Questions: Quadratic Relationships: Mastery Test Type the correct answer in each box. Use numerals instead of words. What is the equation of the parabola passing through the points (0,6), (3,15.6), and (10,-4) ? f(x)= x^2+ x+

Solution Steps

We are given three points through which the parabola passes: \[ (0, 6), \quad (3, 15.6), \quad (10, -4) \]

The general form of a quadratic equation is: \[ f(x) = ax^2 + bx + c \] We need to determine the coefficients \(a\), \(b\), and \(c\) such that the equation passes through the given points.

Substituting the points into the equation gives us the following system of equations:

1. For the point \((0, 6)\): \[ 6 = a(0)^2 + b(0) + c \implies c = 6 \]

2. For the point \((3, 15.6)\): \[ 15.6 = a(3)^2 + b(3) + 6 \implies 15.6 = 9a + 3b + 6 \] Simplifying this, we get: \[ 9a + 3b = 9.6 \implies 3a + b = 3.2 \quad \text{(Equation 1)} \]

3. For the point \((10, -4)\): \[ -4 = a(10)^2 + b(10) + 6 \implies -4 = 100a + 10b + 6 \] Simplifying this, we get: \[ 100a + 10b = -10 \implies 10a + b = -1 \quad \text{(Equation 2)} \]

We now have a system of two equations:

1. \(3a + b = 3.2\) (Equation 1)
2. \(10a + b = -1\) (Equation 2)

To eliminate \(b\), we can subtract Equation 1 from Equation 2: \[ (10a + b) - (3a + b) = -1 - 3.2 \] This simplifies to: \[ 7a = -4.2 \implies a = -\frac{4.2}{7} = -0.60 \]

Now, substituting \(a\) back into Equation 1 to find \(b\): \[ 3(-0.60) + b = 3.2 \implies -1.8 + b = 3.2 \implies b = 5.0 \]

Now that we have \(a\), \(b\), and \(c\): \[ a = -0.60, \quad b = 5.00, \quad c = 6.00 \] The equation of the parabola is: \[ f(x) = -0.60x^2 + 5.00x + 6.00 \]

Final Answer