Describe and apply transformations of functions: vertical and horizontal shifts, reflections, and vertical stretches or compressions, and connect a change in the equation to the change in the graph.

What this topic is asking

The Functions category includes transformations (the F-BF standards): how a change to a function's equation shifts, reflects, or stretches its graph. On the Grade 10 MCAS this is tested with parabolas and other parent functions, asking you to match an equation to its graph or to describe the move from a parent function. The rules are short, but the horizontal-shift direction reverses sign, which is where most errors happen.

Shifts

A shift (translation) slides the graph without changing its shape.

• Vertical shift: adding a constant outside the function, $f(x) + k$, moves the graph up by $k$ (down if $k$ is negative). This matches intuition: every output is raised by $k$.
• Horizontal shift: adding a constant inside the function, $f(x - h)$, moves the graph right by $h$. The sign is opposite to its appearance: $f(x - 3)$ shifts right 3, but $f(x + 3)$ shifts left 3.

The horizontal reversal is the single most tested trap. The reason: to get the same output the parent gave at $x$, the new function needs an input that is $h$ larger, so the whole graph slides right.

Reflections

A reflection flips the graph across an axis:

• $-f(x)$ negates every output, flipping the graph across the x-axis (up becomes down).
• $f(-x)$ negates every input, flipping the graph across the y-axis (left becomes right).

For a parabola $f(x) = x^2$, $-f(x) = -x^2$ opens downward instead of upward. For symmetric functions like $x^2$, a y-axis reflection looks unchanged, but for a line or a shifted parabola it is visible.

Vertical stretches and compressions

Multiplying the function by a constant $a$ scales the outputs:

• $|a| > 1$ is a vertical stretch: the graph is pulled taller, which for a parabola makes it narrower.
• $0 < |a| < 1$ is a vertical compression: the graph is flattened, making a parabola wider.
• A negative $a$ also reflects across the x-axis, on top of the stretch or compression.

Connecting to vertex form

For parabolas, vertex form $y = a(x - h)^2 + k$ collects every transformation: $a$ is the stretch and reflection, $h$ is the horizontal shift (vertex $x$), and $k$ is the vertical shift (vertex $y$). Reading vertex form is therefore the same skill as describing transformations, which is why this topic and quadratic graphs reinforce each other.

Transforming other parent functions

The same rules apply to any parent function, not just $x^2$. For a linear parent $f(x) = x$, the function $g(x) = -x + 3$ is a reflection across the x-axis (the $-x$) shifted up 3. For an absolute-value parent $f(x) = |x|$, the function $h(x) = |x - 2| - 1$ shifts right 2 and down 1, moving the corner point to $(2, -1)$. Recognizing the parent and then applying shifts, reflections, and stretches lets you handle a graph you have not memorized.

A practical reading habit: scan the equation from the inside out. The change grouped with $x$ inside the function is the horizontal shift (sign reversed); a factor multiplying the whole function is the stretch or reflection; a constant added at the end is the vertical shift. Naming them in that order keeps a multi-part transformation organized.

Matching an equation to a graph

A frequent MCAS item gives several candidate equations and one graph, and asks which matches. Work from the most visible feature: locate the vertex or key point on the graph, then check which equation places it there. If the graph of a parabola has its vertex at $(-2, 3)$ and opens downward, the equation must be of the form $y = a(x + 2)^2 + 3$ with $a < 0$. This narrows four options to one quickly, without plotting every point.

Try this

Q1. How does $y = f(x) - 6$ transform $y = f(x)$?

• Cue. Shifts down 6.

Q2. The vertex of $y = (x - 2)^2 + 7$ is where?

• Cue. $(2, 7)$: right 2, up 7.