Transformations of Functions

Q: Why does $f(x + 3)$ move the graph left instead of right?

The change is happening to the input. To get the same output $f$ used to give at $x = 0$, you now have to plug in $x = -3$, because $f(-3 + 3) = f(0)$. Every point needs an input $3$ smaller than before, so the whole graph slides $3$ units left. Matching the inside to $x - h$ handles the sign automatically.

Q: Does the order of transformations matter?

Yes, when a stretch or reflection is combined with a vertical shift. For $g(x) = -2f(x) + 5$, the stretch and reflection happen before the shift up $5$ — the same order of operations you'd use to evaluate the expression. Shifts in different directions (one horizontal, one vertical) can happen in either order.

Q: How can I tell a vertical stretch from a shift on a graph?

Look at the $x$-intercepts and the overall shape. A shift moves every point the same distance and keeps the shape identical. A stretch changes the steepness and leaves any point on the $x$-axis exactly where it was, since multiplying $0$ by $a$ still gives $0$.

A function transformation takes a graph you already know — like $f(x) = x^2$ or $f(x) = |x|$ — and moves, stretches, or flips it. Instead of plotting a new function point by point, you read the transformation off the equation and slide the old graph into place. Every transformation is one of three moves: a shift, a stretch or compression, or a reflection.

The whole topic comes down to one question: is the change inside the function's parentheses or outside? Outside changes act on the output, so they move the graph vertically and behave exactly like you'd expect. Inside changes act on the input, so they move the graph horizontally — and they work backwards from what the sign suggests. Keep that inside-outside split straight and every transformation problem is quick.

Shifts: adding and subtracting

Adding a constant outside the function shifts the graph vertically: $f(x) + k$ moves every point up $k$ units (down if $k$ is negative). This one matches intuition — bigger outputs mean a higher graph.

Adding a constant inside the function shifts the graph horizontally, and the direction is the opposite of the sign: $f(x - h)$ moves the graph right $h$ units, so $f(x - 3)$ is a shift right $3$ and $f(x + 3)$ is a shift left $3$ . To read the direction correctly, match the inside to the pattern $x - h$ : in $f(x + 3)$ , we have $x + 3 = x - (-3)$ , so $h = -3$ and the graph moves left.

In the graph below, the black curve is $f(x) = x^2$ and the blue curve is $g(x) = (x - 2)^2 + 1$ : the same parabola shifted right $2$ and up $1$ . The vertex moved from $(0, 0)$ to $(2, 1)$ , and every other point moved with it.

Stretches and compressions: multiplying

Multiplying the function by a constant changes its shape. Outside the function, $a \cdot f(x)$ stretches the graph vertically when $|a| > 1$ and compresses it toward the $x$ -axis when $0 < |a| < 1$ . Every $y$ -value gets multiplied by $a$ , so $2f(x)$ makes the graph twice as tall at every point.

The $x$ -intercepts do not move under a vertical stretch — a point with $y = 0$ stays at $y = 0$ no matter what you multiply by. That's a fast way to check whether a graph was stretched or shifted: if the intercepts stayed put but the graph got steeper, it was a stretch.

Reflections: the negative sign

A negative sign outside the function reflects the graph across the $x$ -axis: $-f(x)$ flips every point to the opposite side, so a parabola opening up becomes a parabola opening down. A negative sign inside the function, $f(-x)$ , reflects the graph across the $y$ -axis instead — same inside-outside logic as shifts: outside acts vertically, inside acts horizontally.

Transformations stack. The function $g(x) = -2(x + 1)^2 + 5$ takes $f(x) = x^2$ , shifts it left $1$ , stretches it vertically by $2$ , reflects it across the $x$ -axis, and shifts it up $5$ . Read the pieces one at a time and the graph draws itself: the vertex lands at $(-1, 5)$ and the parabola opens down, twice as steep as the parent. The graph below shows the black parent $f(x) = x^2$ and the blue result $g(x)$ .

Worked examples

Example 1: describe a shift

Describe the transformation of $f(x) = x^2$ that produces $g(x) = (x - 4)^2 + 2$ .

Match the inside to the pattern

x - h

x - 4 = x - h, \quad h = 4

A positive

h

shifts the graph right

\text{right } 4

The

+\,2

outside shifts the graph up

\text{up } 2

Check with the vertex:

(0, 0)

moves to

(4, 2)

Answer: Shift right $4$ and up $2$

Example 2: a stretch and a reflection together

Describe the transformation of $f(x) = |x|$ that produces $g(x) = -2|x|$ .

The factor

2

outside stretches the graph vertically by

2

2f(x)

The negative sign reflects it across the

x

-axis

-2f(x)

Track one point:

(3, 3)

is on

f

g(3) = -2|3| = -6

(3, 3)

moves to

(3, -6)

— twice as far from the axis, on the other side

Answer: Vertical stretch by $2$ , then a reflection across the $x$ -axis; the V opens downward

Example 3: write the equation from a description

Write the equation for $f(x) = \sqrt{x}$ shifted left $5$ and down $2$ .

Left

5

means

h = -5

, so the inside is

x - (-5)

\sqrt{x + 5}

Down

2

subtracts

2

outside

\sqrt{x + 5} - 2

Check the starting point:

(0, 0)

f

moves to

(-5, -2)

Answer: $g(x) = \sqrt{x + 5} - 2$

Example 4: track a single point

The point $(2, 5)$ is on the graph of $f$ . Where does it land on the graph of $g(x) = f(x - 3) + 7$ ?

The inside

x - 3

shifts the graph right

3

, so the

x

-coordinate increases by

3

2 + 3 = 5

The

+\,7

outside shifts the graph up

7

, so the

y

-coordinate increases by

7

5 + 7 = 12

Confirm:

g(5) = f(5 - 3) + 7 = f(2) + 7 = 5 + 7

g(5) = 12

Answer: $(5, 12)$

Try one yourself

Common questions

Why does

f(x + 3)

move the graph left instead of right?

The change is happening to the input. To get the same output $f$ used to give at $x = 0$ , you now have to plug in $x = -3$ , because $f(-3 + 3) = f(0)$ . Every point needs an input $3$ smaller than before, so the whole graph slides $3$ units left. Matching the inside to $x - h$ handles the sign automatically.

Does the order of transformations matter?

Yes, when a stretch or reflection is combined with a vertical shift. For $g(x) = -2f(x) + 5$ , the stretch and reflection happen before the shift up $5$ — the same order of operations you'd use to evaluate the expression. Shifts in different directions (one horizontal, one vertical) can happen in either order.

How can I tell a vertical stretch from a shift on a graph?

Look at the $x$ -intercepts and the overall shape. A shift moves every point the same distance and keeps the shape identical. A stretch changes the steepness and leaves any point on the $x$ -axis exactly where it was, since multiplying $0$ by $a$ still gives $0$ .

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Transformations of Functions

Shifts: adding and subtracting

Stretches and compressions: multiplying

Reflections: the negative sign

Worked examples

Example 1: describe a shift

Example 2: a stretch and a reflection together

Example 3: write the equation from a description

Example 4: track a single point

Try one yourself

Common questions

Keep learning

Introduction to Functions

Axis of Symmetry

Inverse Relations & Functions