Graph Transformations and Symmetry in Functions: Vertical, Horizontal, Reflection, and Scaling

What does f(x)+c do to a graph?

Shifts the graph vertically by c units.

What does f(x)-c do to a graph?

Shifts the graph down c units.

What does f(x)+c do when c>0?

Shifts the graph up c units.

What does f(x+c) do to a graph?

Shifts the graph horizontally left c units.

What does f(x-c) do to a graph?

Shifts the graph horizontally right c units.

Why do horizontal shifts seem backwards?

The change is made to the input x, so the direction is reversed.

What does cf(x) do to a graph?

Vertically scales the graph by factor c.

What does cf(x) do when c>1?

Vertically stretches the graph by c.

What does cf(x) do when 0

Vertically shrinks the graph.

What does f(cx) do to a graph?

Horizontally scales the graph by factor 1/c.

What does f(cx) do when c>1?

Horizontally compresses the graph by factor c.

What does f(cx) do when 0

Horizontally stretches the graph by factor 1/c.

What is the key difference between inside and outside changes?

Outside changes are vertical; inside changes are horizontal.

What does -f(x) do to a graph?

Reflects the graph across the x-axis.

What does f(-x) do to a graph?

Reflects the graph across the y-axis.

What point transformation comes from f(x)+c?

(x,y) → (x,y+c)

What point transformation comes from f(x+c)?

(x,y) → (x-c,y)

What point transformation comes from cf(x)?

(x,y) → (x,cy)

What point transformation comes from f(cx)?

(x,y) → (x/c,y)

What happens to (x,y) under -f(x)?

(x,y) → (x,-y)

What happens to (x,y) under f(-x)?

(x,y) → (-x,y)

What does f(x+3) do?

Shifts the graph 3 units left.

What does f(x-3) do?

Shifts the graph 3 units right.

What does f(x)+3 do?

Shifts the graph 3 units up.

What does f(x)-3 do?

Shifts the graph 3 units down.

What does 2f(x) do?

Vertically stretches by a factor of 2.

What does 1/2 f(x) do?

Vertically shrinks by a factor of 2.

What does f(2x) do?

Horizontally compresses by a factor of 2.

What does f(x/2) do?

Horizontally stretches by a factor of 2.

What does -f(x) reflect?

Across the x-axis.

What does f(-x) reflect?

Across the y-axis.

What happens when the scaling constant is negative?

It causes a reflection plus a scaling.

What does f(-2x) do?

Reflects across the y-axis and horizontally compresses by 2.

How do you read 2f(x+1)-1?

Left 1, vertical stretch 2, then down 1.

What is the order for 2f(x+1)-1?

Shift left 1 → stretch vertically by 2 → shift down 1.

Does transformation order always matter?

No; some transformations can be reordered, but others cannot.

Why can order matter?

Some transformations change the result of later transformations.

What is an even function?

One satisfying f(-x)=f(x).

What symmetry does an even function have?

Reflectional symmetry across the y-axis.

What is an odd function?

One satisfying f(-x)=-f(x).

What symmetry does an odd function have?

180° rotational symmetry about the origin.

What must the domain allow for even/odd classification?

If x is in the domain, -x must also be in the domain.

How do you test if a function is even?

Calculate f(-x) and check whether it equals f(x).

How do you test if a function is odd?

Calculate f(-x) and check whether it equals -f(x).

What if f(-x) equals neither f(x) nor -f(x)?

The function is neither even nor odd.

What symmetry does x² have?

Even; it has y-axis symmetry.

What symmetry does x³ have?

Odd; it has 180° rotational symmetry.

What does a 180° rotation about the origin do?

Reflects across both the x-axis and y-axis.

What is the main transformation rule to remember?

Outside = vertical; inside = horizontal.