Function Transformations

🪞 Function Transformations

Part 1 of 7 — Parent Functions & the Big Idea

Topics in This Part

🔑 Key Concept: A transformation takes a known graph — a parent function — and moves, stretches, or flips it without changing its basic shape. Master a handful of parent shapes, and you can graph thousands of functions instantly.

What Is a Transformation?

Every transformation we study is built from a parent function $f(x)$. We change $f$ in one of two places:

$g(x) = a \cdot f\big(b(x - h)\big) + k$

• The numbers outside $f$ — the $a$ and the $+k$ — change the output ($y$). These behave intuitively.
• The numbers inside $f$ — the $b$ and the $-h$ — change the input ($x$). These behave backwards (we'll see why in Part 3).

A First Look

💡 The shape stays the same — a parabola is still a parabola, a line is still a line. Only its position and size change.

The Parent Functions

These are the "starting graphs." Memorizing their basic shape is the foundation of everything that follows.

Each passes through (or starts at) a memorable anchor point. For $x^2$, $\lvert x\rvert$, $x^3$, and $\sqrt{x}$, that anchor is the origin $(0,0)$.

🔑 Key Idea: When you transform a parent function, you are really just relocating its anchor point and adjusting its steepness.

Concept Check 🎯

Match the Parent to Its Shape 🔽

Connect each parent function to its defining feature.

Find the Anchor 🧮

The parent function $f(x) = x^2$ has its vertex (anchor point) at the origin. For each transformed graph, the anchor moves. Just read off where it lands.

1) $g(x) = x^2 + 5$ — the vertex moves to $(0, \,?)$ (enter the $y$-coordinate) 2) $g(x) = (x - 3)^2$ — the vertex moves to $(\,?, 0)$ (enter the $x$-coordinate)

The Plan

You now know the two families:

• Outside changes ($a$, $k$) act on the output $y$ — and behave intuitively.
• Inside changes ($b$, $h$) act on the input $x$ — and behave backwards.

Next up, Part 2 tackles the easy family first: vertical transformations, where everything does exactly what you'd expect.

🪞 Function Transformations

Part 2 of 7 — Vertical Transformations (the intuitive family)

🔑 The Idea: Everything that happens outside $f(x)$ changes the output $y$ — and it does exactly what your intuition expects. Add to go up, multiply to stretch taller, negate to flip over the $x$-axis.

Vertical Shifts

Adding a constant outside $f$ moves the whole graph up or down:

$g(x) = f(x) + k$

• $k > 0$ → shift up by $k$
• $k < 0$ → shift down by $\lvert k \rvert$

Example: $g(x) = \lvert x \rvert - 4$

Start with the V-shape of $\lvert x \rvert$ (vertex at origin), then slide it down $4$. New vertex: $(0, -4)$.

💡 Only the $y$-coordinate changes; the $x$-coordinate is untouched.

Vertical Stretches, Compressions & Reflections

Multiplying $f(x)$ by a constant $a$ scales the output:

$g(x) = a \cdot f(x)$

Example: $g(x) = -2x^2$

The $-$ flips the parabola to open downward; the $2$ makes it twice as steep.

$\text{Point } (1, 1) \text{ on } x^2 \;\longrightarrow\; (1, -2) \text{ on } -2x^2$

⚠️ A negative $a$ does two things: it reflects AND scales by $\lvert a\rvert$. Don't forget the reflection.

Concept Check 🎯

Name the Vertical Transformation 🔽

Each $g$ comes from the parent $f$. Identify what happened.

Track the Point 🧮

The point $(4, 2)$ lies on the parent $f(x) = \sqrt{x}$. Find its image (new coordinates) under each vertical transformation.

1) $g(x) = \sqrt{x} + 3$ → image is $(4, \,?)$ 2) $g(x) = 5\sqrt{x}$ → image is $(4, \,?)$ 3) $g(x) = -\sqrt{x}$ → image is $(4, \,?)$

🪞 Function Transformations

Part 3 of 7 — Horizontal Transformations (the backwards family)

🔑 The Twist: Everything inside $f$ acts on the input $x$ — and it does the opposite of what you'd expect. $f(x - 3)$ shifts right, not left. We'll prove why.

Why "Inside" Is Backwards

Consider $g(x) = f(x - 3)$. To get the same output the parent had at $x = 0$, the new function needs its input to satisfy $x - 3 = 0$, i.e. $x = 3$.

$\text{Parent value at } 0 \;\Longrightarrow\; \text{new graph reaches it at } x = 3 \;\Rightarrow\; \text{shift RIGHT } 3$

So a minus inside shifts right, and a plus inside shifts left — the opposite of the sign.

Example: $g(x) = \sqrt{x + 2}$

This is $f(x + 2)$ with $f = \sqrt{x}$, so it shifts the square-root graph left $2$. Its starting point moves from $(0,0)$ to $(-2, 0)$.

