The parent function f(x) = x2 is translated such that the function g(x) = –x2 + 6x – 5 represents the new function. What is true
2 answers:
Answer:
1. Translation 3 units to the right;
2. Reflection across the x-axis;
3. Translation 4 units up.
Step-by-step explanation:
First, rewrite the function in following way:
Apply such transformations:
1. Translate the graph of the parent function 3 units to the right to get the graph of the function
2. Reflect the graph of the function across the x-axis to get the graph of the function
3. Translate the graph of the function 4 units up to get the graph of the function
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Now, bearing in mind that, for a graph, to be "differentiable", means a "smooth transition", not an abrupt edge or a "cusp", the first subfunction of the piece-wise is just a quadratic, the second subfunction is a linear, well, so the linear needs to hit the quadratic "smoothly", not make an abrupt edge, for that, the linear must continue the quadratic from 2 onwards, since 2 is the range edge.
now, what is the piece-wise when x = 2? well, f(2) = x² or (2)² or 4
that means, the linear needs to hit it at that point, to make it so, let's make it the slope of the linear, the same as the quadratic's.
what's the quadratic's slope? well, simple enough
so we have y = 4x + b
now, recall, for the quadratic, when x =2, y = 4, so for the linear, when x = 2, y = 4 as well, this means
check the picture below.
now, what is the piece-wise when x = 2? well, f(2) = x² or (2)² or 4
that means, the linear needs to hit it at that point, to make it so, let's make it the slope of the linear, the same as the quadratic's.
what's the quadratic's slope? well, simple enough
so we have y = 4x + b
now, recall, for the quadratic, when x =2, y = 4, so for the linear, when x = 2, y = 4 as well, this means
check the picture below.