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Algebra - Transformations of graphs (translations, reflections, stretches)-extended

Grade 10IB

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

🔑Concepts

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Vertical and Horizontal Translations: Adding a constant kk outside the function, y=f(x)+ky = f(x) + k, shifts the graph vertically by kk units. Subtracting a constant hh inside the function argument, y=f(x−h)y = f(x - h), shifts the graph horizontally to the right by hh units.

Graph showing a translation of a parabola 2 units right and 1 unit up.
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Reflections: Multiplying the entire function by −1-1, y=−f(x)y = -f(x), reflects the graph across the xx-axis. Multiplying the input variable by −1-1, y=f(−x)y = f(-x), reflects the graph across the yy-axis.

Graph showing reflection of an exponential function across the x-axis.
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Vertical Stretches: Multiplying the function by a factor aa, y=a⋅f(x)y = a \cdot f(x), stretches the graph vertically away from the xx-axis if a>1a > 1, or compresses it toward the axis if 0<a<10 < a < 1. Each yy-coordinate is multiplied by aa.

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Horizontal Stretches: Multiplying the input xx by a factor bb, y=f(b⋅x)y = f(b \cdot x), results in a horizontal stretch by a factor of 1b\frac{1}{b} toward or away from the yy-axis.

📐Formulae

y=f(x)+k(Vertical Translation)y = f(x) + k \quad \text{(Vertical Translation)}

y=f(x−h)(Horizontal Translation)y = f(x - h) \quad \text{(Horizontal Translation)}

y=−f(x)(Reflection in x-axis)y = -f(x) \quad \text{(Reflection in } x\text{-axis)}

y=f(−x)(Reflection in y-axis)y = f(-x) \quad \text{(Reflection in } y\text{-axis)}

y=a⋅f(x)(Vertical Stretch by factor a)y = a \cdot f(x) \quad \text{(Vertical Stretch by factor } a\text{)}

y=f(b⋅x)(Horizontal Stretch by factor 1b)y = f(b \cdot x) \quad \text{(Horizontal Stretch by factor } \frac{1}{b}\text{)}

💡Examples

Problem 1:

Given the parent function f(x)=x2f(x) = x^2, find the equation of the function g(x)g(x) that results from translating f(x)f(x) by 44 units to the left and 77 units up.

Solution:

g(x)=(x+4)2+7g(x) = (x + 4)^2 + 7

Explanation:

To translate 44 units to the left, we replace xx with (x−(−4))(x - (-4)), which is (x+4)(x + 4). To translate 77 units up, we add 77 to the entire function. Thus, g(x)=(x+4)2+7g(x) = (x + 4)^2 + 7.

Problem 2:

The point P(2,6)P(2, 6) lies on the graph of y=f(x)y = f(x). Determine the new coordinates of point PP on the graph of y=3f(x−1)y = 3f(x - 1).

Solution:

(3,18)(3, 18)

Explanation:

The transformation f(x−1)f(x - 1) shifts the graph 11 unit to the right, so the xx-coordinate becomes 2+1=32 + 1 = 3. The transformation 3f(x)3f(x) is a vertical stretch by a factor of 33, so the yy-coordinate becomes 6×3=186 \times 3 = 18. The new coordinates are (3,18)(3, 18).

Problem 3:

Describe the transformations required to turn the graph of f(x)=xf(x) = \sqrt{x} into g(x)=−2xg(x) = -\sqrt{2x}.

Solution:

  1. Horizontal compression by a scale factor of 12\frac{1}{2} from the yy-axis. 2. Reflection in the xx-axis.

Explanation:

The coefficient 22 inside the square root (2x2x) represents a horizontal stretch with scale factor 1b\frac{1}{b}, where b=2b=2, resulting in a compression. The negative sign outside the function (−f(x)-f(x)) indicates a reflection across the xx-axis.

Problem 4:

Given the function f(x)=∣x∣f(x) = |x|, describe and sketch the transformation g(x)=12∣x∣g(x) = \frac{1}{2}|x|.

V-shaped graph being compressed vertically.

Solution:

The transformation is a vertical stretch (or compression) by a scale factor of 12\frac{1}{2}. For every point (x,y)(x, y) on f(x)f(x), the new coordinates on g(x)g(x) will be (x,12y)(x, \frac{1}{2}y). For example, (2,2)(2, 2) becomes (2,1)(2, 1).

Explanation:

Since the multiplier a=12a = \frac{1}{2} is outside the absolute value sign and 0<a<10 < a < 1, the graph is compressed vertically toward the xx-axis.

Problem 5:

The graph of y=f(x)y = f(x) is shown in the plane. Sketch the graph of y=f(−x)−2y = f(-x) - 2.

A triangular shaped line segment reflected across the y-axis and moved down.

Solution:

  1. Reflect the graph of y=f(x)y = f(x) in the yy-axis to obtain y=f(−x)y = f(-x).
  2. Translate the resulting graph downwards by 22 units to obtain y=f(−x)−2y = f(-x) - 2.

Explanation:

Replacing xx with −x-x performs a horizontal reflection. Subtracting 22 from the result shifts every point down by 22 units.