Quadratic Functions: General Form and Properties

Quadratic functions are polynomial functions of degree 2. This tutorial explores their graphs, properties including vertex and intercepts, and the relationship between a function's graph and solutions to its corresponding equation. Interactive exploration is available through coefficient adjustments.

This tutorial may be better understood with the help of an interactive applet on quadratic functions .

After completing this tutorial, you may proceed to additional quadratic function tutorials or graphing quadratic functions.

A - Definition of a Quadratic Function

A quadratic function \(f\) is defined as:

\[ f(x) = ax^{2} + bx + c \]

where \(a\), \(b\), and \(c\) are real numbers and \(a \neq 0\). Its graph is called a parabola—a U-shaped curve that opens upward if \(a > 0\) and downward if \(a < 0\).

Examples:

  1. \( f(x) = -2x^{2} + x - 1 \)
  2. \( f(x) = x^{2} + 3x + 2 \)

B - Standard Form and Vertex of a Quadratic Function

Any quadratic function can be rewritten in standard (vertex) form:

\[ f(x) = a(x - h)^{2} + k \]

where the point \((h, k)\) is the vertex of the parabola. The values of \(h\) and \(k\) are derived from coefficients \(a, b, c\) as follows:

Derivation of Vertex Coordinates

Starting from the general form and completing the square:

  1. Given: \( f(x) = ax^{2} + bx + c \)
  2. Factor \(a\) from the quadratic and linear terms:
    \( f(x) = a\left[x^{2} + \frac{b}{a}x\right] + c \)
  3. Add and subtract \(\left(\frac{b}{2a}\right)^{2}\) inside the parentheses:
    \( f(x) = a\left[x^{2} + \frac{b}{a}x + \left(\frac{b}{2a}\right)^{2} - \left(\frac{b}{2a}\right)^{2}\right] + c \)
  4. Note that \(x^{2} + \frac{b}{a}x + \left(\frac{b}{2a}\right)^{2} = \left[x + \frac{b}{2a}\right]^{2}\)
  5. Rewrite:
    \( f(x) = a\left[x + \frac{b}{2a}\right]^{2} - a\left(\frac{b}{2a}\right)^{2} + c \)
  6. Simplify:
    \( f(x) = a\left[x + \frac{b}{2a}\right]^{2} - \frac{b^{2}}{4a} + c \)

Thus, the standard form is achieved with:

\[ h = -\frac{b}{2a} \quad \text{and} \quad k = c - \frac{b^{2}}{4a} \]

The vertex \((h, k)\) represents the maximum point (if \(a < 0\)) or minimum point (if \(a > 0\)) of the parabola.

Example: Convert to Standard Form and Find Vertex

Rewrite \( f(x) = -2x^{2} + 4x + 1 \) in standard form and identify its vertex.

Solution:

  1. Given: \( f(x) = -2x^{2} + 4x + 1 \)
  2. Factor \(-2\) from the first two terms:
    \( f(x) = -2(x^{2} - 2x) + 1 \)
  3. Complete the square inside parentheses:
    Coefficient of \(x\) is \(-2\); half is \(-1\); square is \((-1)^2 = 1\).
    \( f(x) = -2\left(x^{2} - 2x + 1 - 1\right) + 1 \)
  4. Regroup:
    \( f(x) = -2\left[(x - 1)^{2} - 1\right] + 1 \)
    \( f(x) = -2(x - 1)^{2} + 2 + 1 \)
    \( f(x) = -2(x - 1)^{2} + 3 \)
  5. Standard form: \( a = -2, h = 1, k = 3 \).
  6. Vertex: \((1, 3)\).

Alternative using formulas:
\( h = -\frac{b}{2a} = -\frac{4}{2(-2)} = 1 \)
\( k = c - \frac{b^{2}}{4a} = 1 - \frac{4^{2}}{4(-2)} = 1 - \frac{16}{-8} = 1 + 2 = 3 \)

C - X-Intercepts of a Quadratic Function

The x-intercepts (roots/zeros) of \( f(x) = ax^{2} + bx + c \) are the real solutions to:

\[ ax^{2} + bx + c = 0 \]

They are found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

The discriminant \( D = b^{2} - 4ac \) determines the nature of the roots:

Examples: Finding X-Intercepts

  1. For \( f(x) = x^{2} + 2x - 3 \):
    Solve \( x^{2} + 2x - 3 = 0 \).
    \( D = 2^{2} - 4(1)(-3) = 16 \)
    \( x_{1} = \frac{-2 + \sqrt{16}}{2} = 1 \), \( x_{2} = \frac{-2 - \sqrt{16}}{2} = -3 \).
    X-intercepts: \((1, 0)\) and \((-3, 0)\).
  2. For \( g(x) = -x^{2} + 2x - 1 \):
    Solve \( -x^{2} + 2x - 1 = 0 \).
    \( D = 2^{2} - 4(-1)(-1) = 0 \)
    \( x = -\frac{b}{2a} = -\frac{2}{2(-1)} = 1 \).
    X-intercept: \((1, 0)\).
  3. For \( h(x) = -2x^{2} + 2x - 2 \):
    Solve \( -2x^{2} + 2x - 2 = 0 \).
    \( D = 2^{2} - 4(-2)(-2) = -12 \) (negative).
    No real x-intercepts.

Practice: Find the X-Intercepts

  1. \( f(x) = x^{2} + x - 2 \)
  2. \( g(x) = 4x^{2} + x + 1 \)
  3. \( h(x) = x^{2} - 4x + 4 \)

Check Answers

D - Y-Intercept of a Quadratic Function

The y-intercept is found by evaluating \( f(0) \):

\[ f(0) = a(0)^{2} + b(0) + c = c \]

Thus, the graph always crosses the y-axis at \((0, c)\).

Examples:

  1. \( f(x) = x^{2} + 2x - 3 \) → \( f(0) = -3 \) → Y-intercept: \((0, -3)\).
  2. \( g(x) = 4x^{2} - x + 1 \) → \( g(0) = 1 \) → Y-intercept: \((0, 1)\).
  3. \( h(x) = -x^{2} + 4x + 4 \) → \( h(0) = 4 \) → Y-intercept: \((0, 4)\).

References and Links