Introduction to Quadratic Functions
Parabolas explained with simple formulas and examples
Quadratic functions are one of the most important ideas in algebra. Their graphs form a smooth U-shaped or upside-down U-shaped curve called a parabola. Once you understand the parts of a quadratic equation, you can quickly identify the direction, vertex, axis of symmetry, and intercepts of its graph.
What Is a Quadratic Function?
A quadratic function is a function whose highest exponent on the variable is 2. The standard form is:
Here, a, b, and c are constants, and a ≠ 0.
Understanding a, b, and c
Each coefficient has a role in the equation. The value of a controls the direction and width of the parabola. The values of b and c help determine the location of the graph.
| Part | Role | Example |
|---|---|---|
| a | Direction and width | In 2x2, a = 2 |
| b | Helps locate the axis of symmetry | In 2x2 + 4x, b = 4 |
| c | Y-intercept | In x2 − 3, c = −3 |
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How the Value of a Changes the Parabola
If a Is Positive
When a > 0, the parabola opens upward. For example:
f(x) = x2
If a Is Negative
When a < 0, the parabola opens downward. For example:
f(x) = −x2
Width and Narrowness
If the absolute value of a is greater than 1, the parabola is narrower. If the absolute value of a is between 0 and 1, the parabola is wider.
The Vertex and Axis of Symmetry
The vertex is the highest or lowest point of a parabola. The graph is symmetric around a vertical line called the axis of symmetry.
Axis of Symmetry Formula
Worked Example: Find the Vertex
Find the vertex of f(x) = x2 − 6x + 5.
a = 1, b = −6 → x = −(−6) ÷ 2(1) = 3
Now substitute x = 3:
f(3) = 32 − 6(3) + 5 = 9 − 18 + 5 = −4
The vertex is (3, −4).
Vertex Form and Graph Shifts
Vertex form makes the location of the vertex easy to read:
The vertex is (h, k).
Worked Example: Read Vertex Form
For f(x) = 2(x − 4)2 + 1, the vertex is (4, 1). The graph opens upward because a = 2, and it is narrower than the parent graph because |2| > 1.
Intercepts of a Quadratic Function
The y-intercept is found by setting x = 0. In standard form, it is the point (0, c). The x-intercepts are the points where the parabola crosses the x-axis, so f(x) = 0.
Example: Find the Y-Intercept
For f(x) = 2x2 + 3x − 5, set x = 0:
f(0) = −5 → y-intercept = (0, −5)
The Discriminant: Predicting X-Intercepts
For the quadratic equation ax2 + bx + c = 0, the discriminant is:
| Discriminant | X-intercepts | Graph meaning |
|---|---|---|
| D > 0 | Two real solutions | Crosses the x-axis twice. |
| D = 0 | One real solution | Touches the x-axis once. |
| D < 0 | No real solutions | Does not cross the x-axis. |
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Common Mistakes to Avoid
Forgetting the sign of a: A negative a means the parabola opens downward.
Misreading x − h: In vertex form, x − h gives a vertex x-coordinate of h.
Using c as the vertex: c gives the y-intercept, not usually the vertex.
Skipping parentheses: Substitute carefully when evaluating f(x).
Practice Problems
Try these questions before opening the answers:
1. Does f(x) = −3x2 + 2 open up or down?
2. What is the axis of symmetry of f(x) = x2 − 8x + 3?
3. What is the vertex of f(x) = (x + 2)2 − 5?
4. How many real x-intercepts can a quadratic have when D = 0?
Tap to reveal the answers
1. Down, because a = −3. 2. x = 4. 3. (−2, −5). 4. One real x-intercept.
Final Takeaway
A quadratic function has the form f(x) = ax2 + bx + c and its graph is a parabola. The sign of a tells you whether it opens up or down, the axis of symmetry is x = −b ÷ 2a, and vertex form a(x − h)2 + k reveals the vertex (h, k)
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