Show that `y=ax^(2)+bx+c` represents a parabola.
Also find equation its axis.

Answer

To show that the equation \( y = ax^2 + bx + c \) represents a parabola and to find the equation of its axis, we can follow these steps:

### Step 1: Write the given equation
We start with the given equation of the curve:
\[ y = ax^2 + bx + c \]

### Step 2: Rearrange the equation
To analyze the equation, we can rearrange the quadratic part:
\[ y = a(x^2 + \frac{b}{a}x) + c \]

### Step 3: Complete the square
Next, we will complete the square for the expression \( x^2 + \frac{b}{a}x \):
1. Take half of the coefficient of \( x \), which is \( \frac{b}{2a} \).
2. Square it to get \( \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} \).
3. Add and subtract this square inside the equation:
\[ y = a\left(x^2 + \frac{b}{a}x + \frac{b^2}{4a^2} - \frac{b^2}{4a^2}\right) + c \]
This simplifies to:
\[ y = a\left(\left(x + \frac{b}{2a}\right)^2 - \frac{b^2}{4a^2}\right) + c \]

### Step 4: Simplify the equation
Now, distribute \( a \) and simplify:
\[ y = a\left(x + \frac{b}{2a}\right)^2 - \frac{ab^2}{4a^2} + c \]
\[ y = a\left(x + \frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right) \]

### Step 5: Identify the standard form of a parabola
The equation now resembles the standard form of a parabola:
\[ y = a\left(x - h\right)^2 + k \]
where \( h = -\frac{b}{2a} \) and \( k = c - \frac{b^2}{4a} \). This confirms that the equation represents a parabola.

### Step 6: Find the equation of the axis
The axis of symmetry for the parabola is given by the vertical line that passes through the vertex, which is at \( x = h \):
\[ x = -\frac{b}{2a} \]

### Conclusion
Thus, we have shown that the equation \( y = ax^2 + bx + c \) represents a parabola, and the equation of its axis is:
\[ x = -\frac{b}{2a} \]