Describe the effect on the graph of the parent function $f(x) = x^2$ when $a$, $h$, and $k$ change in $f(x) = a(x - h)^2 + k$, and transform quadratic functions graphically and algebraically (TEKS A.7B, A.7C).

What this topic is asking

TEKS A.7B and A.7C ask you to connect the parameters $a$, $h$, and $k$ in vertex form to transformations of the parent parabola $f(x) = x^2$, and to write the transformed function. This is reliably tested in the Quadratic Functions category, and it is the fastest way to graph a parabola: start from the parent and shift, reflect, or stretch.

Vertical and horizontal shifts

The parent parabola has its vertex at the origin. The constants $h$ and $k$ slide it.

• $k$ (vertical). $f(x) = x^2 + k$ moves the parabola up by $k$ (or down if $k$ is negative). This matches intuition: adding to the output raises the graph.
• $h$ (horizontal). $f(x) = (x - h)^2$ moves the parabola right by $h$. The direction is opposite to the sign inside the parentheses, which is the most common point of confusion: $(x + 4)^2$ shifts left 4.

So $g(x) = (x - 2)^2 + 5$ is the parent shifted right 2 and up 5, with vertex $(2, 5)$.

Reflections and stretches: the role of a

The parameter $a$ does two jobs at once.

• Sign (reflection). If $a < 0$, the parabola is reflected over the $x$-axis and opens downward.
• Magnitude (stretch or compression). If $|a| > 1$, the parabola is stretched vertically and looks narrower; if $0 < |a| < 1$, it is compressed and looks wider.

Writing the transformed function

Reverse the process to write a function from a description or a vertex. A parabola with vertex $(-2, 5)$, opening down, and the same width as a stretch of 2, is $g(x) = -2(x + 2)^2 + 5$: $h = -2$ gives $(x + 2)$, $k = 5$, and $a = -2$ for the reflection and stretch.

How STAAR examines this topic

• Multiple choice. Identify the transformation from an equation, or match a transformed graph. Horizontal-direction errors are the standard trap.
• Inline choice. Choose reflection, narrower or wider, and shift directions from dropdowns.
• Equation editor. Write the transformed function from a description.

A clarifying idea is that the transformations are read straight off vertex form without any computation: the form is built so that $a$, $h$, and $k$ each map to one geometric move, which is exactly why vertex form is the graphing-friendly representation.

Why the horizontal shift looks backward

The reversed direction of the horizontal shift confuses almost everyone at first, and there is a clean reason for it. The expression $(x - h)$ asks "how far is the input from $h$?" For the transformed graph to behave at $x = h + 2$ the way the parent behaves at $x = 2$, the whole graph must slide so that the action that happened at the origin now happens at $x = h$. Setting the inside equal to zero, $x - h = 0$, gives $x = h$ as the new vertex location, which is why $(x - 3)$ puts the vertex at $x = +3$ and $(x + 3)$, equal to $(x - (-3))$, puts it at $x = -3$. Tying the shift to "where does the inside equal zero?" removes the guesswork.

Tracking a single point through a transformation

A concrete check is to follow one point. The parent passes through $(1, 1)$, since $1^2 = 1$. Under $g(x) = 2(x - 3)^2 + 4$, that point moves: the input shifts right 3 to $x = 4$, and the output is stretched by 2 and raised by 4, giving $2(1) + 4 = 6$, so the image is $(4, 6)$. Tracing one easy point this way verifies the direction of every transformation at once and is a quick way to eliminate wrong answer choices on a matching item.

Try this

Q1. Describe the shift in $g(x) = (x + 5)^2 - 2$. [1 point]

• Cue. Left 5, down 2; vertex $(-5, -2)$.

Q2. Is $g(x) = \frac{1}{2}x^2$ narrower or wider than the parent? [1 point]

• Cue. $|a| = \frac{1}{2} < 1$, so wider (compression).