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This task involves finding the expression of a quadratic function \( f(x) = ax^2 + bx + c \) that satisfies certain conditions given in the problem. Let’s analyze the provided details:

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### **Problem Setup**
1. The function \( f \) is a quadratic polynomial defined by:
   \[ f(x) = ax^2 + bx + c \]
   where \( a, b, c \) are real coefficients.

2. Two specific points are given:
   - \( A(0, 1) \): The function passes through this point.
     \[ f(0) = 1 \]
   - \( B(2, 5) \): The function also passes through this point.
     \[ f(2) = 5 \]

3. The line \( d \) is tangent to the curve \( \mathcal{C} \) at the point \( A(0, 1) \).

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### **Strategy for Solution**

#### Step 1: Use the given points to generate two equations.
- Since \( f(0) = 1 \), substituting into the general form:
  \[ f(0) = a(0)^2 + b(0) + c = c = 1 \]
  Thus, \( c = 1 \).

- Now, use the second point \( B(2, 5) \):
  \[ f(2) = a(2)^2 + b(2) + 1 = 5 \]
  Simplifying:
  \[ 4a + 2b + 1 = 5 \quad \Rightarrow \quad 4a + 2b = 4 \quad (1) \]

#### Step 2: Find the derivative of \( f(x) \) and apply the tangent condition.
The derivative of the function \( f(x) = ax^2 + bx + c \) is:
\[ f'(x) = 2ax + b \]

- At point \( A(0, 1) \), the slope of the tangent line is given by:
  \[ f'(0) = 2a(0) + b = b \]
  This slope \( b \) is the same as the slope of the line \( d \), which passes through points \( A(0, 1) \) and \( B(2, 5) \).

- The slope between \( A(0, 1) \) and \( B(2, 5) \) is:
  \[ \text{slope} = \frac{5 - 1}{2 - 0} = \frac{4}{2} = 2 \]
  Thus, \( b = 2 \).

#### Step 3: Solve for \( a \) using the equations.
Now that we know \( b = 2 \), substitute it into equation (1):
\[ 4a + 2(2) = 4 \]
\[ 4a + 4 = 4 \quad \Rightarrow \quad 4a = 0 \quad \Rightarrow \quad a = 0 \]

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### **Final Expression of the Function**
Thus, the quadratic function is:
\[
f(x) = 0 \cdot x^2 + 2x + 1 = 2x + 1
\]

This means that the quadratic function simplifies to a linear function \( f(x) = 2x + 1 \), as the coefficient \( a \) is zero.

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Do you have any further questions about this solution, or would you like additional details?

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### **Related Questions**
1. What is the general equation of the tangent line to a quadratic function at a given point?
2. How does the slope of a line relate to the derivative of a function?
3. Can a quadratic function ever reduce to a linear function? If so, under what conditions?
4. What methods can be used to solve systems of equations involving multiple variables?
5. What are some applications of tangents in optimization problems?

**Tip:** When working with quadratic functions, always check if the function reduces to a simpler form, like a linear function, under specific conditions.

Find the expression of a quadratic function f(x) = ax^2 + bx + c, where the function passes through points A(0, 1) and B(2, 5), and the line is tangent to the curve at A(0, 1).

Solve for the quadratic function f(x) = ax^2 + bx + c that passes through points A(0, 1) and B(2, 5), with a tangent line at A(0, 1). This problem covers key algebraic concepts, including quadratic functions, derivatives, and the slope formula.

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