4.3 Quadratic Function

Quadratic Functions Overview

Key Characteristics to Determine Graphing

1. Concavity

   • To determine concavity, check the coefficient $a$ in the vertex form of the function. If $a < 0$ (negative), the function is concave down (like a frown). If $a > 0$ (positive), the function is concave up (like a cup).

2. Vertex

   • The vertex of the function can be found as follows:
     • In vertex form: $f(x) = a(x - h)^2 + k$, the vertex is $(h, k)$.
     • In standard form: Calculate $h = -\frac{b}{2a}$ and then find $k$ by plugging $h$ back into the function.

3. Axis of Symmetry

   • The axis of symmetry is given by the line $x = h$, where $h$ is the x-coordinate of the vertex.

4. Intercepts

   • Y-Intercept: Set $x = 0$ and solve for $y$.
   • X-Intercepts: Set $y = 0$ and solve for $x$. A negative value under the square root indicates no real x-intercepts.

5. Domain

   • For all quadratic functions, the domain is all real numbers: $(-\infty, \infty)$.

6. Range

   • Concave up: Range starts from the vertex's y-value, $k$, to infinity (inclusive).
   • Concave down: Range goes from negative infinity to the vertex's y-value, $k$ (inclusive).

7. Increasing/Decreasing Intervals

   • In a concave up graph, the function is increasing from $h$ to $\infty$ and decreasing from $(-\infty, h)$.
   • In a concave down graph, it is the opposite: increasing from $(-\infty, h)$ and decreasing from $h$ to $\infty$.

8. Absolute Maximum/Minimum

   • A concave up function has an absolute minimum at the vertex, while a concave down function has an absolute maximum at the vertex.

Example Summary

• Graphing the function $f(x) = -2(x - 6)^2 - 16$:
  • Concavity: Down (since $a = -2$)
  • Vertex: $(6, -16)$
  • Axis of Symmetry: $x = 6$
  • No x-intercepts; y-intercept: $(0, -88)$
  • Domain: $(-\infty, \infty)$; Range: $(-\infty, -16]$
• For another function $f(x) = 2x^2 - 3x + 2$:
  • Concavity: Up (since $a = 2$)
  • Vertex: $\left(\frac{3}{4}, \frac{7}{8}\right)$
  • Axis of Symmetry: $x = \frac{3}{4}$
  • Y-intercept: $(0, 2)$; No x-intercepts
  • Domain: $(-\infty, \infty)$; Range: $[\frac{7}{8}, \infty)$

Additional Notes

• For complex problems, use the Quadratic Formula when necessary, especially for finding x-intercepts. If the discriminant is negative, there are no real solutions.
• Remember to consider vertex form vs standard form when identifying characteristics of the quadratic function.