Combining Transformations: Building Complex Graphs from Parent Functions

A consistent way to read a transformed function

Many complicated-looking graphs are built by transforming a simpler “parent” function. A standard container for multiple transformations is:

g(x) = a f(b(x - h)) + k

Think of this as taking the parent graph y = f(x) and changing it in layers. To stay consistent, interpret each parameter as follows:

• Inside (horizontal) changes happen to the input of f: b(x - h) affects how x-values are used.
• Outside (vertical) changes happen to the output of f: multiplying by a and adding k affects y-values.

What each parameter does (with the “opposite” rule for horizontal)

Key interpretation habit: horizontal transformations come from what happens to x inside f( ), and they behave “oppositely” (because you are changing the input). Vertical transformations come from what happens outside f( ) and behave “directly.”

Choosing an order for graphing: a reliable workflow

When you need to sketch or analyze g(x) = a f(b(x-h)) + k, use a consistent order so you don’t lose track:

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1. Start with the parent: identify f(x) and recall a few key points on its graph.
2. Apply horizontal changes: handle b and h (inside the function).
3. Apply vertical changes: handle a (multiply outputs).
4. Apply vertical shift: handle k (add to outputs).

This order matches point-mapping neatly: first change x-coordinates (horizontal), then change y-coordinates (vertical), then shift up/down.

Mapping key points step-by-step (the most dependable method)

Instead of trying to redraw an entire curve, transform a set of key points from the parent graph. Suppose (x, y) is on y = f(x), meaning y = f(x). For

g(x) = a f(b(x - h)) + k

the corresponding point (X, Y) on y = g(x) is:

X = x / b + h and Y = a y + k

This single mapping rule automatically includes reflections and stretches/compressions (because b and a can be negative or fractional).

Example 1: Multiple transformations of a quadratic

Let f(x) = x^2 and

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

Identify transformations:

• b = 3: horizontal compression by factor 1/3
• h = 1: shift right 1
• a = -2: reflect across x-axis and vertical stretch by 2
• k = 4: shift up 4

Map key points from y = x^2. Use common points: (-2,4), (-1,1), (0,0), (1,1), (2,4).

Compute X = x/3 + 1 and Y = -2y + 4:

Graphing move: plot these transformed points and draw the parabola through them. The vertex is at (1, 4) (the image of the parent vertex (0,0)).

Quick check using evaluated points

Pick one or two x-values on the new graph and verify by direct evaluation of g.

For example, the mapped point (4/3, 2) should satisfy g(4/3) = 2:

g(4/3) = -2 f(3(4/3 - 1)) + 4 = -2 f(3(1/3)) + 4 = -2 f(1) + 4 = -2(1^2) + 4 = 2

It matches.

Working both directions: equation ↔ transformation description

From equation to description

Procedure:

1. Identify the parent function f.
2. Rewrite the inside as b(x - h) if possible (factor out a constant from the x-expression).
3. Read off a, b, h, k.
4. State horizontal changes (from inside), then vertical changes (outside).

Example 2: Describe transformations for g(x) = 3| -2(x + 4) | - 1.

• Parent: f(x) = |x|
• Inside: -2(x + 4) = -2(x - (-4)) so b = -2, h = -4
• Outside: a = 3, k = -1

Transformations:

• Shift left 4 (because h = -4)
• Horizontal compression by factor 1/2 and reflection across the y-axis (because b = -2)
• Vertical stretch by 3 (because a = 3)
• Shift down 1 (because k = -1)

Note: For |x|, reflecting across the y-axis does not change the shape (it is symmetric), but it still matters as a transformation step for other parents.

From description to equation

Procedure:

1. Start with the parent f(x).
2. Build the inside first: decide b and h to create f(b(x - h)).
3. Then apply vertical scale/reflection: multiply by a.
4. Finally apply vertical shift: add k.

Example 3: Start with f(x) = √x. Apply: horizontal stretch by 2, shift right 3, reflect across x-axis, shift up 5.

• Horizontal stretch by 2 means multiply input by 1/2: use b = 1/2.
• Shift right 3: use (x - 3).
• Reflect across x-axis: a = -1.
• Shift up 5: k = 5.

Equation:

g(x) = -√((1/2)(x - 3)) + 5

You may also write g(x) = -√((x - 3)/2) + 5.

Cumulative exercises (with built-in checks)

Exercise set A: Identify and map points

A1. Let f(x) = |x| and g(x) = -f(2(x - 3)) + 1.

• (i) List a, b, h, k.
• (ii) Map the parent key points (-2,2), (-1,1), (0,0), (1,1), (2,2) using X = x/b + h, Y = ay + k.
• (iii) Check: verify your mapped point corresponding to parent (1,1) by evaluating g(X) at your computed X.

A2. Let f(x) = 1/x and g(x) = 2 f(-(x + 1)) - 3.

• (i) Identify transformations.
• (ii) Map these parent points: (1,1), (2,1/2), (-1,-1), (-2,-1/2).
• (iii) Check: choose one mapped point (X,Y) and confirm g(X)=Y by direct substitution.

Exercise set B: Graphing order practice (horizontal then vertical then shifts)

B1. Start with f(x) = x^2. Create g(x) = a f(b(x - h)) + k that does the following in order: horizontal compression by 1/4, shift left 2, vertical stretch by 3, shift down 5.

• (i) Write the equation.
• (ii) Map the parent vertex and two additional parent points.
• (iii) Check: evaluate g at the x-value of your mapped vertex to confirm the y-value.

B2. Rewrite g(x) = -2(x + 6)^2 + 7 in the form a f(b(x - h)) + k with f(x)=x^2. Then state the transformations and map the parent point (1,1).

Exercise set C: Write equations from descriptions (increasing complexity)

C1. Parent f(x)=|x|. Transformations: reflect across y-axis, horizontal stretch by 3, shift right 2, vertical compression by 1/2, shift up 4. Write g(x) in the form a f(b(x-h)) + k.

C2. Parent f(x)=√x. Transformations: shift left 1, horizontal compression by 2, reflect across x-axis, vertical stretch by 4, shift down 3. Write g(x).

C3. Parent f(x)=1/x. Transformations: shift right 5, reflect across y-axis, vertical stretch by 2, shift up 1. Write g(x). Then do a check by evaluating your function at one convenient x-value (avoid values that make the denominator zero) and confirm it matches the transformed point you would expect from mapping.

Exercise set D: Mixed “equation ↔ description” with verification

D1. A function is defined by g(x)=3f(-2(x-4))+6. Without knowing the exact parent, describe the transformations applied to f. Then, if the point (-1, 2) lies on y=f(x), find the corresponding point on y=g(x) using mapping. Check: verify by showing that plugging your new x-value into g produces the new y-value in terms of f.

D2. Suppose f(0)=1 and f(2)=5. Define g(x)=-f((x-3)/2)+4.

• (i) Find two points on g that correspond to the given points on f by mapping.
• (ii) Check: evaluate g(3) and g(7) directly using the given values of f and confirm they match your mapped y-values.