Graphing Quadratic Functions: Understanding $f(x) = x^2 - 8x + 15$

Key Concepts

• Quadratic Function: In the form $ax^2 + bx + c$.
• Axis of Symmetry: A vertical line that divides the parabola into two mirror images.
  • Formula: $x = -\frac{b}{2a}$
• Vertex: The point where the parabola changes direction; either a maximum or minimum.
  • Lies on the axis of symmetry.
• Opens Up or Down: Determined by the sign of $a$. If $a > 0$, the parabola opens up.

Steps to Graph $f(x) = x^2 - 8x + 15$

1. Identify $a$, $b$, and $c$:

   • $a = 1$, $b = -8$, $c = 15$.

2. Find the Axis of Symmetry:

   • Formula: $x = -\frac{b}{2a}$
   • Calculation: $x = -\frac{-8}{2 \times 1} = 4$
   • Graph this as a vertical dotted line at $x = 4$.

3. Determine if Parabola Opens Up or Down:

   • $a = 1$ (positive), so the parabola opens up.
   • The vertex is a minimum point.

4. Find the Vertex:

   • The x-value is on the line of symmetry: $x = 4$
   • Calculate $f(4)$ to find the y-value:
     • $f(4) = 4^2 - 8(4) + 15$
     • $f(4) = 16 - 32 + 15 = -1$
   • Vertex at $(4, -1)$.

5. Create a Table of Values:

   • Choose points to the left and right of the vertex to plot.
   • Calculate values for $f(x)$:
     • For $x = 3$: $f(3) = 3^2 - 8(3) + 15 = 9 - 24 + 15 = 0$
     • For $x = 2$: $f(2) = 2^2 - 8(2) + 15 = 4 - 16 + 15 = 3$
   • Use symmetry to find points:
     • $x = 5$ mirrors $x = 3$: $f(5) = 0$
     • $x = 6$ mirrors $x = 2$: $f(6) = 3$

6. Graph the Points and Parabola:

   • Plot points: $(2, 3)$, $(3, 0)$, $(4, -1)$, $(5, 0)$, $(6, 3)$.
   • Draw the parabola through these points.

Tips

• Symmetry: Use symmetry to minimize calculation by reflecting points across the axis of symmetry.
• Vertex: Always check the point on the axis of symmetry to determine the vertex first.
• Range Selection: Choose x-values close to the vertex to simplify calculations, especially for large or small values of $a$.