Grade by Grade
A. Characteristics of Functions · Functions and Equivalent Expressions

Combining Transformations: Sketching y = af(k(x − d)) + c

A1.9 A1.8About 75 minutes
Learning goals
  • I can identify a, k, d and c in an equation and describe each transformation in order.
  • I can use the mapping rule (x, y) → (x/k + d, ay + c) to sketch a transformed parent function.
  • I can factor k out of the bracket before identifying d.
  • I can state the domain, range and any asymptotes of a transformed function, and write an equation from a description.

Words to know

y = af(k(x − d)) + c
The general form of a transformed function: a and k stretch, compress or reflect; d and c translate.
combined mapping rule
(x, y) → (x/k + d, ay + c).
parameter
A number in an equation, such as a, k, d or c, that can change to make a family of functions.
key points
Easy points on a parent function, used to sketch its transformations.

Putting it all together

A figure skater's routine combines a few basic moves: a spin, a jump, a glide. Each one is simple on its own; the skill is in doing them in the right order. In this lesson you will combine all four transformations in one equation:

y = af(k(x − d)) + c

With the four parent functions and these four parameters you can sketch a huge number of graphs by hand, and state their domain and range without making a table of values from scratch.

y = a f( k (x − d) ) + cavertical stretchor compression;a < 0: flip in x-axiskhorizontal stretchor compression (1/k);k < 0: flip in y-axisdhorizontal shiftright if d > 0,left if d < 0cvertical shiftup if c > 0,down if c < 0Order: stretches, compressions and reflections (a, k) first, then translations (d, c)mapping rule: (x, y) → (x/k + d, ay + c)
Each parameter has one job. a and c are outside the function and act on y; k and d are inside and act on x.

The order matters

Just as with order of operations (BEDMAS), apply the transformations in this order:

  1. Stretches, compressions and reflections (a and k).
  2. Translations (d and c).

The combined mapping rule does this automatically:

(x, y) → (x/k + d, ay + c)

For x: first divide by k, then add d. For y: first multiply by a, then add c.

Important: k must be factored out of the bracket. For y = √(2x − 6), write √(2(x − 3)) first: k = 2 and d = 3, not d = 6.

Worked example

Example 1: A transformed square root

Describe the transformations of y = √x that give y = −2√(x + 1) + 3. Then sketch it and state the domain and range.

  1. Match to y = af(k(x − d)) + c: a = −2, k = 1, d = −1, c = 3.
  2. Transformations: vertical stretch by a factor of 2; reflection in the x-axis; translation 1 left and 3 up.
  3. Mapping rule: (x, y) → (x − 1, −2y + 3).
  4. Parent points: (0, 0) → (−1, 3); (1, 1) → (0, 1); (4, 2) → (3, −1); (9, 3) → (8, −3).
  5. The graph starts at (−1, 3) and goes down to the right.
  6. D = {x ∈ ℝ | x ≥ −1}, R = {y ∈ ℝ | y ≤ 3}.
  7. Check one point in the equation: x = 3 gives −2√4 + 3 = −4 + 3 = −1 ✓.

Starts at (−1, 3), passes (0, 1), (3, −1), (8, −3). D: x ≥ −1; R: y ≤ 3.

xy−22610−4−224(−1, 3)(0, 1)(3, −1)(8, −3)parent y = √xy = −2√(x + 1) + 3a = −2: stretch by 2,flip in x-axisd = −1: left 1c = 3: up 3(x, y) → (x − 1, −2y + 3)D = {x ∈ ℝ | x ≥ −1}R = {y ∈ ℝ | y ≤ 3}
The parent y = √x (faint) is stretched by 2, flipped over the x-axis, then moved 1 left and 3 up.

A table makes it tidy

Make three columns: parent point (x, y), x/k + d, ay + c. Fill each row with the same two calculations. It is the fastest and safest way to sketch a combined transformation.

Worked example

Example 2: Factor out k first

Sketch y = √(2x − 6) and state its domain and range.

