Quadratic Functions and Their Properties: Study Notes

Quadratic Functions

Graphing and Analyzing Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically written in the form $f(x) = ax^2 + bx + c$. Their graphs are called parabolas, which can open upwards or downwards depending on the leading coefficient.

• Standard Form: $f(x) = a(x - h)^2 + k$, where $(h, k)$ is the vertex of the parabola.

• General Form: $f(x) = ax^2 + bx + c$

Key Properties of Quadratic Functions

• Direction of Opening: Determined by the sign of $a$ in $f(x) = a(x - h)^2 + k$.

  • If $a > 0$, the parabola opens upward.

  • If $a < 0$, the parabola opens downward.

• Vertex: The highest or lowest point on the graph, given by $(h, k)$ in standard form. In general form, the vertex is at: $h = -\frac{b}{2a}$ $k = f(h)$

• Axis of Symmetry: The vertical line passing through the vertex, $x = h$.

• Y-intercept: The point where the graph crosses the y-axis, found by evaluating $f(0)$.

• X-intercepts (Roots): The points where $f(x) = 0$. These can be found by factoring, completing the square, or using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Domain: All real numbers, $(-\infty, \infty)$.

• Range: Depends on the direction of opening:

  • If $a > 0$, range is $[k, \infty)$.

  • If $a < 0$, range is $(-\infty, k]$.

• Maximum or Minimum Value: The value $k$ at the vertex. If the parabola opens upward, $k$ is the minimum; if downward, $k$ is the maximum.

Example: Analyzing $f(x) = -\frac{1}{2}(x - 3)^2 + 12$

• Direction of Opening: $a = -\frac{1}{2} < 0$, so the parabola opens downward.

• Vertex: $(h, k) = (3, 12)$

• Axis of Symmetry: $x = 3$

• Y-intercept: $f(0) = -\frac{1}{2}(0 - 3)^2 + 12 = -\frac{1}{2}(9) + 12 = -4.5 + 12 = 7.5$

• X-intercepts: Set $f(x) = 0$: $-\frac{1}{2}(x - 3)^2 + 12 = 0$ $-\frac{1}{2}(x - 3)^2 = -12$ $(x - 3)^2 = 24$ $x - 3 = \pm \sqrt{24} = \pm 2\sqrt{6}$ $x = 3 \pm 2\sqrt{6}$

• Domain: $(-\infty, \infty)$

• Range: Since the parabola opens downward, range is $(-\infty, 12]$

• Maximum Value: $12$at$x = 3$

Summary Table: Properties of $f(x) = -\frac{1}{2}(x - 3)^2 + 12$

Property	Value
Direction of Opening	Downward
Vertex	$(3, 12)$
Axis of Symmetry	$x = 3$
Y-intercept	$(0, 7.5)$
X-intercepts	$x = 3 \pm 2\sqrt{6}$
Domain	$(-\infty, \infty)$
Range	$(-\infty, 12]$
Maximum Value	$12$