Horizontal Stretches, Compressions & Reflections

Multiplying the input by $b$ scales horizontally — also backwards:

$g(x) = f(bx)$

Example: $g(x) = f(2x)$

Multiplying the input by $2$ compresses the graph horizontally by a factor of $\tfrac{1}{2}$ — the graph reaches each output at half the $x$-distance.

⚠️ Inside is backwards twice over: $b = 2$ does NOT stretch by $2$; it compresses by $\tfrac{1}{2}$. And $f(-x)$ reflects over the $y$-axis (compare $-f(x)$, which reflected over the $x$-axis in Part 2).

Concept Check 🎯

Inside or Outside? Which Way? 🔽

Classify each transformation of the parent $f$.

Track the Point (Horizontally) 🧮

The point $(6, 9)$ lies on a parent function $f$. Find the $x$-coordinate of its image under each horizontal transformation. (The $y$-coordinate stays $9$ for all of these.)

1) $g(x) = f(x - 2)$ → image $x$-coordinate $= \,?$ 2) $g(x) = f(2x)$ → image $x$-coordinate $= \,?$ 3) $g(x) = f(-x)$ → image $x$-coordinate $= \,?$

🪞 Function Transformations

Part 4 of 7 — Combining Transformations & Order

🔑 The Goal: Read a function like $g(x) = -2(x-3)^2 + 1$ and list every transformation, in the correct order, that turns the parent into it.

The General Form

$g(x) = a \cdot f\big(b(x - h)\big) + k$

Read off the four numbers:

Example: $g(x) = -2(x - 3)^2 + 1$

Here $a = -2$, $h = 3$, $k = 1$ (and $b = 1$). So, starting from $f(x) = x^2$:

• Reflect over the $x$-axis and stretch vertically by $2$ (from $a = -2$)
• Shift right $3$ (from $h = 3$)
• Shift up $1$ (from $k = 1$)

Vertex moves from $(0,0)$ to $(3, 1)$, and the parabola opens down.

Why Order Matters

When transformations affect the same direction, order changes the result. A safe, reliable order:

1. Horizontal inside work: factor out $b$, handle reflections/stretches, then the shift $h$.
2. Vertical outside work: apply $a$ (stretch/reflect), then add $k$.

⚠️ Classic trap: stretch-then-shift $\ne$ shift-then-stretch. For $g(x) = 2f(x) + 3$, you must multiply first, then add $3$. Doing it backwards, $2(f(x)+3) = 2f(x) + 6$, gives the wrong graph.

Example: applying $g(x) = 2\sqrt{x} - 3$ to the point $(4, 2)$

$y: \quad 2 \cdot \underbrace{2}_{\text{stretch}} = 4, \qquad 4 - 3 = 1$

So $(4, 2)$ maps to $(4, 1)$ — multiply by $2$ first, subtract $3$ second.

Concept Check 🎯

Decode Each Parameter 🔽

For $g(x) = -\dfrac{1}{2}(x + 6)^2 - 4$ built from $f(x) = x^2$, identify what each piece does.

Apply Two Steps in Order 🧮

The point $(9, 3)$ lies on $f(x) = \sqrt{x}$. Apply $g(x) = -2\sqrt{x} + 1$ to find the image.

1) First, the stretch & reflection $-2 \cdot 3 = \,?$ (the intermediate $y$-value) 2) Then add $1$: final $y$-coordinate $= \,?$ 3) The $x$-coordinate of the image is $= \,?$ (no horizontal change)

🪞 Function Transformations

Part 5 of 7 — Mapping Points & Reading Graphs

🔑 Skill: Given any point on the parent, instantly produce its image — and run it in reverse to recover the parent's points from a transformed graph. This is exactly what AP free-response asks.

The Coordinate Mapping Rule

For $g(x) = a\,f\big(b(x - h)\big) + k$, a point $(x_0, y_0)$ on $f$ maps to:

$\left(\dfrac{x_0}{b} + h,\;\; a\,y_0 + k\right)$

In words:

• New $x$: divide the old $x$ by $b$, then add $h$.
• New $y$: multiply the old $y$ by $a$, then add $k$.

Example: $g(x) = 3f(2x - 4) + 1$

First rewrite the inside in $b(x-h)$ form: $2x - 4 = 2(x - 2)$, so $b = 2$, $h = 2$, $a = 3$, $k = 1$.

A point $(6, 5)$ on $f$ maps to:

$\left(\dfrac{6}{2} + 2,\;\; 3 \cdot 5 + 1\right) = (3 + 2,\; 16) = (5, 16)$

💡 Always factor out $b$ first. $f(2x - 4)$ is NOT a shift of $4$; it is $f\big(2(x-2)\big)$ — a compression by $\tfrac12$ and a shift right $2$.

