Quadratic Functions: Graphs, Properties, and Applications

Quadratic Functions

Introduction to Quadratic Functions

A quadratic function is a polynomial function of degree 2, typically written as $f(x) = ax^2 + bx + c$, where $a \neq 0$. The graph of a quadratic function is a parabola, which may open upward or downward depending on the sign of $a$.

• Vertex: The highest or lowest point on the parabola.

• Axis of symmetry: A vertical line passing through the vertex, dividing the parabola into two symmetric halves.

• Intercepts: Points where the graph crosses the x-axis (x-intercepts) and y-axis (y-intercept).

Quadratic Function in Vertex Form

Definition and Properties

The vertex form of a quadratic function is:

$f(x) = a(x - h)^2 + k$

• The vertex is at the point $(h, k)$.

• The axis of symmetry is the line $x = h$.

• If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

Graphing a Quadratic Function in Vertex Form

1. Determine the direction: If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

2. Find the vertex: The vertex is $(h, k)$.

3. Find x-intercepts: Solve $a(x - h)^2 + k = 0$ for $x$.

4. Find the y-intercept: Compute $f(0)$.

5. Plot points: Plot the vertex, intercepts, and additional points as needed. Connect with a smooth curve.

Example: Suppose $f(x) = -2(x - 1)^2 + 4$.

• The parabola opens downward ($a = -2 < 0$).

• Vertex: $(1, 4)$.

• To find x-intercepts, solve $-2(x - 1)^2 + 4 = 0$.

• y-intercept: $f(0) = -2(0 - 1)^2 + 4 = -2(1) + 4 = 2$.

Quadratic Function in General Form

Definition and Properties

The general form of a quadratic function is:

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

• The vertex's x-coordinate is $x = -\frac{b}{2a}$.

• The y-coordinate is $f\left(-\frac{b}{2a}\right)$.

• If $a > 0$, the parabola opens upward; if $a < 0$, it opens downward.

Graphing a Quadratic Function in General Form

1. Determine the direction: Check the sign of $a$.

2. Find the vertex: $x = -\frac{b}{2a}$, $y = f\left(-\frac{b}{2a}\right)$.

3. Find x-intercepts: Solve $ax^2 + bx + c = 0$.

4. Find the y-intercept: $f(0) = c$.

5. Plot points: Plot the vertex, intercepts, and additional points as needed. Connect with a smooth curve.

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

• Opens downward ($a = -1$).

• Vertex: $x = -\frac{4}{2 \times -1} = 2$, $y = f(2) = -(2)^2 + 4 \times 2 - 3 = -4 + 8 - 3 = 1$.

• x-intercepts: Solve $-x^2 + 4x - 3 = 0$.

• y-intercept: $f(0) = -3$.

Minimum and Maximum Values of Quadratic Functions

Finding Extrema

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

• If $a > 0$, the function has a minimum at $x = -\frac{b}{2a}$.

• If $a < 0$, the function has a maximum at $x = -\frac{b}{2a}$.

• The minimum or maximum value is $f\left(-\frac{b}{2a}\right)$.

Example: For $f(x) = 2x^2 - 8x + 5$:

• $a = 2 > 0$; function has a minimum.

• Minimum at $x = -\frac{-8}{2 \times 2} = 2$.

• Minimum value: $f(2) = 2(2)^2 - 8 \times 2 + 5 = 8 - 16 + 5 = -3$.

Domain and Range of Quadratic Functions

Determining Domain and Range

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

• Range:

  • If $a > 0$, range is $[k, \infty)$ where $k$ is the minimum value.

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

Example: For $f(x) = 3x^2 + 2x + 1$, vertex at $x = -\frac{2}{6} = -\frac{1}{3}$, $y = f(-\frac{1}{3})$.

Applications: Maximizing or Minimizing Quadratic Functions

Strategy for Solving Optimization Problems

1. Identify the quantity to be maximized or minimized.

2. Express this quantity as a function of one variable, using problem constraints.

3. Rewrite the function in standard quadratic form.

4. Find the vertex to determine the maximum or minimum value.

5. Interpret the result in the context of the problem.

Example: Maximizing Area

Problem: You have 120 feet of fencing to enclose a rectangular region. What dimensions maximize the area?

• Let $x$ = length, $y$ = width. Perimeter: $2x + 2y = 120 \implies x + y = 60 \implies y = 60 - x$.

• Area: $A = x \cdot y = x(60 - x) = 60x - x^2$.

• This is a quadratic function opening downward ($a = -1$).

• Maximum at $x = -\frac{b}{2a} = -\frac{60}{2 \times -1} = 30$.

• So $y = 60 - 30 = 30$.

• Maximum area: $A = 30 \times 30 = 900$ square feet.

Summary Table: Forms and Properties of Quadratic Functions

Form	Equation	Vertex	Axis of Symmetry	Direction
Vertex Form	$f(x) = a(x - h)^2 + k$	$(h, k)$	$x = h$	Up if $a > 0$, Down if $a < 0$
General Form	$f(x) = ax^2 + bx + c$	$\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)$	$x = -\frac{b}{2a}$	Up if $a > 0$, Down if $a < 0$