Question
Graph the function. How is the graph a translation of f(x) = x2?
y = (x –1)2 + 3
(1 point)
Responses
graphThe function has a minimum at the point left-parenthesis 1 comma negative 3 right-parenthesis and passes through the points left-parenthesis 0 comma negative 2 right-parenthesis and left-parenthesis 2 comma negative 2 right-parenthesis.
f(x)
translated down 3 unit(s) and translated to the right 1 unit(s)
Image with alt text: graph The function has a minimum at the point left-parenthesis 1 comma negative 3 right-parenthesis and passes through the points left-parenthesis 0 comma negative 2 right-parenthesis and left-parenthesis 2 comma negative 2 right-parenthesis. f(x) translated down 3 unit(s) and translated to the right 1 unit(s)
graphThe function has a minimum at the point left-parenthesis negative 1 comma 3 right-parenthesis and passes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma 4 right-parenthesis.
f(x)
translated down 3 unit(s) and translated to the left 1 unit(s)
Image with alt text: graph The function has a minimum at the point left-parenthesis negative 1 comma 3 right-parenthesis and passes through the points left-parenthesis negative 2 comma 4 right-parenthesis and left-parenthesis 0 comma 4 right-parenthesis. f(x) translated down 3 unit(s) and translated to the left 1 unit(s)
graphThe function has a minimum at the point left-parenthesis negative 1 comma negative 3 right-parenthesis and passes through the points left-parenthesis negative 3 comma 1 right-parenthesis and left-parenthesis 1 comma 1 right-parenthesis.
f(x)
translated up 3 unit(s) and translated to the left 1 unit(s)
Image with alt text: graph The function has a minimum at the point left-parenthesis negative 1 comma negative 3 right-parenthesis and passes through the points left-parenthesis negative 3 comma 1 right-parenthesis and left-parenthesis 1 comma 1 right-parenthesis. f(x) translated up 3 unit(s) and translated to the left 1 unit(s)
graphThe function has a minimum at the point left-parenthesis 1 comma 3 right-parenthesis and passes through the points left-parenthesis 0 comma 4 right-parenthesis and left-parenthesis 2 comma 4 right-parenthesis.
f(x)
translated up 3 unit(s) and translated to the right 1 unit(s)
Image with alt text: graph The function has a minimum at the point left-parenthesis 1 comma 3 right-parenthesis and passes through the points left-parenthesis 0 comma 4 right-parenthesis and left-parenthesis 2 comma 4 right-parenthesis. f(x) translated up 3 unit(s) and translated to the right 1 unit(s)
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The correct response is:
Image with alt text: graph The function has a minimum at the point left-parenthesis 1 comma negative 3 right-parenthesis and passes through the points left-parenthesis 0 comma negative 2 right-parenthesis and left-parenthesis 2 comma negative 2 right-parenthesis. f(x) translated down 3 unit(s) and translated to the right 1 unit(s)
The correct response is:
The graph of the function y = (x – 1)² + 3 is a translation of the function f(x) = x². It is translated 1 unit to the right and 3 units up. The graph has a minimum at the point (1, -3) and passes through the points (0, -2) and (2, -2).
Image:
[Graph of y = (x - 1)² + 3]
[Graph Image Description: The graph of the function y = (x - 1)² + 3 is shown. It has a minimum at the point (1, -3) and passes through the points (0, -2) and (2, -2). The graph is a translation of the function f(x) = x², translated 1 unit to the right and 3 units up.]
To graph the function y = (x - 1)^2 + 3, we can start by graphing the parent function f(x) = x^2. The parent function f(x) = x^2 is a simple parabola with its vertex at the origin (0, 0) and opening upwards.
To graph the function y = (x - 1)^2 + 3, we need to understand how it is a translation of the parent function f(x) = x^2. The terms (x - 1) and +3 in the equation indicate the translation.
In the equation y = (x - 1)^2 + 3, the term (x - 1) represents a horizontal translation. The value inside the parentheses, x - 1, has the effect of shifting the graph horizontally to the right by 1 unit. This means that every point on the graph will move 1 unit to the right compared to the parent function.
The term +3 represents a vertical translation. The constant term, +3, moves the entire graph vertically upwards by 3 units. This means that every point on the graph will move 3 units upwards compared to the parent function.
Combining these translations, we can determine that the graph of y = (x - 1)^2 + 3 will have its vertex shifted 1 unit to the right and 3 units upwards compared to the vertex of the parent function f(x) = x^2.
Therefore, the graph of y = (x - 1)^2 + 3 will have a minimum at the point (1, 3) and will pass through the points (0, 2) and (2, 2). It is translated down 3 units and translated to the right 1 unit compared to the parent function f(x) = x^2.
You can visualize this graph by plotting these points on a coordinate plane and connecting them with a smooth curve.