To determine which graph represents \( g(x) = (x - 2)^2 - 3 \), let's analyze the transformation applied to \( f(x) = x^2 \).
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Find the vertex: The given function \( g(x) = (x - 2)^2 - 3 \) is in vertex form, \( g(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. For \( g(x) \):
- \( h = 2 \)
- \( k = -3 \)
Therefore, the vertex of \( g(x) \) is at the point \((2, -3)\).
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Find the direction: The coefficient in front of the squared term is positive (1), meaning that the parabola opens upwards.
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Identify points: To find other points on the graph, we can substitute some values into \( g(x) \):
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When \( x = 2 \):
\[ g(2) = (2 - 2)^2 - 3 = 0 - 3 = -3 \quad (\text{Vertex at } (2, -3)) \] -
When \( x = 1 \):
\[ g(1) = (1 - 2)^2 - 3 = 1 - 3 = -2 \] -
When \( x = 3 \):
\[ g(3) = (3 - 2)^2 - 3 = 1 - 3 = -2 \] -
When \( x = 0 \):
\[ g(0) = (0 - 2)^2 - 3 = 4 - 3 = 1 \] -
When \( x = 4 \):
\[ g(4) = (4 - 2)^2 - 3 = 4 - 3 = 1 \]
This tells us that the points \((0, 1)\) and \((4, 1)\) both lie on the graph of \( g(x) \).
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Match with given options: Now let's match \( g(x) \) with the provided graph options:
- The first option has a vertex at \((2, -3)\) and points \((0, 1)\) and \((4, 1)\).
Therefore, the correct graph representing \( g(x) = (x - 2)^2 - 3 \) is the first option titled "On a coordinate plane, a parabola opens up. It goes through (0, 1), has a vertex at (2, negative 3), and goes through (4, 1)."
2 answers