Problem 61 - Algebra 1

Question

You can obtain a graphical representation of the relationship \(2^{1 / 2}=\sqrt{2}\) by investigating the graph of \(f(x)=2^{x}\) a. Graph \(f(x)=2^{x}\) b. Use the Trace feature to find values of \(f\) when \(x=\frac{1}{2}\) c. Compare the value from part (b) with the value of \(\sqrt{2}\).

Step-by-Step Solution

Answer

On graphing \(f(x)=2^{x}\) and tracing \(x=\frac{1}{2}\), we get \(f(\frac{1}{2})\) approximately equal to 1.414. This value is the same as \(\sqrt{2}\), showing that \(2^{\frac{1}{2}}\) is indeed equal to \(\sqrt{2}\).

Step 1: Graph the Function f(x)=2^x

Start by graphing the function \(f(x)=2^{x}\). You can use a graphing tool or calculator for this. The graph should show that the function is increasing and that it crosses the y-axis at 1.

Step 2: Use the Trace Feature for x=1/2

Next, use the Trace feature to find values of \(f\) when \(x=\frac{1}{2}\). The Trace feature allows you to move along the curve of the graph and obtain the y-coordinate corresponding to the chosen x-coordinate. When you trace to \(x=\frac{1}{2}\), note the corresponding value of \(f(x)\).

Step 3: Compare the Value with √2

Finally, compare the obtained value from Step 2 with \(\sqrt{2}\). The square root of 2 is approximately 1.414, and you'll observe that \(f(x)\) at \(x=\frac{1}{2}\) also has the same numerical value.

Key Concepts

Graphing Functions

Graphing functions is an essential skill in mathematics that helps visualize the behavior of different types of functions. When graphing an exponential function, such as \( f(x) = 2^x \), understanding its characteristics becomes easier. This function is an example of an exponential curve, meaning it exhibits a rapid rate of increase as \( x \) becomes larger.
The graph of \( f(x) = 2^x \) has some distinct properties:

• It passes through the point (0, 1) because any number to the power of zero is 1.
• The graph is always above the x-axis and approaches zero but never actually touches the x-axis.
• The rate of increase becomes steeper as \( x \) increases.

To graph it, you can use a graphing calculator. A clear visualization helps in understanding how the output changes concerning different input values (x-values) and can provide insight into practical scenarios where exponential growth is observed.

Radical Expressions

Radical expressions involve roots, such as square roots or cube roots. They typically appear in the form \( \sqrt{a} \). In the exercise, the expression \( \sqrt{2} \) is key to understanding the relationship with the exponential function \( 2^{1/2} \).
Radical expressions are often equated with fractional exponents. Specifically, \( 2^{1/2} \) is mathematically equivalent to \( \sqrt{2} \). This equivalence allows us to switch between notation whenever it suits the problem or context best. Some important points to remember are:

• Both \( \sqrt{a} \) and \( a^{1/2} \) denote the same number.
• Radicals and exponents follow common algebraic laws, making them useful for solving equations.
• Understanding these expressions is crucial for solving more complex mathematical problems.

In the exercise, realizing that \( 2^{1/2} = \sqrt{2} \) helps confirm that the value found using graphing is indeed the same as the well-known square root of 2.

Trace Feature

The trace feature is a useful tool found in many graphing calculators and software. It allows you to explore a graph by moving along its curve and viewing the corresponding coordinate values for any chosen x-value.
Using the trace feature can help:

• Find specific function values without calculating them by hand.
• Observe how the function behaves at various points along the graph.
• Verify numerical values for better accuracy by direct observation.

In the context of the problem, the trace feature was used to determine the value of \( f(x) \) for \( x=\frac{1}{2} \). This provided a direct way to see that \( f(x) \) yields approximately 1.414 at this point, which matches \( \sqrt{2} \). Using this feature is particularly helpful when analyzing graphs of complex functions where precise calculations are needed.