How to Graph a Quadratic Function
What is a quadratic function?
A quadratic function is a mathematical function that can be written in the standard form f(x) = ax² + bx + c (where a is not zero). When you graph it, it creates a smooth, U-shaped curve called a parabola.
Key Features to Identify
- Vertex: The lowest or highest point on the parabola (the turning point).
- Axis of Symmetry: The invisible vertical line passing through the vertex that splits the parabola into two mirror images.
- x-intercepts (roots): The points where the parabola crosses the x-axis.
- y-intercept: The point where the parabola crosses the y-axis.
Step-by-Step Method
- Find the vertex: Use the formula x = -b / (2a) to find the x-coordinate. Then plug that x back into the equation to find the y-coordinate.
- Find the y-intercept: Substitute 0 for x. The y-intercept is simply the 'c' value in the standard form.
- Find the x-intercepts: Set f(x) = 0 and solve using the quadratic formula, factoring, or completing the square.
- Plot and draw: Plot these key points on a graph and draw a smooth U-shaped curve connecting them, making sure it is symmetric around the axis of symmetry.
Worked Example
Graph f(x) = x² - 4x + 3
- Vertex: Here, a = 1, b = -4, c = 3.
x = -(-4) / (2 * 1) = 4 / 2 = 2.
Plug in x = 2: y = (2)² - 4(2) + 3 = 4 - 8 + 3 = -1.
Vertex is at (2, -1). - y-intercept: Plug in x = 0.
y = 3.
y-intercept is at (0, 3). - x-intercepts: Set x² - 4x + 3 = 0.
This factors to (x - 3)(x - 1) = 0.
So x = 3 and x = 1.
x-intercepts are at (1, 0) and (3, 0).
Try it: Graph x² - 4x + 3 to see the parabola
Frequently Asked Questions
What shape does a quadratic function make?
A quadratic function forms a U-shaped curve called a parabola. If 'a' is positive, it opens upwards. If 'a' is negative, it opens downwards.
How do you find the vertex of a parabola?
The x-coordinate of the vertex is found using the formula x = -b / (2a). Then, plug this x-value back into the original function to find the y-coordinate.
What is the axis of symmetry?
The axis of symmetry is the vertical line that divides the parabola exactly in half. It passes through the vertex and its equation is x = -b / (2a).
How do you find x-intercepts of a quadratic?
You can find x-intercepts (also called roots or zeros) by setting y = 0 and solving for x. You can solve by factoring, completing the square, or using the quadratic formula.
Does a² + bx + c always have two roots?
No. Depending on the discriminant (b² - 4ac), a quadratic can have two real roots, one real root (if the vertex touches the x-axis), or no real roots (if it never crosses the x-axis).
Related Resources