Write quadratic functions when given real solutions and graphs of their related equations, and write quadratic functions that fit data sets using vertex form or standard form (TEKS A.6C, A.8B).

What this topic is asking

TEKS A.6C and A.8B ask you to write a quadratic function, the reverse of graphing. You build it from its real solutions (zeros), from its graph (vertex and a point), or from a data set (best fit, with technology). The key connection is that a zero corresponds to a factor, which links this dot point to solving by factoring.

From zeros: factored form

A zero of a quadratic is where it crosses the $x$-axis, and each zero comes from a factor set to zero. So a zero at $x = r$ corresponds to the factor $(x - r)$.

The factor uses the opposite sign of the zero: $x = -1$ gives $(x + 1)$. Without more information, $a = 1$ is the simplest choice; a given point determines $a$ exactly.

From a graph: vertex form

When you can read the vertex from a graph, vertex form is fastest.

From data: regression

When given a table of data that follows a parabola, the calculator's quadratic regression returns $f(x) = ax^2 + bx + c$ (or you can fit vertex form if the vertex is known). A.8B explicitly allows technology here. The interpretive step is reading $a$, the vertex, or the zeros from the fitted model and relating them to the situation.

Connecting the forms

The three forms describe the same parabola and convert into one another. Factored form shows the zeros; vertex form shows the maximum or minimum; standard form shows the $y$-intercept. Expanding $f(x) = a(x - r_1)(x - r_2)$ or $a(x - h)^2 + k$ returns standard form, a useful check that your function is right.

How STAAR examines this topic

• Multiple choice. Choose the function with given zeros (factor-sign trap) or matching a graph.
• Equation editor. Build the function in factored or vertex form from zeros, a vertex, or a point.
• Drag and drop. Assemble factors or parameters into a form template.

A clarifying idea is that a single extra point fixes $a$: the zeros or the vertex give the shape and position, but only a point off the axis tells you how wide or narrow the parabola is, which is why these problems always supply one.

Choosing the right form for the information given

The fastest write comes from matching the form to what the problem hands you. If you are given zeros (or $x$-intercepts on a graph), reach for factored form, because each zero drops straight into a factor. If you are given the vertex (the turning point, a maximum or minimum), reach for vertex form, because $h$ and $k$ are read directly. If you are given a table of data, use regression to get standard form. Trying to force the wrong form, for example starting from standard form when you know the vertex, makes you solve for three unknowns instead of one.

A worked check by expansion

After writing a function, expanding it back to standard form verifies the work and often reveals the $y$-intercept as a bonus. From $f(x) = (x + 2)(x - 5)$, expanding gives $x^2 - 3x - 10$, so the $y$-intercept is $-10$ and the function is confirmed to have the intended zeros at $-2$ and $5$. If the expansion does not reproduce the features you started from, a sign or the value of $a$ is wrong. This expand-and-check habit is especially valuable on equation-editor items, where the answer must be exact and there is no list of choices to sanity-check against.

Try this

Q1. Write a quadratic with zeros $x = 4$ and $x = -3$ (take $a = 1$). [1 point]

• Cue. $f(x) = (x - 4)(x + 3)$.

Q2. A parabola has vertex $(0, -5)$ and passes through $(2, 3)$. Find $a$. [1 point]

• Cue. $3 = a(2)^2 - 5 \Rightarrow 4a = 8 \Rightarrow a = 2$.