Graphing Quadratic Functions

Graphing quadratic functions provides a visual representation of their behavior. Quadratic functions take the form:

$$ f(x) = ax^2 + bx + c $$

The graph of a quadratic function is a parabola, which can open upwards (if \( a > 0 \)) or downwards (if \( a < 0 \)). Key features when graphing include identifying the vertex, axis of symmetry, intercepts, and direction of opening.

Steps for Graphing a Quadratic Function

  1. Identify the Vertex: Use \( x = -\frac{b}{2a} \) to find the x-coordinate of the vertex. Substitute this value into \( f(x) \) to find the y-coordinate.
  2. Find the Axis of Symmetry: The axis of symmetry is the vertical line passing through the vertex, with equation \( x = -\frac{b}{2a} \).
  3. Determine the Y-intercept: The y-intercept occurs where \( x = 0 \), so \( f(0) = c \).
  4. Plot Additional Points: Choose x-values on either side of the vertex, substitute them into the function, and plot the corresponding points to shape the parabola.
  5. Draw the Parabola: Using the points, draw a smooth curve through the vertex and other points, extending in the direction the parabola opens (up or down).

Example

Graph the quadratic function:

$$ f(x) = x^2 - 4x + 3 $$

Solution:

  • Find the vertex: \( x = -\frac{-4}{2 \cdot 1} = 2 \)
  • Substitute \( x = 2 \) into \( f(x) \): \( f(2) = (2)^2 - 4(2) + 3 = -1 \)
  • Vertex: \( (2, -1) \)
  • Axis of symmetry: \( x = 2 \)
  • Y-intercept: \( f(0) = 3 \)
  • Additional points: For \( x = 1 \), \( f(1) = 0 \); for \( x = 3 \), \( f(3) = 0 \)
  • Draw the parabola with vertex at \( (2, -1) \), y-intercept at \( (0, 3) \), and symmetry about \( x = 2 \).

Practice Questions

  1. Graph the function \( f(x) = -x^2 + 2x + 3 \).
    Solution

    For \( f(x) = -x^2 + 2x + 3 \):

    • Vertex: \( x = -\frac{2}{2 \cdot (-1)} = 1 \)
    • Substitute \( x = 1 \) into \( f(x) \): \( f(1) = -(1)^2 + 2(1) + 3 = 4 \)
    • The vertex is \( (1, 4) \)
    • Axis of symmetry: \( x = 1 \)
    • Y-intercept: \( f(0) = 3 \)
    • Plot points and draw a downward-opening parabola.
  2. Graph the function \( f(x) = 2x^2 - 8x + 6 \).
    Solution

    For \( f(x) = 2x^2 - 8x + 6 \):

    • Vertex: \( x = -\frac{-8}{2 \cdot 2} = 2 \)
    • Substitute \( x = 2 \) into \( f(x) \): \( f(2) = 2(2)^2 - 8(2) + 6 = -2 \)
    • The vertex is \( (2, -2) \)
    • Axis of symmetry: \( x = 2 \)
    • Y-intercept: \( f(0) = 6 \)
    • Plot points and draw an upward-opening parabola.
  3. Graph the function \( f(x) = -3x^2 + 6x - 2 \).
    Solution

    For \( f(x) = -3x^2 + 6x - 2 \):

    • Vertex: \( x = -\frac{6}{2 \cdot (-3)} = 1 \)
    • Substitute \( x = 1 \) into \( f(x) \): \( f(1) = -3(1)^2 + 6(1) - 2 = 1 \)
    • The vertex is \( (1, 1) \)
    • Axis of symmetry: \( x = 1 \)
    • Y-intercept: \( f(0) = -2 \)
    • Plot points and draw a downward-opening parabola.
→Quadratic Inequalities