A quadratic equation is an equation where the highest power of the variable is 2 (square).
Examples:
- x² + 3x + 2 = 0
- x² – 9 = 0
- 2x² + 5 = 0
Important point:
- A quadratic equation does NOT make a straight line.
- When we draw, it forms a curved shape.
What Does a Quadratic Equation Look Like on a Graph?
If you draw a quadratic equation on a graph, you get a U-shaped curve called a parabola.
The curve can be:
- U shape (opens up) – when x² is positive
- n shape (opens down) – when x² is negative
U-Shaped Parabola
- A U-shaped parabola appears when the coefficient of x² is positive.
- Example equation:
y = x²
- This equation causes the curve to open upwards.
Table of Values to Plot:
x | y = x² |
–3 | 9 |
–2 | 4 |
–1 | 1 |
0 | 0 |
1 | 1 |
2 | 4 |
3 | 9 |
Example graph:
Explanation for the above graph:
- The above graph forms a U-shaped curve called parabola.
Why is it U-Shaped?
1. The number in front of x² is positive
In y = x², the positive sign makes the parabola open upwards.
2. The lowest point is at x = 0
At x = 0 → y = 0.
This is the bottom of the curve.
Everything else is higher, so the curve rises on both sides.
3. As you move away from 0, y becomes bigger:
Examples:
x = 1 → y = 1
x = 2 → y = 4
x = 3 → y = 9
The further you go, the higher the curve goes forming the U shape.
4. The graph is the same on both sides:
x² is symmetrical, so the left side and right side match perfectly, making a smooth U.
Important Features of the U-Shape (Parabola):
✔ Lowest point (Vertex)
The very bottom of the parabola is called the vertex.
For y = x², the vertex is at (0, 0).
✔ Symmetry:
The parabola is symmetrical.
Whatever happens on the right side of the y-axis also happens on the left side.
✔ Opens Upwards:
Because the coefficient of x² is positive, the graph opens up (like a smile 😀).
If it were negative, it would open down (sad face ☹️).
U-shape real-life example:
- A valley between two hills forms a natural U shape — the lowest point is in the middle, and both sides go up.
- A satellite dish.
- A bowl shape.
What Is an N-Shaped Parabola?
- An n-shaped parabola is the graph of a quadratic equation where the x² term is negative.
- Example equation:
y = –x²
This negative sign in front of x² causes the curve to open downwards.
Table of Values to Plot:
x | y = -x² |
–3 | -9 |
–2 | -4 |
–1 | -1 |
0 | 0 |
1 | -1 |
2 | -4 |
3 | -9 |
Plotting these points forms the N-shaped parabola.
Example graph:
Explanation for the above graph:
- The above graph forms an N-shaped curve..
Why is it N-Shaped?
1. The number in front of x² is negative
In y = –x², the minus sign makes the parabola open downwards.
2. Squaring always makes numbers positive, but the minus sign turns them negative.
Example:
x | x² | –x² |
–3 | 9 | –9 |
–2 | 4 | –4 |
–1 | 1 | –1 |
No matter what x you choose, –x² gives a negative y. This pulls the graph downward on both sides.
3. The highest point is at x = 0
- At x = 0 → y = 0
- This is the top of the graph. This is the top of the curve.
- Everything else is lower, so the curve goes down on both sides.
4. As you move away from 0, y becomes smaller (more negative):
Examples:
x = 1 → y = –1
x = 2 → y = –4
x = 3 → y = –9
The further you go, the lower the curve drops — forming the n shape.
5. The graph is the same on both sides:
–x² is symmetrical, so the left and right sides match perfectly, making a smooth n-shape.
N-shape real-life example:
- An upside-down arch
- The shape of a rainbow (approximate)
- The curve of a slide going down
How quadratic equations are different from linear equations:
Linear Equations | Quadratic Equations |
Highest power of x is 1 | Highest power of x is 2 |
Graph is a straight line | Graph is a curved parabola |
One solution | Two solutions (sometimes one or none) |
Easy to solve | Needs factoring, formula, or other methods |
Why Do We Learn Quadratic Equations?
Quadratic equations are used in many real-life situations:
- Physics → throwing objects, falling motion
- Area problems → finding side lengths
- Business → profit calculations
- Engineering → designing bridges and arches
- Any situation involving curves
They help us understand how things move, curve, and change.
Summary:
If the highest power of a variable is 2, the equation is quadratic.
- Graphs of quadratics are parabolas.
- Positive x² → U-shape
- Negative x² → n-shape
- Quadratics are used in real-life curves and motions
Practice question:
1. Which of the following are quadratic equations?
a) y = 2x + 3
b) y = x² – 4
c) y = 5x – 7
d) y = –3x² + 2x – 1
2. Identify the shape of the graph:
a) y = 4x² → ______
b) y = –2x² → ______
3. As x moves away from 0 in y = x², the y-values _______ .
a) Increase
b) Decrease
4. Which equation will create a U-shape?
a) y = –4x²
b) y = 3x²
c) y = –x²
5. If x = 3, what is the value of:
a) y = x²
b) y = –x²
FAQs on Quadratic Equations
Why is the graph of a quadratic a curve and not a straight line?
- Because the value of x² changes faster than x.
- This makes the graph bend into a U shape or n shape, not a straight line.
What is a parabola?
- A parabola is the curved shape made when you graph a quadratic equation.
- It can open upwards or downwards.
Why are parabolas symmetrical?
- Because x² gives the same value for both +x and –x.
- This creates a mirror image on each side of the graph.