  1. Factor: 2x − 6 = 2(x − 3). So y = √(2(x − 3)): k = 2, d = 3, a = 1, c = 0.
  2. Transformations: horizontal compression by a factor of 1/2, then 3 right.
  3. Rule: (x, y) → (x/2 + 3, y).
  4. (0, 0) → (3, 0); (1, 1) → (3.5, 1); (4, 2) → (5, 2); (9, 3) → (7.5, 3).
  5. D = {x ∈ ℝ | x ≥ 3}, R = {y ∈ ℝ | y ≥ 0}.
  6. Check: x = 5 gives √(10 − 6) = √4 = 2 ✓. If you had used d = 6, the graph would wrongly start at (6, 0).

Starts at (3, 0); D: x ≥ 3; R: y ≥ 0.

Worked example

Example 3: A transformed parabola

Sketch y = −3(x + 2)² + 4 using a mapping rule. State the domain and range.

  1. a = −3, k = 1, d = −2, c = 4. Rule: (x, y) → (x − 2, −3y + 4).
  2. (−2, 4) → (−4, −8); (−1, 1) → (−3, 1); (0, 0) → (−2, 4); (1, 1) → (−1, 1); (2, 4) → (0, −8).
  3. The vertex is (−2, 4) and the parabola opens down, three times as steep as y = x².
  4. D = {x ∈ ℝ}, R = {y ∈ ℝ | y ≤ 4}.

Vertex (−2, 4), opening down; R: y ≤ 4.

Worked example

Example 4: A transformed reciprocal function

Sketch y = 2/(x − 1) + 1 and state the domain, range and asymptotes.

  1. Write it as 2f(x − 1) + 1 with f(x) = 1/x: a = 2, d = 1, c = 1.
  2. Asymptotes move with the translation: x = 1 and y = 1. (The stretch does not move them.)
  3. Rule: (x, y) → (x + 1, 2y + 1). (1, 1) → (2, 3); (2, 0.5) → (3, 2); (0.5, 2) → (1.5, 5); (−1, −1) → (0, −1); (−2, −0.5) → (−1, 0).
  4. D = {x ∈ ℝ | x ≠ 1}, R = {y ∈ ℝ | y ≠ 1}.

Asymptotes x = 1, y = 1; branches through (2, 3), (3, 2) and (0, −1), (−1, 0).

xy−4−345−4−3−2246(2, 3)(3, 2)(0, −1)(−1, 0)(1.5, 5)x = 1y = 1y = 2/(x − 1) + 1= 2f(x − 1) + 1, f(x) = 1/xstretch by 2, right 1, up 1D = {x ∈ ℝ | x ≠ 1}R = {y ∈ ℝ | y ≠ 1}
y = 2/(x − 1) + 1: draw the dashed asymptotes first, then the mapped points, then the two branches.
Worked example

Example 5: A reflection in the y-axis

Sketch y = √(4 − x) − 2 and state the domain and range.

  1. Rewrite the inside: 4 − x = −x + 4 = −(x − 4). So y = √(−(x − 4)) − 2: k = −1, d = 4, c = −2.
  2. Transformations: reflection in the y-axis, then 4 right and 2 down.
  3. Rule: (x, y) → (−x + 4, y − 2). (0, 0) → (4, −2); (1, 1) → (3, −1); (4, 2) → (0, 0); (9, 3) → (−5, 1).
  4. The graph starts at (4, −2) and goes up to the left.
  5. D = {x ∈ ℝ | x ≤ 4}, R = {y ∈ ℝ | y ≥ −2}. Check the domain from the equation: 4 − x ≥ 0 gives x ≤ 4 ✓.

Starts at (4, −2), passes (3, −1), (0, 0), (−5, 1). D: x ≤ 4; R: y ≥ −2.

Worked example

Example 6: Writing the equation

Write the equation of the function: y = 1/x is stretched vertically by a factor of 3, reflected in the x-axis, and translated 2 left and 4 up. State its asymptotes.

  1. a = −3 (stretch by 3 and reflect), d = −2, c = 4, k = 1.
  2. y = −3f(x + 2) + 4 = −3/(x + 2) + 4.
  3. Asymptotes: x = −2 and y = 4.

y = −3/(x + 2) + 4, with asymptotes x = −2 and y = 4.

