Problem 64 - College Algebra with Corequisite Support

Question

For the following exercises, write a formula for the function g that results when the graph of a given toolkit function is transformed as described. The graph of \(f(x)=\sqrt{x}\) is reflected over the \(x\) -axis and horizontally stretched by a factor of 2 .

Step-by-Step Solution

Answer

The function is \(g(x) = -\sqrt{\frac{x}{2}}\).

Step 1: Understand the Transformation

Identify the given toolkit function, which is the square root function, and recognize the two transformations: 1) reflection over the \(x\)-axis, and 2) horizontal stretch by a factor of 2.

Step 2: Reflection over the x-axis

Reflecting a function over the \(x\)-axis changes the sign of the function. For the given function \(f(x) = \sqrt{x}\), the reflection would be \(g(x) = -\sqrt{x}\).

Step 3: Horizontal Stretch by a Factor of 2

A horizontal stretch by a factor of 2 involves substituting \(\frac{x}{2}\) for \(x\) in the function. Thus, applying the horizontal stretch to \(-\sqrt{x}\), we get \(g(x) = -\sqrt{\frac{x}{2}}\).

Step 4: Write the Final Function

Combine the effects of both transformations to determine the final function. The function \(g(x) = -\sqrt{\frac{x}{2}}\) is the result of reflecting \(\sqrt{x}\) over the \(x\)-axis and horizontally stretching it by a factor of 2.

Key Concepts

Square Root Function

A square root function is a type of mathematical function written in the form \( f(x) = \sqrt{x} \). This function generates a curve when graphed, forming the upper half of a sideways parabola. The square root function only accepts non-negative values for \( x \), since you cannot obtain a real-numbered result from a negative square root under regular circumstances. It starts at the origin (0, 0) and moves upwards as \( x \) increases.

• Key starting point: \( (0,0) \)
• Zero slope at the start, increasing slowly
• Graph shape: Half side of parabola

When adding modifications such as reflections or stretches, it's essential to remember the original graph's characteristics. Knowing where the square root function starts and how it rises can help in understanding more complex transformations. These transformations alter its standard look, offering new insights into how functions can change dramatically with simple adjustments.

Reflection

Reflection in mathematics refers to flipping a function over an axis. For example, reflecting over the \( x \)-axis changes a function by flipping it upside down. This means that for any function \( f(x) \), the reflected function becomes \( g(x) = -f(x) \). Here, the reflection adjusts the vertical position of the graph, effectively changing the sign of all the function's output values.
Applying this to our square root function, \( \sqrt{x} \), reflection over the \( x \)-axis turns it into \( -\sqrt{x} \). Notice:

• Original value becomes negative: Points above the \( x \)-axis shift below.
• The general shape of the graph remains the same but is inverted.

Reflections can be useful for understanding the behavior of inverse functions or analyzing the symmetry within a graph. They are significant in geometrical transformations and have a critical role in various applied aspects across mathematical modeling and physics.

Horizontal Stretch

A horizontal stretch changes the width of a function's graph. To achieve a stretch by a specific factor, you replace \( x \) with \( \frac{x}{k} \), where \( k \) is your stretching factor. If \( k \) is greater than 1, as in our case with a factor of 2, it elongates the graph horizontally. This means it takes the function longer to reach the same height as before. Given \( f(x) = \sqrt{x} \), altering it to \( f(x) = \sqrt{\frac{x}{2}} \) will make the graph stretch wider.

• The graph covering more horizontal distance for the same values of \( y \).
• Each point on the graph moves farther from the y-axis.

Understanding the impact of horizontal stretches helps in visually interpreting how functions can transform across a graph. Such scaling transformations are pivotal when fitting functions to match real-world data or in graphical illustrations to find how variables dynamically change.