using the y=ax^2 + bx+ c solve it using the points (400,150)(440,175) and (450, 150)

To solve for the equation of a quadratic function in the form \( y = ax^2 + bx + c \) using the points \( (400, 150) \), \( (440, 175) \), and \( (450, 150) \), we will follow these steps:

1. **Substitute the given points into the equation**:
   Each point will give us an equation by substituting the values of \(x\) and \(y\) into the general form.

   For the point \( (400, 150) \):
   \[
   150 = a(400)^2 + b(400) + c
   \]

   For the point \( (440, 175) \):
   \[
   175 = a(440)^2 + b(440) + c
   \]

   For the point \( (450, 150) \):
   \[
   150 = a(450)^2 + b(450) + c
   \]

2. **Solve the system of equations**:
   These three equations will allow us to solve for the unknowns \( a \), \( b \), and \( c \).

Let's calculate the exact values and solve for \(a\), \(b\), and \(c\).The solution to the quadratic equation \( y = ax^2 + bx + c \) using the points \( (400, 150) \), \( (440, 175) \), and \( (450, 150) \) is:

\[
a = -\frac{1}{16}, \quad b = \frac{425}{8}, \quad c = -11100
\]

Thus, the quadratic equation is:

\[
y = -\frac{1}{16}x^2 + \frac{425}{8}x - 11100
\]

Would you like further details, or do you have any questions?

Here are some related questions:
1. How can you determine the vertex of the parabola from the equation?
2. What is the significance of the coefficient \( a \) in the quadratic equation?
3. How do you find the x-intercepts of the parabola?
4. What role do the points play in determining the shape of the parabola?
5. How does the quadratic formula relate to this equation?

**Tip**: The sign of \( a \) determines whether the parabola opens upwards (positive \( a \)) or downwards (negative \( a \)).

Learn how to solve the quadratic equation y = ax^2 + bx + c using points (400, 150), (440, 175), and (450, 150) with a step-by-step solution. This problem involves applying the quadratic equation formula and solving a system of equations. Suitable for high school students.

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