Answer:
Option D: g(x) = (x - 2)² + 1
Step-by-step explanation:
Definitions:
The equation that represents the parent function of a parabola can be represented by f(x) = x², which is a quadratic (or square) function. It is one of the basic elementary functions commonly used in the field of Mathematics. Performing mathematical operations on a parent function that results into a new function represents the transformation of the parent graph.
Vertical translation: y = x² + k
This represents the vertical shift of the parent graph of a quadratic function.
- k > 0 ⇒ The parent graph of a quadratic function shifts by k units upward.
- k < 0 ⇒ The parent graph of a quadratic function shifts by |k | units downward.
Horizontal translation: y = (x + h )²
This represents the horizontal shift of the parent graph of a quadratic function.
- h > 0 ⇒ The parent graph of a quadratic function shifts by h units to the right.
- h < 0 ⇒ The parent graph of a quadratic function shifts by |h | units to the left.
Solution:
The given graph shows the translation of the parent function, where the vertex occurs at point, (2, 1). Since the graph is upward-facing, then it means that the vertex is its minimum point on the graph.
If we substitute the values of the vertex, (h, k) into the vertex form of a quadratic function, g(x) = a(x - h)² + k:
Vertex = (2, 1)
g(x) = a(x - 2)² + 1
We can solve for the value of "a" by choosing another point from the graph, (1, 2). Substitute these coordinates into the vertex form:
g(x) = a(1 - 2)² + 1
2 = a(1 - 2)² + 1
2 = a(-1)² + 1
2 = a(1) + 1
Subtract 1 from both sides to isolate "a":
2 - 1 = a(1) + 1 - 1
1 = a
Therefore, the quadratic function that represents the given graph is:
g(x) = (x - 2)² + 1, which matches Option D.
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Additional Notes on the constant, "a":
The constant, "a" in the vertex form, g(x) = a(x - h)² + k determines the vertical stretch, compression, and reflection (across the x-axis) of the graph.
- | a | > 1 ⇒ The graph of a parabola opens upward. It also represents the vertical stretch of the graph (appears narrower than the parent graph).
- 0 < a < 1 ⇒ This represents the vertical compression of the graph (appears wider than the parent graph).
- f(x) = -ax² ⇒ This represents the reflection of the graph across the x-axis.
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Keywords:
Parabola
Transformations of Quadratic Functions
Quadratic equations
Parent Function
Vertex form
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Learn more about transformations of quadratic functions here:
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