Problem 35 Graph. $$ x=2^{-y} $$... [FREE SOLUTION] (2024)

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In this exercise, you are given an exponential equation: \[ x = 2^{-y} \]. Exponential equations are equations where the variable is in the exponent. This means that the equation shows how one variable changes exponentially with respect to another. Here, the variable \(y\) is the exponent.

To better understand exponential equations, it's important to know that they usually have a constant base raised to a variable exponent. Examples can be \( 2^y \) or \( 3^{-y} \). These types of equations are very common in scientific contexts, like biological growth, radioactive decay, and financial calculations.

The specific equation given, \( x = 2^{-y} \), shows that as \( y \) increases, \( x \) decreases because of the negative exponent. This is contrary to what you would see in a positive exponential equation.

Graphing functions is a way to visualize the relationship between variables. For the exponential equation \( x = 2^{-y} \), you will create a graph on a coordinate plane to show how \( x \) changes as \( y \) varies.

• First, determine some key points to plot. Calculate values of \( x \) for different values of \( y \).
• For \( y = 0 \), \( x = 2^0 = 1 \).
• For \( y = 1 \), \( x = 2^{-1} = 0.5 \).
• For \( y = -1 \), \( x = 2 \).

After you identify key points, you can plot them on the coordinate plane. Connecting these points will give you the shape of the graph. For our equation, the curve should be a decreasing exponential curve. This means it starts with higher values of \( x \) when \( y \) is negative and approaches zero as \( y \) becomes more positive.

Graphing functions helps in understanding behaviors and trends, making equations easier to comprehend.

Natural logarithms, denoted as \( \ln \), are a crucial concept in solving exponential equations. They use the constant \( e \) (approximately 2.71828) as the base of the logarithm. One key property is that the natural logarithm and the exponential function are inverses of each other.

In our equation \( x = 2^{-y} \), using natural logarithms helps to solve for \( y \). Here’s how you do it:

• Take the natural logarithm of both sides: \[ \ln(x) = \ln(2^{-y}) \]
• Apply the logarithm rules: \[ \ln(x) = -y \ln(2) \]
• Finally, solve for \( y \): \[ y = -\frac{\ln(x)}{\ln(2)} \]

Using natural logarithms helps convert exponential equations into linear ones, making them easier to solve. It’s a powerful tool used not only in algebra but also in calculus, finance, and many other fields.

Understanding the coordinate plane is essential when plotting functions. The coordinate plane is a two-dimensional surface formed by two perpendicular lines: the x-axis (horizontal) and the y-axis (vertical).

Each point on the plane is determined by an ordered pair \((x, y)\). For example, the point \((1, 0)\) lies at 1 on the x-axis and 0 on the y-axis.

When graphing the given equation \( x = 2^{-y} \), you place each key point on this plane. For example:

• \((1, 0)\)
• \((0.5, 1)\)
• \((2, -1) \)

You’ll then connect these points to form the curve.

This coordinate system is fundamental for visualizing and better understanding mathematical relationships between variables. Whether you’re dealing with simple lines or more complicated curves, knowing how to use the coordinate plane is essential.