Example Question - function transformations
Here are examples of questions we've helped users solve.
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.
Understanding Function Transformations through Graphs
The image presents a math problem with two graphs. The graph of a function f(x) is shown in gray, and the graph of another function g(x) is shown in pink. We are informed that g(x) has the same shape as f(x), which implies that g(x) is a transformation of f(x). The transformation appears to involve both a horizontal shift to the right and a vertical shift downward. Specifically, the point (0,1) on f(x) has been mapped to the point (3,0) on g(x). This indicates a horizontal shift of 3 units to the right and a vertical shift of 1 unit down. Now, let's examine the provided options to determine the correct equation for g(x): A. \( g(x) = f(x − 3)^2 − 1 \): Incorrect because this suggests f(x) is squared after a shift, which is inconsistent with uniform scaling. B. \( g(x) = f(x − 1)^2 − 3 \): Incorrect because the inner subtraction should reflect a horizontal shift of 3 units, not 1. C. \( g(x) = f(x − 3) − 3 \): Almost correct, but it shows an extra vertical shift down by 3 units instead of 1 unit. D. \( g(x) = f(x − 3) − 1 \): Correct because this represents a rightward shift of 3 units (x − 3) and a downward shift of 1 unit (− 1). Therefore, the correct answer is: D. \( g(x) = f(x − 3) − 1 \)
Solving Transformed Square Root Functions
The equation provided appears to be in the form of a transformed square root function: y = A√(B(x - C)) + D Where: - A represents the vertical stretch or compression and reflection over the x-axis. - B represents the horizontal stretch or compression and reflection over the y-axis. - C represents the horizontal shift (left or right). - D represents the vertical shift (up or down). For the given equation: y = 2√(x + 3) + 4 We need to find the values of A, B, C, and D. It's important to note that the instruction specifies to find the OPPOSITE of what you see in the equation for B and C. That means we take the negative or opposite value of what we see next to x for C (usually given as x - C, but here we have x + 3, so C will be the opposite of 3) and potentially the opposite of the B value, though there's no B visible, implying B is 1 (the default multiplicative identity). Hence, the values are: A = 2 (because of the vertical stretch by a factor of 2) B = 1 (the value in front of the "(x + 3)" as there's no visible coefficient, it's understood to be 1) C = -3 (the OPPOSITE of +3, indicating a horizontal shift 3 units to the LEFT) D = 4 (the vertical shift upwards by 4 units) So, we have: A = 2 B = 1 C = -3 D = 4
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