Question

The graphical representation of a quadratic polynomial $ax^2 + bx + c$ is a _____.

• A. Hyperbola
• B. Circle
• C. Parabola
• D. Straight line

Hint:

Always remember: \[ \text{Degree 1} \rightarrow \text{Straight Line} \] \[ \text{Degree 2} \rightarrow \text{Parabola} \] \[ \text{Degree 3} \rightarrow \text{Cubic Curve} \] The graph of every quadratic equation is always a parabola.

Answer 1

The Correct Option is C

Concept: A quadratic polynomial is a polynomial of degree 2. The general form of a quadratic polynomial is: \[ y=ax^2+bx+c \] where:
• \(a,b,c\) are constants
• \(a \neq 0\) The graph of every quadratic polynomial is called a parabola. The shape of the parabola depends on the value of \(a\):
• If \(a>0\), the parabola opens upward.
• If \(a<0\), the parabola opens downward.

Step 1: Understand the degree of the polynomial.
The highest power of \(x\) in the polynomial \[ ax^2+bx+c \] is \(2\). Therefore, it is a quadratic polynomial.

Step 2: Recall the graph associated with degree 2 equations.
We know:
• Linear polynomial \(\rightarrow\) Straight line
• Quadratic polynomial \(\rightarrow\) Parabola
• Circle has equation involving both \(x^2\) and \(y^2\)
• Hyperbola has a completely different standard equation Hence, the graph of \[ y=ax^2+bx+c \] is always a parabola.

Step 3: Verify with examples.
Example 1: \[ y=x^2 \] Its graph is a U-shaped parabola. Example 2: \[ y=-x^2 \] Its graph is an inverted parabola. Thus, every quadratic polynomial represents a parabola. Therefore, the correct answer is: \[ \boxed{\text{Parabola}} \]