krit.club logo

Functions - Transformations of Graphs

Grade 11IB

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

•

Vertical Translation: A transformation of the form y=f(x)+ky = f(x) + k shifts the entire graph vertically. If k>0k > 0, the graph moves upwards; if k<0k < 0, it moves downwards. Visually, every point (x,y)(x, y) on the original function is mapped to a new point (x,y+k)(x, y+k), meaning the shape of the graph remains identical but its vertical position changes relative to the x-axis.

•

Horizontal Translation: A transformation of the form y=f(x−h)y = f(x - h) shifts the graph horizontally. If h>0h > 0, the graph moves to the right; if h<0h < 0, the graph moves to the left. Visually, the curve slides along the x-axis, mapping any point (x,y)(x, y) to (x+h,y)(x+h, y). Note that the sign inside the brackets is opposite to the direction of movement.

•

Vertical Stretch and Compression: The transformation y=af(x)y = a f(x) scales the graph vertically by a factor of aa. If ∣a∣>1|a| > 1, the graph undergoes a vertical stretch away from the x-axis. If 0<∣a∣<10 < |a| < 1, it undergoes a vertical compression toward the x-axis. Visually, the x-intercepts remain fixed as 'anchor points' while all other points move further from or closer to the x-axis.

•

Horizontal Stretch and Compression: The transformation y=f(bx)y = f(bx) scales the graph horizontally by a scale factor of 1b\frac{1}{b}. If ∣b∣>1|b| > 1, the graph is horizontally compressed toward the y-axis. If 0<∣b∣<10 < |b| < 1, the graph is horizontally stretched away from the y-axis. Visually, the y-intercept remains fixed while the width of the features of the graph (like peaks or loops) is altered.

•

Reflection in the Axes: A transformation y=−f(x)y = -f(x) reflects the graph across the x-axis, effectively flipping it vertically (mapping (x,y)(x, y) to (x,−y)(x, -y)). A transformation y=f(−x)y = f(-x) reflects the graph across the y-axis, creating a mirror image horizontally (mapping (x,y)(x, y) to (−x,y)(-x, y)).

•

Combined Transformations: When a function undergoes multiple changes, such as y=af(b(x−h))+ky = a f(b(x-h)) + k, the transformations should typically be applied in a specific order: stretches and reflections first, followed by translations. Visually, this ensures that the scaling happens relative to the original axes before the entire shape is moved to its final location.

•

Invariant Points: During specific transformations, certain points on the graph do not move. In a vertical stretch y=af(x)y = a f(x), points on the x-axis are invariant. In a horizontal stretch y=f(bx)y = f(bx), points on the y-axis are invariant. Identifying these points helps in accurately sketching the transformed curve.

📐Formulae

y=f(x)+ky = f(x) + k (Vertical translation by vector (0k)\begin{pmatrix} 0 \\ k \end{pmatrix})

y=f(x−h)y = f(x - h) (Horizontal translation by vector (h0)\begin{pmatrix} h \\ 0 \end{pmatrix})

y=af(x)y = a f(x) (Vertical stretch by scale factor aa)

y=f(bx)y = f(bx) (Horizontal stretch by scale factor 1b\frac{1}{b})

y=−f(x)y = -f(x) (Reflection in the x-axis)

y=f(−x)y = f(-x) (Reflection in the y-axis)

(x,y)→(xb+h,ay+k)(x, y) \rightarrow (\frac{x}{b} + h, ay + k) (General point mapping for y=af(b(x−h))+ky = a f(b(x-h)) + k)

💡Examples

Problem 1:

Given the function f(x)=x2f(x) = x^2, describe the sequence of transformations required to obtain the graph of g(x)=−2(x+3)2−5g(x) = -2(x + 3)^2 - 5 and find the new coordinates of the vertex.

Solution:

Step 1: Identify the transformations. The function is in the form g(x)=af(x−h)+kg(x) = a f(x-h) + k. Here, a=−2a = -2, h=−3h = -3, and k=−5k = -5. Step 2: List transformations in order:

  1. Horizontal translation 3 units to the left (h=−3h = -3).
  2. Vertical stretch by a factor of 22 and a reflection in the x-axis (due to a=−2a = -2).
  3. Vertical translation 5 units down (k=−5k = -5). Step 3: Apply to the vertex. The original vertex of f(x)=x2f(x) = x^2 is (0,0)(0, 0). Applying the horizontal shift: (0−3,0)=(−3,0)(0 - 3, 0) = (-3, 0). Applying the stretch/reflection: (−3,−2×0)=(−3,0)(-3, -2 \times 0) = (-3, 0). Applying the vertical shift: (−3,0−5)=(−3,−5)(-3, 0 - 5) = (-3, -5).

Explanation:

We use the standard form af(b(x−h))+ka f(b(x-h)) + k to extract transformation parameters. The vertex (0,0)(0,0) is a useful reference point that directly reflects translations (h,k)(h, k) when the base function is x2x^2.

Problem 2:

The point P(4,12)P(4, 12) lies on the graph of y=f(x)y = f(x). Find the coordinates of the corresponding point P′P' on the graph of y=12f(2x)+3y = \frac{1}{2} f(2x) + 3.

Solution:

Step 1: Identify the horizontal transformation. The input is 2x2x, so b=2b = 2. The horizontal scale factor is 1b=12\frac{1}{b} = \frac{1}{2}. The new x-coordinate is x′=4×12=2x' = 4 \times \frac{1}{2} = 2. Step 2: Identify the vertical transformations. The function is multiplied by a=12a = \frac{1}{2} and then k=3k = 3 is added. The new y-coordinate is y′=(12×12)+3y' = (12 \times \frac{1}{2}) + 3. Step 3: Calculate y′y'. y′=6+3=9y' = 6 + 3 = 9. Step 4: State final coordinates. P′=(2,9)P' = (2, 9).

Explanation:

This solution uses the mapping rule (x,y)→(xb,ay+k)(x, y) \rightarrow (\frac{x}{b}, ay + k). We apply the horizontal stretch factor to the x-coordinate and the vertical stretch and translation to the y-coordinate.