Problem 66 - Algebra 2

Question

REVIEW If the equation \(y=3^{x}\) is graphed, which of the following values of \(x\) would produce a point closest to the \(x\) -axis? $$ \begin{array}{l}{\mathrm{F} \frac{3}{4}} \\ {\mathrm{G} \frac{1}{4}} \\\ {\mathrm{H} 0} \\ {\mathrm{J}-\frac{3}{4}}\end{array} $$

Step-by-Step Solution

Answer

\(x = -\frac{3}{4}\) produces a point closest to the \(x\)-axis.

Step 1: Understand the Problem

We are given the equation \(y=3^{x}\) and need to determine which value of \(x\) from the given options produces a point closest to the \(x\)-axis. The \(x\)-axis corresponds to \(y=0\), so we want the smallest value of \(y\).

Step 2: Calculate for \(x = \frac{3}{4}\)

Substitute \(x = \frac{3}{4}\) into the equation \(y=3^{x}\):\[ y = 3^{\frac{3}{4}} \approx 2.279 \]This is the distance from the \(x\)-axis as we're calculating the corresponding \(y\) value.

Step 3: Calculate for \(x = \frac{1}{4}\)

Substitute \(x = \frac{1}{4}\) into the equation:\[ y = 3^{\frac{1}{4}} \approx 1.316 \]

Step 4: Calculate for \(x = 0\)

Substitute \(x = 0\) into the equation:\[ y = 3^{0} = 1 \]

Step 5: Calculate for \(x = -\frac{3}{4}\)

Substitute \(x = -\frac{3}{4}\) into the equation:\[ y = 3^{-\frac{3}{4}} = \frac{1}{3^{\frac{3}{4}}} \approx 0.439 \]

Step 6: Compare Results

Now compare the \(y\)-values. The values we calculated are approximately: 2.279, 1.316, 1, and 0.439. The smallest \(y\)-value, which is closest to the \(x\)-axis, corresponds to \(x = -\frac{3}{4}\).

Key Concepts

Graphing Exponential Functions

Graphing exponential functions can seem challenging, but it's much easier when you break it down step by step. As seen in the exercise, the exponential function is given as \( y = 3^x \). This function represents growth, where each increase in \( x \) leads to a multiplicative increase in \( y \).
To graph this function, you plot points by substituting different values of \( x \) into the equation to find corresponding \( y \) values. This helps illustrate how rapidly \( y \) increases as \( x \) becomes more positive.

• For \( x = 0 \), \( y = 1 \), because any number raised to the power of zero equals one.
• For positive values of \( x \), say \( x = 1 \), \( y = 3^1 = 3 \).
• As \( x \) becomes negative, such as \( x = -1 \), \( y = 3^{-1} = \frac{1}{3} \), showing decay with negative \( x \).

These calculations allow us to sketch a curve that begins near \( y = 0 \) and rapidly rises as \( x \) increases. Understanding the graph helps visualize how the function behaves over various \( x \) values, including which points lie closest to the x-axis.

Equation Solving

Equation solving is essential to graphing functions and understanding their behavior. In this exercise, our task is to find which value of \( x \) brings the resulting point closest to the \( x \)-axis on the graph of \( y = 3^x \). Remember, the \( x \)-axis represents where \( y = 0 \).
However, for exponential functions like \( y = 3^x \), \( y \) won't ever actually reach zero since \( 3^x \) never equals zero. Instead, we look for the smallest \( y \).

• Calculate \( y \) for each option: for \( x = -\frac{3}{4} \), \( y \approx 0.439 \); for \( x = 0 \), \( y = 1 \); and so on.
• Compare these values to find the smallest.
• The smallest \( y \) is the point closest to the \( x \)-axis.

This process simplifies understanding of how small changes in \( x \) affect \( y \) in exponential functions.

Coordinate Geometry

Coordinate geometry allows us to visualize mathematical relationships by plotting them on a graph. Here, the focus is on locating points on a plane for the function \( y = 3^x \). Each \( x \) value produces a specific \( y \). This method provides a visual representation, helping us to solve the exercise.
The task requires calculating \( y \) for different \( x \) values, then plotting these points. Let's understand how this helps in coordinate geometry:

• For various \( x \), compute \( y \) using the equation \( y = 3^x \).
• Mark these \( (x, y) \) points on the graph plane.
• Identify which point is nearest to the \( x \)-axis (where \( y \) is smallest).

The visual nature of coordinate geometry makes it easier to grasp these concepts. By plotting and comparing, one identifies that \( x = -\frac{3}{4} \) results in a point closest to the \( x \)-axis due to its smallest \( y \) value. Thus, coordinate geometry becomes an indispensable tool in solving such exponential problems.