Quadratic Functions
Explore standard, vertex and factored forms of quadratics, find the axis of symmetry and vertex, and use the discriminant.
Three Forms of Quadratic Functions
A quadratic function is a polynomial of degree 2. Its graph is a parabola. There are three standard ways to express a quadratic:
Standard Form
y = ax² + bx + c
Reveals the y-intercept (c) and allows use of the quadratic formula.
Vertex Form
y = a(x - h)² + k
Reveals the vertex (h, k) and makes transformations clear.
Factored Form
y = a(x - p)(x - q)
Reveals the x-intercepts (roots) at x = p and x = q.
The value of a determines the shape: if a > 0, the parabola opens upward (concave up); if a < 0, it opens downward (concave down). The larger |a|, the narrower the parabola.
Key Features of a Parabola
Every parabola has an axis of symmetry and a vertex (the turning point).
Parabola Diagram
Axis of Symmetry
x = -b/2a
The vertical line through the vertex that divides the parabola into two mirror images.
Discriminant
Δ = b² - 4ac
Δ > 0: two real roots; Δ = 0: one repeated root; Δ < 0: no real roots.
The vertex can be found by substituting x = -b/(2a) back into the equation. If the quadratic is in vertex form y = a(x - h)² + k, the vertex is simply (h, k).
Sketching Quadratics
To sketch a parabola, identify these key features:
- Direction: Does it open up (a > 0) or down (a < 0)?
- Vertex: Find using x = -b/(2a) or by completing the square.
- Y-intercept: Set x = 0 to find the point (0, c).
- X-intercepts: Solve ax² + bx + c = 0 using factoring or the quadratic formula.
- Axis of symmetry: Draw the vertical line x = -b/(2a).
The Quadratic Formula
x = -b ± √(b² - 4ac)/2a
Use this formula to find the x-intercepts (roots) when factoring is difficult.
Key Vocabulary
Parabola
The U-shaped curve that is the graph of every quadratic function.
Vertex
The turning point of a parabola — a minimum when a > 0, a maximum when a < 0.
Discriminant
The expression Δ = b² - 4ac that determines the number and type of roots.
Axis of Symmetry
The vertical line x = -b/(2a) that divides the parabola into two symmetrical halves.
Worked Examples
Find the vertex and axis of symmetry of y = 2x² - 8x + 3.
Step 1: Identify a = 2, b = -8, c = 3.
Step 2: Axis of symmetry: x = -(-8)/(2×2) = 8/4 = 2.
Step 3: y-coordinate of vertex: y = 2(2)² - 8(2) + 3 = 8 - 16 + 3 = -5.
Answer: Vertex = (2, -5), axis of symmetry is x = 2.
Determine the nature of the roots of x² + 4x + 7 = 0.
Step 1: Identify a = 1, b = 4, c = 7.
Step 2: Calculate Δ = b² - 4ac = 16 - 28 = -12.
Step 3: Since Δ < 0, the equation has no real roots.
Answer: The parabola does not cross the x-axis — it lies entirely above it (since a > 0).
Convert y = x² - 6x + 5 to factored form.
Step 1: Find two numbers that multiply to 5 and add to -6: → -1 and -5.
Step 2: Factor: y = (x - 1)(x - 5).
Step 3: The x-intercepts are x = 1 and x = 5.
Answer: y = (x - 1)(x - 5), with roots at x = 1 and x = 5.
Knowledge Check
Select the correct answer for each question. Click "Check Answer" to see if you are right.
Question 1
What is the axis of symmetry of y = x² - 4x + 1?
Question 2
What is the discriminant of 3x² + 2x - 5 = 0?
Question 3
The quadratic y = -(x - 3)² + 4 has its vertex at:
Question 4
If the discriminant of a quadratic is zero, how many real roots does it have?
Question 5
What is the y-intercept of y = 2x² - 3x + 5?
Key Concepts Summary
- ●A quadratic function has three forms: standard (y = ax² + bx + c), vertex (y = a(x-h)² + k), and factored (y = a(x-p)(x-q)).
- ●The axis of symmetry is x = -b/(2a) and passes through the vertex.
- ●The discriminant Δ = b² - 4ac determines the number of real roots.
- ●If a > 0 the parabola opens upward (minimum); if a < 0 it opens downward (maximum).
- ●The quadratic formula x = (-b ± √Δ) / (2a) finds the roots of any quadratic.