Problem 64
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 refl cted over the \(x\) -axis and horizontally stretched by a factor of 2 .
Step-by-Step Solution
Verified Answer
The function is \( g(x) = -\sqrt{\frac{x}{2}} \).
1Step 1: Identify the Base Function
The original function is given as \( f(x) = \sqrt{x} \), which represents the square root function.
2Step 2: Apply the Reflection Transformation
Reflecting the graph over the \( x \)-axis means multiplying the function by \(-1\). Therefore, the function becomes \(-\sqrt{x}\).
3Step 3: Apply the Horizontal Stretch Transformation
A horizontal stretch by a factor of 2 means replacing \( x \) with \( \frac{x}{2} \) in the function. Thus, updating the equation from Step 2, \(-\sqrt{x}\) becomes \(-\sqrt{\frac{x}{2}}\).
4Step 4: Write the Transformed Function Formula
The transformed function \( g(x) \) is \( g(x) = -\sqrt{\frac{x}{2}} \).
Key Concepts
Reflection TransformationHorizontal StretchSquare Root Function
Reflection Transformation
In the world of mathematics, particularly in function transformations, the concept of reflection plays a vital role. When we say a function is reflected over the \(x\)-axis, it means we invert the output values. This is equivalent to taking the negative of the function itself. For example, if you have the square root function \(f(x) = \sqrt{x}\), reflecting it over the \(x\)-axis will change the function to \(-\sqrt{x}\).
Why does this happen? Imagine the graph of \(\sqrt{x}\), which starts at the origin and curves upwards. By multiplying the entire function by \(-1\), each of those points flips to the opposite side of the \(x\)-axis, creating a mirror image below the axis. This transformation is crucial to visualize as it allows you to understand how changes affect graphs. Remember that reflecting over the \(x\)-axis changes all the positive \(y\)-values into negative ones, effectively flipping the graph upside down.
Why does this happen? Imagine the graph of \(\sqrt{x}\), which starts at the origin and curves upwards. By multiplying the entire function by \(-1\), each of those points flips to the opposite side of the \(x\)-axis, creating a mirror image below the axis. This transformation is crucial to visualize as it allows you to understand how changes affect graphs. Remember that reflecting over the \(x\)-axis changes all the positive \(y\)-values into negative ones, effectively flipping the graph upside down.
Horizontal Stretch
Understanding horizontal stretches can initially be tricky, but breaking it down makes it simpler. A horizontal stretch changes the 'width' of a function's graph along the \(x\)-axis. Mathematically, for a horizontal stretch by a factor of \(2\), you replace \(x\) in \(f(x)\) with \(\frac{x}{2}\).
Let's look at \(f(x) = \sqrt{x}\) for illustration. With a horizontal stretch factor of \(2\), this becomes \(\sqrt{\frac{x}{2}}\). This substitution acts like stretching the graph from left to right, effectively spreading it out. This transformation affects every point on the graph equally, making it appear as though it expands along the \(x\)-axis.
Remember, transformations like these are intuitive once you practice visualizing these changes. A helpful way to confirm your understanding is to think about key points. Notice how these points move farther from the \(y\)-axis, emphasizing the "stretch" in horizontal stretch.
Let's look at \(f(x) = \sqrt{x}\) for illustration. With a horizontal stretch factor of \(2\), this becomes \(\sqrt{\frac{x}{2}}\). This substitution acts like stretching the graph from left to right, effectively spreading it out. This transformation affects every point on the graph equally, making it appear as though it expands along the \(x\)-axis.
Remember, transformations like these are intuitive once you practice visualizing these changes. A helpful way to confirm your understanding is to think about key points. Notice how these points move farther from the \(y\)-axis, emphasizing the "stretch" in horizontal stretch.
Square Root Function
The square root function, \(f(x) = \sqrt{x}\), is one of the basic tools in your mathematical toolkit. This function has a unique shape, starting at the origin \((0,0)\) and curving to the right. Its domain is \(x \geq 0\) due to the square root's restriction to non-negative numbers.
Visualizing this can help make sense of transformations applied to it. A key aspect of the square root function is its gentle slope, which allows it to gradually increase. When applied transformations like reflection and horizontal stretch, the original shape gets altered.
By reflecting it over the \(x\)-axis, \(\sqrt{x}\) becomes \(-\sqrt{x}\), flipping the curve downward. With a horizontal stretch, \(\sqrt{x}\) turns into \(\sqrt{\frac{x}{2}}\), widening its reach along the \(x\)-axis. By mastering the square root function's fundamental properties, you're better equipped to tackle more complex transformations.
Visualizing this can help make sense of transformations applied to it. A key aspect of the square root function is its gentle slope, which allows it to gradually increase. When applied transformations like reflection and horizontal stretch, the original shape gets altered.
By reflecting it over the \(x\)-axis, \(\sqrt{x}\) becomes \(-\sqrt{x}\), flipping the curve downward. With a horizontal stretch, \(\sqrt{x}\) turns into \(\sqrt{\frac{x}{2}}\), widening its reach along the \(x\)-axis. By mastering the square root function's fundamental properties, you're better equipped to tackle more complex transformations.
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