Symmetry of a Graph

1 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Reflecting outputs and reflecting inputs produce different graph symmetries. Track a generic point to understand the transformation rather than relying on one picture.

Method

$-f(x)$ reflects across the horizontal axis. $f(-x)$ reflects across the vertical axis. $-f(-x)$ reflects through the origin. Reflection across $y=x$ exchanges coordinates and describes an inverse relation.

Worked example

If $(2,5)$ lies on $f$ , then $(2,-5)$ lies on $-f$ , $(-2,5)$ lies on $f(-x)$ , and $(-2,-5)$ lies on $-f(-x)$ . For the even function $x^2$ , horizontal input reflection leaves the graph unchanged, but output reflection gives a downward-opening parabola.

Common mistake

The graph $y=-x^2$ is not a right-to-left reflection of $y=x^2$ ; it is an up-to-down reflection.

Check your understanding

How is $y=-\sqrt x$ obtained from $y=\sqrt x$ ?

Show answer

Reflect vertically across the horizontal axis; the domain remains $x\ge0$ .

Explore

Move and scale a graph

Try this. Change one slider at a time. Positive h moves the vertex right; k moves it up; negative a reflects the curve and a = 0 makes it constant.

Vertical multiplier a1Horizontal shift h0Vertical shift k0

y = 1(x − (0))² + (0). Vertex (0, 0); opens up. Range y ≥ 0.

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

If a graph is symmetric about the y-axis and contains (2,5), which point must it contain?

(2,−5)

(−2,−5)

(−2,5)

Hint 1 · Find a starting point

Reflection in the y-axis changes the horizontal coordinate.

Hint 2 · Take the next step

Keep the vertical coordinate unchanged.

Show the reasoning

Answer: (−2,5)

The reflected point is (−2,5).

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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