What are the differences between horizontal and vertical transformations?

Horizontal: Affect the x-values, inside the function. | Vertical: Affect the y-values, outside the function.

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What are the differences between horizontal and vertical transformations?

Horizontal: Affect the x-values, inside the function. | Vertical: Affect the y-values, outside the function.

What are the differences between growth and decay exponential functions?

Growth: Base > 1, function increases as x increases. | Decay: 0 < Base < 1, function decreases as x increases.

What are the differences between product and power properties?

Product: Adding exponents when multiplying same bases. | Power: Multiplying exponents when raising a power to a power.

Compare $b^x$ and $b^{-x}$ .

$b^x$ : Exponential growth if b > 1, decay if 0 < b < 1. | $b^{-x}$ : Exponential decay if b > 1, growth if 0 < b < 1.

Compare horizontal translation and vertical stretch.

Horizontal translation: Shifts the graph left or right. | Vertical stretch: Changes the steepness of the graph.

Compare horizontal and vertical dilations.

Horizontal: Affects the x-axis, compression or stretch. | Vertical: Affects the y-axis, stretch or compression.

Compare $f(x) = b^{x+k}$ and $f(x) = b^x + k$ .

$f(x) = b^{x+k}$ : Horizontal shift. | $f(x) = b^x + k$ : Vertical shift.

Compare $f(x) = b^{cx}$ and $f(x) = cb^x$ .

$f(x) = b^{cx}$ : Horizontal dilation. | $f(x) = cb^x$ : Vertical dilation.

Compare reflection about the x-axis and y-axis.

x-axis: Changes the sign of the output. | y-axis: Changes the sign of the input.

Compare $b^{1/2}$ and $(b)^{2}$ .

$b^{1/2}$ : Square root of b. | $(b)^{2}$ : Square of b.

Simplify $2^{2x} * 2^{x-1}$ .

Add the exponents: $2^{2x + (x-1)} = 2^{3x-1}$ .

Simplify $(4^{x+1})^2$ .

Multiply the exponents: $4^{2(x+1)} = 4^{2x+2}$ .

Simplify $5^{-x}$ .

Use the negative exponent property: $5^{-x} = \frac{1}{5^x}$ .

Evaluate $9^{1/2}$ .

Find the square root: $9^{1/2} = \sqrt{9} = 3$ .

Rewrite $f(x) = 3^{x-2}$ in the form $a cdot 3^x$ .

Use the product property: $3^{x-2} = 3^x cdot 3^{-2} = \frac{1}{9} cdot 3^x$ .

Rewrite $g(x) = 16^{3x}$ in the form $a^x$ .

Use the power property: $16^{3x} = (16^3)^x = 4096^x$ .

Describe the transformation from $y = 2^x$ to $y = 2^{x+3}$ .

Horizontal translation 3 units to the left.

Describe the transformation from $y = 3^x$ to $y = 2 cdot 3^x$ .

Vertical stretch by a factor of 2.

Describe the transformation from $y = 4^x$ to $y = 4^{-x}$ .

Reflection over the y-axis.

Describe the transformation from $y = 5^x$ to $y = (1/5)^x$ .

Reflection over the y-axis.

Explain the product property.

When multiplying exponential terms with the same base, add the exponents: $b^m * b^n = b^{m+n}$ .

Explain the power property.

When raising an exponential term to a power, multiply the exponents: $(b^m)^n = b^{mn}$ .

Explain the negative exponent property.

A negative exponent indicates the reciprocal of the base raised to the positive value of the exponent: $b^{-n} = \frac{1}{b^n}$ .

Explain exponential unit fractions.

An exponential unit fraction, like $b^{1/k}$ , represents the kth root of b. The value of $b^{1/k}$ is the kth root of b, when it exists.

Explain the relationship between horizontal translation and vertical dilation.

A horizontal translation of an exponential function, $f(x) = b^{x+k}$ , is equivalent to a vertical dilation, $f(x) = ab^x$ , where $a = b^k$ .

Explain the relationship between horizontal dilation and changing the base.

A horizontal dilation of an exponential function, $f(x) = b^{cx}$ , is equivalent to changing the base, $f(x) = (b^c)^x$ .

Explain the effect of reflecting an exponential function over the y-axis.

Reflecting the graph of $f(x) = b^x$ over the y-axis gives the graph of $f(x) = b^{-x} = \frac{1}{b^x}$ .

Why is the base of an exponential function restricted to positive numbers not equal to 1?

If the base were negative, the function would oscillate between positive and negative values. If the base were 1, the function would be constant. If the base were 0, the function would be undefined for negative exponents.

Explain the importance of understanding transformations of exponential functions.

Understanding transformations helps analyze and predict the behavior of exponential functions under various conditions, which is crucial for modeling real-world scenarios.

Explain why $(-4)^{1/2}$ does not have a real value.

The square root of a negative number is not a real number because no real number multiplied by itself yields a negative result.