The reference sheet gives the standard form f(x) equals ax squared plus bx plus c, the vertex form f(x) equals a times (x minus h) squared plus k, the axis of symmetry x equals negative b over 2a, and the quadratic formula. Knowing which form reveals which feature, vertex form for the turning point, factored form for the zeros, standard form for the y-intercept, is the core skill.

How do you find the vertex of a quadratic?

From vertex form a times (x minus h) squared plus k, the vertex is (h, k) read directly, watching the sign so that (x minus 3) gives h equal to 3 and (x plus 3) gives h equal to negative 3. From standard form, find the axis of symmetry x equals negative b over 2a, then substitute that x back into the function to get the vertex's y-coordinate.

How do the parameters a, h, and k transform the parabola?

Starting from the parent x squared: h shifts horizontally, opposite to the sign shown, so x plus 4 moves left 4; k shifts vertically, up for positive and down for negative; and a reflects and scales, with a negative a reflecting over the x-axis, absolute value of a greater than 1 narrowing the parabola, and a fraction widening it. The vertex moves to (h, k).

Why is the range of a quadratic bounded but the domain is not?

A parabola extends left and right forever, so its domain is all real numbers. But it turns around at its vertex, so its outputs reach a single extreme and reverse: if it opens up the range is y greater than or equal to the minimum, and if it opens down the range is y less than or equal to the maximum. The vertex's y-coordinate is the boundary of the range.

How do you write a quadratic from its zeros?

Use factored form f(x) equals a times (x minus r1) times (x minus r2), where a zero at x equals r gives the factor (x minus r), the opposite sign of the zero. So zeros at negative 2 and 5 give (x plus 2)(x minus 5). The leading coefficient a, which sets the width, is found from a second point on the curve if one is given; otherwise a equals 1 is the simplest choice.

Which form should you use to write a quadratic from a graph?

Use vertex form if you can read the vertex from the graph: write a times (x minus h) squared plus k with the vertex as (h, k), then substitute another point on the curve to solve for a. Use factored form if the x-intercepts are clear instead. Either way, a second known point is what fixes the value of a.

TexasMaths

STAAR Algebra I: a complete guide to graphing and representing quadratic functions

A deep-dive STAAR Algebra I guide to graphing and representing quadratic functions, the function side of the Quadratic Functions and Equations reporting category (about 25 percent of the test). Covers graphing parabolas and key attributes, transformations of the parent function, domain and range, and writing quadratics from zeros, a vertex, or data.

Generated by Claude Opus 4.816 min readA.6, A.7Updated 2026-06-02

Reviewed by: AI editorial process; not yet individually human-reviewed

Jump to a section

What this category demands
Graphing and key attributes
Transformations of the parent function
Domain, range, and representations
Writing quadratics
How this category is examined
Check your knowledge

What this category demands

The Quadratic Functions and Equations reporting category (TEKS A.6, A.7, A.8) is about 25 percent of the STAAR Algebra I test, second only to the linear categories, and it is where the Meets and Masters standards are usually decided. This guide covers the function side, graphing and representing parabolas; the solving side (factoring, square roots, completing the square, the formula, and applications) is in the companion module. Each dot-point page has its own practice: graphing quadratic functions and key attributes, transformations of quadratic functions, domain, range, and representations, and writing quadratic functions.

Graphing and key attributes

A quadratic graphs as a parabola. Its key attributes are the vertex (turning point), the axis of symmetry $x = \frac{-b}{2a}$ (reference sheet), the zeros or $x$ -intercepts (where $y = 0$ ), the $y$ -intercept $(0, c)$ , and the maximum or minimum (the vertex's $y$ -coordinate). The sign of $a$ sets the direction: up for $a > 0$ (minimum), down for $a < 0$ (maximum). In vertex form the vertex is $(h, k)$ ; from standard form, find the axis, then substitute to get the vertex.

Transformations of the parent function

From the parent $f(x) = x^2$ , vertex form $a(x - h)^2 + k$ encodes three moves. $h$ shifts horizontally, opposite to the sign shown; $k$ shifts vertically; $a$ reflects (negative opens down) and scales ( $|a| > 1$ narrows, $|a| < 1$ widens). Reading these straight off vertex form is the fast route to a graph.

