Example Question - vertical stretch
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Transformation Matrix Involving Vertical Stretch and Clockwise Rotation
<p>We first calculate the transformation matrix for a vertical stretch with scale factor \( k \):</p> <p>\[ A_{\text{stretch}} = \left( \begin{array}{cc} 1 & 0 \\ 0 & k \end{array} \right) \]</p> <p>Next, we calculate the transformation matrix for a clockwise rotation through \( \frac{\pi}{4} \) radians about the origin \( O \):</p> <p>\[ A_{\text{rotation}} = \left( \begin{array}{cc} \cos\left(\frac{\pi}{4}\right) & \sin\left(\frac{\pi}{4}\right) \\ -\sin\left(\frac{\pi}{4}\right) & \cos\left(\frac{\pi}{4}\right) \end{array} \right) = \left( \begin{array}{cc} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array} \right) \]</p> <p>The combined transformation matrix \( A \) for both transformations is given by the product of the individual matrices:</p> <p>\[ A = A_{\text{stretch}} \cdot A_{\text{rotation}} \]</p> <p>\[ A = \left( \begin{array}{cc} 1 & 0 \\ 0 & k \end{array} \right) \cdot \left( \begin{array}{cc} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \end{array} \right) \]</p> <p>\[ A = \left( \begin{array}{cc} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{k\sqrt{2}}{2} & \frac{k\sqrt{2}}{2} \end{array} \right) \]</p> <p>Using the given transformation, we can now map the point \( (1, k) \) to \( (2\sqrt{2}, \sqrt{2}) \) using matrix multiplication with \( A \):</p> <p>\[ \left( \begin{array}{cc} \frac{\sqrt{2}}{2} & \frac{\sqrt{2}}{2} \\ -\frac{k\sqrt{2}}{2} & \frac{k\sqrt{2}}{2} \end{array} \right) \left( \begin{array}{c} 1 \\ k \end{array} \right) = \left( \begin{array}{c} 2\sqrt{2} \\ \sqrt{2} \end{array} \right) \]</p> <p>\[ \left( \begin{array}{c} \frac{\sqrt{2}}{2} \cdot 1 + \frac{\sqrt{2}}{2} \cdot k \\ -\frac{k\sqrt{2}}{2} \cdot 1 + \frac{k\sqrt{2}}{2} \cdot k \end{array} \right) = \left( \begin{array}{c} 2\sqrt{2} \\ \sqrt{2} \end{array} \right) \]</p> <p>Solving the system of equations:</p> <p>\[ \frac{\sqrt{2}}{2} + \frac{k\sqrt{2}}{2} = 2\sqrt{2} \]</p> <p>\[ -\frac{k\sqrt{2}}{2} + \frac{k^2\sqrt{2}}{2} = \sqrt{2} \]</p> <p>From the first equation, we get:</p> <p>\[ 1 + k = 4 \]</p> <p>\[ k = 3 \]</p> <p>We only need the first equation to find the value of \( k \) because the problem gives information that the transformation is vertical, implying that \( k \) is a scalar affecting only the second coordinate's magnitude and not its sign. The second equation provided in the operation is redundant since we have the value for \( k \). Hence, \( k = 3 \).</p>
Identifying Transformed Exponential Function
The question asks which function g(x) represents the exponential function f(x) = 5^x after a vertical stretch by a factor of 2 and a reflection across the x-axis. Let's break down the transformations: 1. Vertical stretch by a factor of 2: This means we multiply the original function f(x) by 2. So the function becomes 2 * 5^x. 2. Reflection across the x-axis: To reflect a function across the x-axis, we take the negative of the function. So the function now becomes -2 * 5^x. Now, let's look at the answer choices and see which one matches our transformed function: A. g(x) = 2 * 5^(-x) - This represents a horizontal reflection across the y-axis instead of a vertical reflection and stretch. So, this is incorrect. B. g(x) = -2 * 5^(-x) - This represents a reflection across the x-axis and y-axis. So, this is incorrect. C. g(x) = 2 * 5^x - This represents a vertical stretch but no reflection across the x-axis. So, this is incorrect. D. g(x) = -2 * 5^x - This matches our description of a vertical stretch by a factor of 2 and a reflection across the x-axis. Therefore, the correct answer is D: g(x) = -2 * 5^x.
Transformation of Exponential Function
To solve this problem, we'll consider the transformations required step by step. The given function is f(x) = 5^x, and we are asked to apply the following transformations: 1. Vertical stretch by a factor of 2. 2. Reflection across the x-axis. 1. A vertical stretch by a factor of 2 will multiply the function by 2. This does not affect the exponent, so the new function after this transformation will be: h(x) = 2 * f(x) = 2 * 5^x 2. A reflection across the x-axis means we need to take the negative of the function. So the final function after this reflection will be: g(x) = -h(x) = -2 * 5^x This matches option B: g(x) = -2(5^x).
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