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2.4 · Product Property
State the Product Property of Exponents.
bm · bn = ?
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2.4 · Power Property
State the Power Property of Exponents.
(bm)n = ?
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2.4 · Negative Exponent
State the Negative Exponent Property.
b−n = ?
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2.4 · Horizontal → Vertical
Every horizontal translation bx+h is equivalent to what vertical dilation form?
Think: Product Property in reverse
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2.4 · Apply: Rewrite
Rewrite f(x) = 4x+2 as a vertical dilation with no horizontal shift.
4x+2 = ?
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2.4 · Apply: Rewrite
Rewrite g(x) = 2x+3 as a vertical dilation with no horizontal shift.
2x+3 = ?
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2.4 · Apply: Negative Shift
Rewrite h(x) = 3x−2 as a vertical dilation with no horizontal shift.
3x−2 = ?
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2.4 · Apply: Coefficient
Rewrite k(x) = 4 · 3x+2 with no horizontal shift.
4 · 3x+2 = ?
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✓ Power Property
(bm)n = bmn
Raise a power to a power → multiply the exponents. Example: (53)x = 53x
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✓ Product Property
bm · bn = bm+n
Same base? Add the exponents. Example: 53 · 58 = 511
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✓ Horizontal → Vertical
a · bx where a = bh
The shift h becomes a vertical factor. Example: 4x+2 = 16 · 4x because a = 42 = 16.
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✓ Negative Exponent
b−n = 1/bn
Negative exponent → reciprocal. Example: x−3 = 1/x3 and 3x−2 = (1/9) · 3x.
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✓ Rewrite 2x+3
8 · 2x
2x+3 = 23 · 2x = 8 · 2x. The factor a = 23 = 8.
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✓ Rewrite 4x+2
16 · 4x
4x+2 = 42 · 4x = 16 · 4x. The factor a = 42 = 16.
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✓ Rewrite 4 · 3x+2
36 · 3x
4 · 3x+2 = 4 · 32 · 3x = 4 · 9 · 3x = 36 · 3x
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✓ Rewrite 3x−2
(1/9) · 3x
3x−2 = 3−2 · 3x = (1/9) · 3x. Negative shift gives a fraction.
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2.4 · Horizontal Dilation → Base Change
How do you rewrite a horizontal dilation bcx as a single exponential with exponent x?
bcx = ?
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2.4 · Apply: Base Change
Rewrite y = 92x as an exponential with exponent x only.
92x = ?
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2.4 · Identify Transformation
What transformation does f(x) = 2x/3 represent on y = 2x?
Compare the exponent to the parent function
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2.4 · Identify Transformation
What transformation does f(x) = 5x−1 represent on y = 5x?
Compare to the standard horizontal shift form
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✓ Rewrite 92x
81x
92x = (92)x = 81x. Use the Power Property to absorb the 2 into the base.
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✓ Horizontal Dilation → Base
bcx = (bc)x
Power Property: absorb the dilation factor c into a new base. Example: 92x = (92)x = 81x.
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✓ Identify: 5x−1
Horizontal shift right 1 unit
f(x) = 5x−1 → y = 5x shifted right 1. Also equivalent to f(x) = (1/5) · 5x.
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✓ Identify: 2x/3
Horizontal dilation by factor 3
f(x) = 2(1/3)x → horizontal stretch by 3. Equivalent to (21/3)x = ∛2 x.