Domain, range, and representations

A quadratic's domain is all real numbers (unless a context restricts it), but its range is bounded by the vertex: $y \ge k$ when it opens up, $y \le k$ when it opens down. The three representations highlight different features, the graph the vertex and intercepts, factored form the zeros, vertex form the extremum, and a real-world model clips the domain and range to sensible values.

Writing quadratics

To write from zeros, use factored form $a(x - r_1)(x - r_2)$ , each factor the opposite sign of its zero. To write from a graph, use vertex form with the vertex and a second point to find $a$ . To write from data, use quadratic regression. A second point off the axis is what fixes $a$ .

How this category is examined

Multiple choice. Find the vertex, axis, range, or a transformation; choose a function from zeros or a graph. Sign errors on $b$ and $h$ are the standard traps.
Hot spot. Plot the vertex, zeros, or other points; placement must be exact.
Equation editor. Write a quadratic in factored or vertex form.
Inline choice. Choose opens up or down, maximum or minimum, narrower or wider.

Check your knowledge

Work these as you would for credit on the redesigned test.

Find the axis of symmetry of $f(x) = x^2 - 8x + 3$ . (1 point)
State the vertex of $f(x) = (x - 5)^2 + 2$ . (1 point)
Describe the transformation in $g(x) = (x + 3)^2 - 6$ from the parent. (1 point)
Is $g(x) = -4x^2$ narrower or wider than the parent, and which way does it open? (1 point)
State the range of $f(x) = x^2 + 6$ . (1 point)
State the domain of $f(x) = 2x^2 - x + 1$ . (1 point)
Write a quadratic with zeros $x = -1$ and $x = 6$ (take $a = 1$ ). (1 point)
A parabola has vertex $(2, 1)$ and passes through $(4, 9)$ . Write it in vertex form. (2 points)

Solutions

Step 1: Q1. Find the axis of symmetry

Use $x = \frac{-b}{2a}$ with $a = 1$ and $b = -8$ . The negative of a negative gives a positive:

x = \frac{-(-8)}{2(1)} = 4

Step 2: Q2. Read the vertex from vertex form

In $f(x) = (x - h)^2 + k$ , the vertex is $(h, k)$ read directly. Here $h = 5$ and $k = 2$ :

\text{Vertex } (5, 2)

Step 3: Q3. Identify the transformations

In $g(x) = (x + 3)^2 - 6$ , the $+3$ inside shifts left by $3$ (opposite to the sign) and the $-6$ outside shifts down by $6$ . The vertex moves from $(0, 0)$ to $(-3, -6)$ .

Step 4: Q4. Determine the direction and width

For $g(x) = -4x^2$ , the leading coefficient $a = -4$ . Because $|a| = 4 > 1$ , the parabola is narrower than the parent, and because $a < 0$ , it opens downward.

Step 5: Q5. State the range of an upward parabola

In $f(x) = x^2 + 6$ , the constant $+6$ shifts the vertex to $(0, 6)$ . Because $a = 1 > 0$ the parabola opens up, so the minimum is the vertex $y$ -value:

\text{Range: } y \ge 6

Step 6: Q6. State the domain of a quadratic

Every quadratic with no stated context has inputs that extend infinitely left and right:

\text{Domain: all real numbers}

Step 7: Q7. Write a quadratic from its zeros

Each zero at $r$ gives a factor $(x - r)$ , so a zero at $-1$ gives $(x + 1)$ and a zero at $6$ gives $(x - 6)$ . With $a = 1$ :

f(x) = (x + 1)(x - 6)

Step 8: Q8. Write the function from a vertex and a point

Start from vertex form with the given vertex $(2, 1)$ , then substitute the second point $(4, 9)$ to solve for $a$ :

f(x) = a(x - 2)^2 + 1; \quad 9 = a(4 - 2)^2 + 1 \Rightarrow 8 = 4a \Rightarrow a = 2

f(x) = 2(x - 2)^2 + 1

Sources & how we know this

STAAR Algebra I Assessed Curriculum — Texas Education Agency (2024)
STAAR Algebra I Reference Materials — Texas Education Agency (2024)