Quadratic Functions and Graphs

A quadratic function is a polynomial function of degree 2. When you graph a quadratic function, it creates a symmetric, U-shaped curve called a parabola.

Forms of Quadratic Functions

Quadratic functions can be written in three main forms, each revealing different key features of the parabola:

1. Standard Form: $y = ax^2 + bx + c$
   • The $y$-intercept is easily found at $(0, c)$.
2. Vertex Form: $y = a(x - h)^2 + k$
   • The vertex (the highest or lowest point) is explicitly given as $(h, k)$.
3. Factored Form: $y = a(x - r_1)(x - r_2)$
   • The $x$-intercepts (roots or zeros) are $r_1$ and $r_2$.

Key Features of a Parabola

To accurately sketch a parabola, you need to identify its core features:

• Direction of Opening: If $a > 0$, the parabola opens upwards (like a smile). If $a < 0$, it opens downwards (like a frown).
• Vertex: The turning point of the parabola.
  • In standard form, the $x$-coordinate of the vertex is found using $x = -\frac{b}{2a}$. Plug this $x$ value back into the equation to find the corresponding $y$-coordinate.
  • In vertex form, the vertex is simply $(h, k)$.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror images. Its equation is always $x = h$ (the $x$-coordinate of the vertex).
• Intercepts:
  • $y$-intercept: Set $x = 0$ and solve for $y$.
  • $x$-intercepts: Set $y = 0$ and solve for $x$ by factoring, using the quadratic formula, or completing the square.

Example: Converting Forms and Graphing

Problem: Convert $y = x^2 - 6x + 5$ to vertex form, then identify its vertex and axis of symmetry.

Step 1: Complete the square to find the vertex form. Group the $x$ terms together: $y = (x^2 - 6x) + 5$

Take half of the $x$ coefficient (which is $-6$), divide by $2$ to get $-3$, and square it to get $9$. Add and subtract this value inside the equation to keep it balanced: $y = (x^2 - 6x + 9) - 9 + 5$

Rewrite the perfect square trinomial and simplify the constants: $y = (x - 3)^2 - 4$

Step 2: Identify the features. Now that the equation is in the vertex form $y = a(x - h)^2 + k$, we can easily read the key features:

• Vertex: $(h, k) = (3, -4)$
• Axis of Symmetry: The vertical line $x = 3$
• Direction: Since $a = 1$ (which is positive), the parabola opens upwards.

To graph this function, plot the vertex at $(3, -4)$, draw a dashed vertical line for the axis of symmetry at $x = 3$, mark the $y$-intercept at $(0, 5)$ (found from the standard form), and sketch a smooth, symmetric U-shaped curve through these points.