To find the equation of the quadratic function in the form \( y = ax^2 + bx + c \) that passes through the points \( (0, 1) \), \( (4, 17) \), and \( (-1, 7) \), we can set up a system of equations using these points.
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From the point \( (0, 1) \): \[ 1 = a(0)^2 + b(0) + c \implies c = 1 \]
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From the point \( (4, 17) \): \[ 17 = a(4)^2 + b(4) + c \implies 17 = 16a + 4b + 1 \implies 16a + 4b = 16 \implies 4a + b = 4 \quad (1) \]
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From the point \( (-1, 7) \): \[ 7 = a(-1)^2 + b(-1) + c \implies 7 = a - b + 1 \implies a - b = 6 \quad (2) \]
Now we have a system of two equations:
- \( 4a + b = 4 \) (Equation 1)
- \( a - b = 6 \) (Equation 2)
Next, we can solve this system of equations. From Equation (2), we can express \( b \) in terms of \( a \):
\[ b = a - 6 \]
Now substitute \( b \) into Equation (1):
\[ 4a + (a - 6) = 4 \] \[ 4a + a - 6 = 4 \] \[ 5a - 6 = 4 \] \[ 5a = 10 \implies a = 2 \]
Now, substitute \( a = 2 \) back into Equation (2) to find \( b \):
\[ 2 - b = 6 \implies -b = 4 \implies b = -4 \]
Now we have \( a = 2 \), \( b = -4 \), and \( c = 1 \). Thus, the equation of the quadratic function is:
\[ y = 2x^2 - 4x + 1 \]
Final answer: \[ y = 2x^2 - 4x + 1 \]