Use vertex form to describe transformations of the parent function and to read the vertex, direction, and stretch of a quadratic (A.FGR, Functional and Graphical Reasoning).

What this topic is asking

This Functional and Graphical Reasoning (A.FGR) standard studies how the graph of $y = x^2$ moves and stretches, using vertex form $f(x) = a(x - h)^2 + k$, which displays every transformation at a glance. The Georgia Milestones EOC asks you to read the vertex $(h, k)$ directly, to describe the shifts, reflection, and stretch, and to match an equation to a graph. The single most error-prone part is the horizontal shift, which goes opposite the sign inside the parentheses. Mastering vertex form makes both graphing and the connection to completing the square far easier.

Vertex form displays the vertex

Vertex form is built so the vertex is visible.

For $g(x) = (x - 4)^2 + 3$: $h = 4$, $k = 3$, so the vertex is $(4, 3)$. No formula computation is needed, which is the advantage of vertex form over standard form for reading the vertex.

The four transformations

Comparing $f(x) = a(x - h)^2 + k$ to the parent $y = x^2$:

• Horizontal shift: $h$ moves the graph left or right, opposite the sign inside. $(x - 4)$ shifts right 4; $(x + 1)$ shifts left 1.
• Vertical shift: $k$ moves the graph up (if positive) or down (if negative).
• Reflection: if $a < 0$, the parabola is reflected over the x-axis (opens downward).
• Vertical stretch or compression: $|a|$ scales the graph; $|a| > 1$ makes it narrower, $0 < |a| < 1$ makes it wider.

Why the horizontal shift looks backward

The horizontal shift confuses almost everyone at first, so it is worth the reasoning. Consider $g(x) = (x - 4)^2$. The vertex of the parent $y = x^2$ is where the inside equals zero, namely $x = 0$. For $g$, the inside $(x - 4)$ equals zero when $x = 4$, so the vertex has moved to $x = 4$, a shift to the right. The minus sign inside produces a shift in the positive direction because you need a larger $x$ to make the inside zero again. Likewise $(x + 1)$ is zero at $x = -1$, a shift left. So the rule "opposite the sign inside" is not arbitrary: it comes from asking what input now produces the vertex. Holding onto "set the inside to zero to find $h$" makes the direction obvious every time.

Connecting to completing the square

Vertex form and standard form describe the same parabola, and completing the square is the bridge from standard form $ax^2 + bx + c$ to vertex form $a(x - h)^2 + k$. So if a problem gives standard form and asks for the vertex, you can either use $x = \frac{-b}{2a}$ or complete the square to reach vertex form. The two routes agree, and choosing vertex form is convenient when you also need to describe the transformations, while the axis-of-symmetry formula is faster when you only need the vertex coordinates. This connection is why the transformations topic sits right next to completing the square in the course.

How the Milestones examines this topic

• Multiple choice. Read the vertex from vertex form, or identify a described transformation.
• Hot spot / graphing. Match a vertex-form equation to its parabola, or plot the transformed vertex.
• Constructed response. Describe all transformations from the parent and state the vertex.

Try this

Q1. State the vertex and direction of $g(x) = 3(x - 2)^2 - 7$. [1 point]

• Cue. Vertex $(2, -7)$; $a = 3 > 0$, opens up (narrower than parent).

Q2. Describe the shift in $h(x) = (x + 5)^2 + 2$. [1 point]

• Cue. Left 5 (opposite the $+5$) and up 2; vertex $(-5, 2)$.