54.	The figure shows the graph of the quadratic function y = x2 + bx + c,	y
where b and c are constants. (Given that have two pt (0,5), (6,5)), the graph opens upward)
(a)	Find the values of b and c.
(b)	Find the minimum value of y.


To solve the given problem, we need to find the constants \( b \) and \( c \) of the quadratic function \( y = x^2 + bx + c \), given the points \( (0,5) \) and \( (6,5) \).

### Part (a): Find the values of \( b \) and \( c \)

1. **Using the point \( (0,5) \):**

   Since \( (0,5) \) lies on the graph, we can substitute \( x = 0 \) and \( y = 5 \) into the equation:

   \[ 5 = 0^2 + b(0) + c \]
   \[ c = 5 \]

2. **Using the point \( (6,5) \):**

   Since \( (6,5) \) also lies on the graph, we substitute \( x = 6 \) and \( y = 5 \):

   \[ 5 = 6^2 + b(6) + c \]
   \[ 5 = 36 + 6b + 5 \]
   \[ 5 = 41 + 6b \]
   \[ 6b = 5 - 41 \]
   \[ 6b = -36 \]
   \[ b = -6 \]

Therefore, the values of the constants are:
\[ b = -6 \]
\[ c = 5 \]

### Part (b): Find the minimum value of \( y \)

The minimum value of \( y \) for a quadratic function \( y = ax^2 + bx + c \) (where \( a > 0 \)) occurs at the vertex of the parabola. The x-coordinate of the vertex can be found using the formula:

\[ x = -\frac{b}{2a} \]

For our quadratic function \( y = x^2 - 6x + 5 \), \( a = 1 \) and \( b = -6 \):

\[ x = -\frac{-6}{2(1)} = \frac{6}{2} = 3 \]

To find the minimum value of \( y \), we substitute \( x = 3 \) back into the quadratic equation:

\[ y = (3)^2 - 6(3) + 5 \]
\[ y = 9 - 18 + 5 \]
\[ y = -4 \]

Therefore, the minimum value of \( y \) is:
\[ y = -4 \]

Do you want any further details or have any questions?

Here are some questions you might want to ask next:
1. How do you find the vertex of a quadratic function?
2. What is the significance of the vertex in determining the minimum or maximum value of a quadratic function?
3. How can you determine if a quadratic function opens upwards or downwards?
4. What are some methods to solve quadratic equations?
5. How does the value of the discriminant affect the nature of the roots of a quadratic equation?

**Tip:** When working with quadratic functions, identifying key points such as the vertex and the intercepts can provide valuable insights into the graph and help solve related problems effectively.

Learn how to find the values of b and c in the quadratic function y = x^2 - 6x + 5, given specific points. Also, calculate the minimum value of y using vertex formula. Suitable for high school students studying quadratic functions.

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