To find the constant term \( c \) of the quadratic function in standard form \( y = ax^2 + bx + c \), we can use the information given about the vertex and the y-intercept.
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Identifying the vertex form: The vertex form of a quadratic function is given by: \[ y = a(x - h)^2 + k \] where \( (h, k) \) is the vertex. In this case, the vertex is \( (1, -3) \), so we can substitute \( h = 1 \) and \( k = -3 \) to get: \[ y = a(x - 1)^2 - 3 \]
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Finding the value of \( a \): We know that the function crosses the y-axis at \( (0, -1) \), which means when \( x = 0 \), \( y = -1 \). We can substitute these values into the equation: \[ -1 = a(0 - 1)^2 - 3 \] This simplifies to: \[ -1 = a(1) - 3 \] \[ -1 = a - 3 \] Adding 3 to both sides: \[ a = 2 \]
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Substituting \( a \) back into the vertex form: Now we can substitute \( a \) back into the quadratic function: \[ y = 2(x - 1)^2 - 3 \]
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Expanding to find standard form: Now we expand the equation to get it into standard form: \[ y = 2(x^2 - 2x + 1) - 3 \] \[ y = 2x^2 - 4x + 2 - 3 \] \[ y = 2x^2 - 4x - 1 \]
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Identifying the constant term \( c \): In the standard form \( y = ax^2 + bx + c \), we have: \[ c = -1 \]
Therefore, the value of the constant term \( c \) is \( \boxed{-1} \).
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