Reading a Transformed Graph (Reverse Direction)

If you're handed the graph of $g$ and asked about $f$, undo each step:

$x_0 = b\,(x_g - h), \qquad y_0 = \dfrac{y_g - k}{a}$

Example

$g(x) = 2f(x - 1) - 3$ passes through $(4, 7)$. Which point is on $f$?

• Undo the vertical: $y_0 = \dfrac{7 - (-3)}{2} = \dfrac{10}{2} = 5$
• Undo the horizontal: $x_0 = 1 \cdot (4 - 1) = 3$

So $(3, 5)$ is on $f$.

✅ Check forward: $f(3) = 5 \Rightarrow g(4) = 2(5) - 3 = 7$ ✓

Concept Check 🎯

Map the Point 🧮

The point $(8, 4)$ lies on $f$. Apply $g(x) = 2f\!\left(\tfrac{1}{2}x\right) - 3$. Use new $x = \dfrac{x_0}{b} + h$ and new $y = a\,y_0 + k$ with $a = 2,\ b = \tfrac12,\ h = 0,\ k = -3$.

1) New $x$-coordinate $= \,?$ 2) New $y$-coordinate $= \,?$

Reverse-Engineer the Parent 🔽

$g(x) = -f(x + 2) + 4$ passes through the point $(1, 6)$. Recover the matching point on $f$.

🪞 Function Transformations

Part 6 of 7 — Symmetry & Modeling

🔑 Connection: Reflections lead straight into even/odd symmetry, and shifts/stretches let you build a model to fit real data. This is where transformations earn their keep on the AP exam.

Even & Odd Symmetry

Symmetry is a transformation that leaves the graph unchanged:

Examples

• $f(x) = x^2$ is even: $f(-x) = (-x)^2 = x^2 = f(x)$. Reflecting over the $y$-axis gives the same parabola.
• $f(x) = x^3$ is odd: $f(-x) = (-x)^3 = -x^3 = -f(x)$. A $y$-axis reflection then an $x$-axis reflection returns the original.
• $f(x) = \lvert x \rvert$ is even; $f(x) = \dfrac{1}{x}$ is odd.

💡 If a point $(a, b)$ is on an even function, so is $(-a, b)$. If $(a, b)$ is on an odd function, so is $(-a, -b)$.

Concept Check 🎯

Modeling with Transformations

To fit a parent function to data, choose $a, h, k$ so the anchor and scale match.

Example: a quadratic model

A ball's height (ft) over time $t$ (s) peaks at $20$ ft at time $t = 2$, starting from the parent $f(t) = t^2$.

Because the peak is a maximum, the parabola opens down ($a < 0$). The vertex is $(2, 20)$, so $h = 2$, $k = 20$:

$H(t) = a(t - 2)^2 + 20$

If at $t = 0$ the height is $0$: $\;0 = a(0-2)^2 + 20 = 4a + 20 \Rightarrow a = -5$.

$\boxed{H(t) = -5(t - 2)^2 + 20}$

✅ Check: $H(2) = 20$ (the peak) and $H(0) = -5(4) + 20 = 0$ ✓.

Build the Model 🔽

A V-shaped cost curve from $f(x) = \lvert x \rvert$ has its minimum at the point $(4, 3)$ and opens upward. Choose the correct parameters for $g(x) = a\lvert x - h\rvert + k$.

Solve for the Stretch 🧮

A parabola opening downward has vertex $(1, 8)$ and passes through $(3, 0)$. It has the form $g(x) = a(x - 1)^2 + 8$.

1) Substitute the point: $0 = a(3 - 1)^2 + 8$. What is $(3-1)^2$? $= \,?$ 2) Solve $0 = 4a + 8$ for $a$: $a = \,?$

🪞 Function Transformations

Part 7 of 7 — Mixed Mastery & Exit Quiz

You can now (1) identify parent functions, (2) apply vertical transformations, (3) apply backwards horizontal transformations, (4) combine them in order, (5) map points both ways, and (6) use symmetry and modeling. Time to put it all together.

Quick Reference

For $g(x) = a\,f\big(b(x - h)\big) + k$:

Point mapping: $(x_0, y_0) \to \left(\dfrac{x_0}{b} + h,\; a\,y_0 + k\right)$.

⚠️ Top two traps: (1) inside changes are backwards; (2) always factor out $b$ before reading the shift $h$.

Mixed Practice 🎯

Mixed Drill 🧮

The function $g(x) = 4(x + 1)^2 - 9$ is built from $f(x) = x^2$.

1) Vertex $x$-coordinate $= \,?$ 2) Vertex $y$-coordinate $= \,?$ 3) Vertical stretch factor (the value of $a$) $= \,?$

Exit Quiz ✅

Answer all three to finish the lesson.