Just for fun

Mathematicians could have used the letters a, b, c, d in order for the four parameters. Instead, the Ontario curriculum uses a, k, d and c. Somewhere, the letter b is still waiting by the phone.

Hands-on activity

Transformation cards

You need: 8 index cards (or paper squares), a pencil, grid paper and a partner.

  1. On four cards write a parent function: x, x², √x, 1/x. On four more write transformations: "vertical stretch by 2", "reflect in the x-axis", "3 left", "2 up".
  2. Shuffle each pile. Draw one parent and two transformation cards.
  3. Write the equation of the transformed function, make a mapping table with at least four points, sketch the graph, and state the domain and range.
  4. Your partner checks one point by substituting it into your equation.

What to look for: for example, √x with "reflect in the x-axis" and "3 left" gives y = −√(x + 3): points (−3, 0), (−2, −1), (1, −2), (6, −3); D: x ≥ −3; R: y ≤ 0. Substituting x = 1 gives −√4 = −2 ✓.

Your turn

1

Describe the transformations of f(x) = x² that give y = 2(x − 3)² − 5.

2

Describe the transformations of f(x) = √x that give y = √(−x) + 1.

3

Write the mapping rule for y = 4f(x + 5) − 1.

4

Write the mapping rule for y = f(3(x − 2)).

5

Find the image of (4, 2) under y = −f(2(x − 1)) + 6.

6

State the domain and range of y = 3√(x − 2) − 4.

7

State the domain, range and asymptotes of y = −1/(x + 4) + 2.

More practice: warm-up

1

In y = af(k(x − d)) + c, which two parameters change x-values?

2

Which are done first: translations or stretches?

3

What is the vertex of y = −(x + 6)² + 2?

4

Where does y = √(x + 3) − 7 start?

5

Write y = √(5x + 10) in the form √(k(x − d)).

More practice: on your own

1

Sketch y = (1/2)√(x − 2) + 1 using four mapped points, and state the domain and range.

2

Sketch y = √(3x + 9) using mapped points, and state the domain.

3

Sketch y = −2/(x − 3) using mapped points and state the asymptotes.

4

Write the equation: y = √x compressed horizontally by a factor of 1/4, reflected in the x-axis, then moved 1 right.

5

The point (−2, 6) is on y = f(x). Find its image on y = (1/3)f(−(x + 4)) − 5.

Challenge

1

Sketch y = 2√(−x + 3) − 1 and state its domain and range. Check one point.

2

Find the equation of the form y = a√(x − d) + c that starts at (2, −3) and passes through (6, 5).

3

A reciprocal function y = a/(x − d) + c has asymptotes x = −1 and y = 2 and passes through (1, 0). Find a.

Check your understanding

Quiz

1.

What is the correct mapping rule for y = af(k(x − d)) + c?

2.

For y = √(3x − 12), what are k and d?

3.

What is the range of y = −4√(x + 2) + 1?

4.

What are the asymptotes of y = 5/(x + 3) − 2?

5.

The point (2, 4) on y = x² maps to which point on y = −(x − 1)² + 3?

6.

Which equation is y = √x reflected in the y-axis and then translated 5 right?

For parents and teachers

Big idea: Expectation A1.9 asks students to sketch y = af(k(x − d)) + c by applying several transformations to the parent functions and to state the domain and range. A1.8 (the roles of a, k, d and c) is brought together here.

How to help: Ask your teen to name each parameter's job before sketching ("a stretches and flips up/down, k stretches and flips left/right, d slides sideways, c slides up/down"). Then ask for the mapping rule. Finally, have them check one point in the original equation.

Common difficulties:

  • Not factoring k out of the bracket (√(2x − 6) has d = 3, not 6).
  • Translating before stretching, which gives wrong points.
  • Forgetting that a reflection in the y-axis reverses the domain of √x (x ≤ something).
  • Stating the range of a reflected √x as y ≥ c instead of y ≤ c.