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Topic 1.12 · Vertical Translation
What does g(x) = f(x) + k do to the graph of f?
g(x) = f(x) + k
Outside the parentheses → affects y
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Topic 1.12 · Horizontal Translation
What does g(x) = f(x + h) do to the graph of f?
g(x) = f(x + h)
Inside → OPPOSITE of what it looks like
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Topic 1.12 · Vertical Dilation
What does g(x) = a·f(x) do? What if a < 0?
g(x) = a · f(x)
Outside → exact. Sign matters!
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Topic 1.12 · Horizontal Dilation
What does g(x) = f(bx) do? What if b < 0?
g(x) = f(bx)
Inside → OPPOSITE. Factor = 1/|b|
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Topic 1.12 · Describing Transforms
Describe ALL transformations in:
g(x) = 2f(x − 3) + 1
Work through each part in order
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Topic 1.12 · Describing Transforms
Describe ALL transformations in:
n(x) = −4m(2x)
Inside first, then outside
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Topic 1.12 · Domain & Range
f has domain −3 ≤ x ≤ 5 and range 1 ≤ y ≤ 3. Find domain and range of g(x) = 2f(x+3) − 4.
g(x) = 2f(x + 3) − 4
Set x+3 equal to each boundary
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Topic 1.12 · Table Problem
Using the table, h(x) = 3f(2x) − 1. Find h(−1) and h(2).
f: x=−2→1, x=4→3
Substitute into h, then look up f
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✓ Horizontal Translation
Shifts LEFT h units (if h > 0)
Inside → OPPOSITE. f(x+3) shifts left 3. f(x−3) shifts right 3.
The graph moves in the opposite direction of the sign.
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✓ Vertical Translation
Shifts graph UP k units
Outside → EXACT. f(x)+k adds k to every y-value.
If k<0, shifts down. Example: f(x)−2 shifts down 2.
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✓ Horizontal Dilation
Dilation by 1/|b|; b<0 → reflect y-axis
Inside → OPPOSITE. f(2x) compresses by ½ (not stretch ×2).
f(−x) reflects over the y-axis.
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✓ Vertical Dilation
Dilation by |a|; a<0 → reflect x-axis
Outside → EXACT. 3·f(x) stretches vertically ×3.
−2·f(x) stretches ×2 AND reflects over the x-axis.
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✓ n(x) = −4m(2x)
Horiz ×½ · Vert ×4 · Reflect x-axis
Inside: b=2 → horiz dilation ×½.
Outside: |a|=4 → vert dilation ×4.
a=−4<0 → reflects over x-axis.
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✓ g(x) = 2f(x − 3) + 1
Right 3 · Vert ×2 · Up 1
(x−3) inside → opposite of −3 → right 3.
Outside 2 → vert dilation ×2.
Outside +1 → up 1.
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✓ Table: h(−1) and h(2)
h(−1) = 2 · h(2) = 8
h(−1) = 3f(−2)−1 = 3(1)−1 = 2
h(2) = 3f(4)−1 = 3(3)−1 = 8
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✓ Domain & Range of g
Domain: −6 ≤ x ≤ 2 · Range: −2 ≤ y ≤ 2
Domain: x+3=−3→x=−6; x+3=5→x=2.
Range: 2(1)−4=−2; 2(3)−4=2.