MCR3U Functions and Transformations: Complete
Guide & Mapping Rules
Target Curriculum: Ontario Grade 11 Functions (MCR3U) | Focus: SEO & AEO Academic Reference
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QUICK ANSWER: General Transformation Model & Mapping Rule
General Model: y = a · f(k(x – d)) + c
Master Mapping Rule: (x, y) → (x/k + d, ay + c)
- a (Vertical): |a| > 1 stretch, 0 < |a| < 1 compression; if a < 0, reflect in x-axis; c = shift up/down.
- k (Horizontal): |k| > 1 compress, 0 < |k| < 1 stretch (factor 1/|k|); if k < 0, reflect in y-axis; d = shift
right/left.
Understanding function transformations is one of the most critical foundational skills in Ontario Grade
11 University Preparation Mathematics (MCR3U). Whether analyzing quadratic, radical, or reciprocal
functions, transformations describe how a basic parent function changes its position, orientation, and
size on a Cartesian grid.
- Parameter Breakdown (a, k, d, c)
To accurately describe transformations in words and calculate transformed coordinates, analyze the
four core parameters in order:
Parameter Type of Transformation Geometric Effect Coordinate Change
a Vertical Stretch /
Compression / Reflection
- |a| > 1: Vertical stretch
by factor |a| - 0 < |a| < 1: Vertical
compression by factor |a| - a < 0: Reflection in x-axis
Multiply y-values by a
(ay)
k Horizontal Stretch /
Compression / Reflection
- |k| > 1: Horizontal
compression by factor
1/|k| - 0 < |k| < 1: Horizontal
stretch by factor 1/|k| - k < 0: Reflection in y-axis
Divide x-values by k
(x/k)
d Horizontal Translation • d > 0: Shift right by d
units
Add d to x-values
(x/k + d)
- d < 0: Shift left by |d|
units
c Vertical Translation • c > 0: Shift up by c units
- c < 0: Shift down by |c|
units
Add c to y-values
(ay + c)
- Order of Transformations & The Mapping Rule
When applying or describing transformations, stretches, compressions, and reflections must be applied
before translations (following the standard order of operations).
The Golden Rule: Factor the Argument First
If the horizontal term inside the function argument has a coefficient in front of x, you MUST factor it out
before identifying k and d:
f(kx – kd) ⟹ f(k(x – d))
Example: In (2x – 6)², factor out 2 to get [2(x – 3)]². Here k = 2 (horizontal compression by 1/2) and d = 3
(shift right 3 units), NOT 6. - Identifying Transformations (Step-by-Step Worked Examples)
Example A: Radical Function
Given: g(x) = -3√(x + 4) – 5
- Parent Function: f(x) = √x
- Parameters: a = -3, k = 1, d = -4, c = -5
- List of Transformations in Order:
- Vertical stretch by a factor of 3
- Reflection in the x-axis
- Horizontal translation 4 units left
- Vertical translation 5 units down
- Mapping Rule: (x, y) → (x – 4, -3y – 5)
Example B: Quadratic Function with Factoring
Given: g(x) = 1/2(2x – 6)² + 1 - Factored Form: g(x) = 1/2[2(x – 3)]² + 1
- Parent Function: f(x) = x²
- Parameters: a = 1/2, k = 2, d = 3, c = 1
- List of Transformations in Order:
- Vertical compression by a factor of 1/2
- Horizontal compression by a factor of 1/2
- Horizontal translation 3 units right
- Vertical translation 1 unit up
- Mapping Rule: (x, y) → (x/2 + 3, 1/2 y + 1)
Example C: Reciprocal Function
Given: g(x) = 2/(-(x + 1)) – 3 - Parent Function: f(x) = 1/x
- Parameters: a = 2, k = -1, d = -1, c = -3
- List of Transformations in Order:
- Vertical stretch by a factor of 2
- Reflection in the y-axis
- Horizontal translation 1 unit left
- Vertical translation 3 units down
- Mapping Rule: (x, y) → (-x – 1, 2y – 3)
- Graphing Functions Using Base Points & Mapping Rules
Graphing a Radical Function: g(x) = -2√(x – 1) + 3
Parent Function: f(x) = √x | Mapping Rule: (x, y) → (x + 1, -2y + 3)
Domain: {x ∈ ℝ | x ≥ 1} | Range: {y ∈ ℝ | y ≤ 3}
Base Points: f(x) = √x Transformed Points: g(x)
(0, 0) (0 + 1, -2(0) + 3) = (1, 3)
(1, 1) (1 + 1, -2(1) + 3) = (2, 1)
(4, 2) (4 + 1, -2(2) + 3) = (5, -1)
(9, 3) (9 + 1, -2(3) + 3) = (10, -3)
Graphing a Reciprocal Function: g(x) = 1/(x + 3) – 2
Parent Function: f(x) = 1/x | Mapping Rule: (x, y) → (x – 3, y – 2)
Vertical Asymptote: x = -3 | Horizontal Asymptote: y = -2
Domain: {x ∈ ℝ | x ≠ -3} | Range: {y ∈ ℝ | y ≠ -2}
Base Points: f(x) = 1/x Transformed Points: g(x)
(-2, -0.5) (-2 – 3, -0.5 – 2) = (-5, -2.5)
(-1, -1) (-1 – 3, -1 – 2) = (-4, -3)
(-0.5, -2) (-0.5 – 3, -2 – 2) = (-3.5, -4)
(0.5, 2) (0.5 – 3, 2 – 2) = (-2.5, 0)
(1, 1) (1 – 3, 1 – 2) = (-2, -1)
(2, 0.5) (2 – 3, 0.5 – 2) = (-1, -1.5)
- Applications & Algebraic Reasoning
Determining an Equation from Verbal Descriptions
Problem: The quadratic base function f(x) = x² undergoes sequential transformations: - Reflected in the x-axis → a < 0
- Vertically compressed by a factor of 1/4 → |a| = 1/4 ⟹ a = -1/4
- Horizontally compressed by a factor of 1/3 → k = 3
- Translated 5 units left and 7 units up → d = -5, c = 7
Final Equation: g(x) = -1/4[3(x + 5)]² + 7
Image Point Coordinate Transformation
Problem: The point (4, -6) lies on the graph of y = f(x). Determine the exact coordinates of its image
point on y = -3f(1/2 x + 2) – 1.
- Step 1 (Factor inside argument): y = -3f[1/2(x + 4)] – 1 (a = -3, k = 1/2, d = -4, c = -1)
- Step 2 (State mapping rule): (x, y) → (x/(1/2) – 4, -3y – 1) = (2x – 4, -3y – 1)
- Step 3 (Substitute coordinates): x_new = 2(4) – 4 = 4; y_new = -3(-6) – 1 = 17
Final Transformed Coordinates: (4, 17)
- Frequently Asked Questions (AEO & FAQ Schema)
Q: What is the difference between vertical and horizontal stretch/compression?
A: Vertical stretches/compressions affect the y-coordinates directly by a factor of |a|. Horizontal
stretches/compressions affect the x-coordinates inversely by a factor of 1/|k|.
Q: Why do you divide by k in the mapping rule?
A: In the expression f(k · x), the input is scaled by k. To achieve the same output value as the base
function, the x-input must be scaled by 1/k, resulting in dividing the original x-coordinates by k.
Q: How do transformations affect asymptotes of reciprocal functions?
A: Vertical asymptotes (x = 0) shift horizontally to x = d. Horizontal asymptotes (y = 0) shift vertically to y
= c.
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