Quadratic Functions and Their Graphs: Concepts, Properties, and Applications

Quadratic Functions and Graphs

Definition of a Quadratic Function

A quadratic function is a polynomial function of degree 2, and it can be written in the standard form:

• Standard Form: $P(x) = ax^2 + bx + c$, where a, b, and c are real numbers and a ≠ 0.

• The graph of a quadratic function is called a parabola.

Completing the Square

Completing the square is a method used to rewrite a quadratic function in vertex form:

• Vertex Form: $P(x) = a(x - h)^2 + k$

• This form makes it easy to identify the vertex and the direction in which the parabola opens.

Steps to Complete the Square:

1. Divide each side of the equation by a so the coefficient of $x^2$ is 1.

2. Add $-c/a$ to each side.

3. Add to each side the square of half the coefficient of $x$: $\left(\frac{b}{2a}\right)^2$.

4. Factor the right side as the square of a binomial and combine terms on the left.

5. Isolate the term involving $P(x)$ on the left.

6. Multiply each side by a to return to the original scale.

Example: Complete the square for $P(x) = -x^2 - 6x - 8$.

• $P(x) = -x^2 - 6x - 8$

• $-P(x) = x^2 + 6x + 8$

• $-P(x) - 8 = x^2 + 6x$

• $-P(x) - 8 + 9 = x^2 + 6x + 9$

• $-P(x) + 1 = (x + 3)^2$

• $-P(x) = (x + 3)^2 - 1$

• $P(x) = -(x + 3)^2 + 1$

Characteristics of the Graph of $P(x) = a(x - h)^2 + k$

The graph of a quadratic function in vertex form has several important features:

• Vertex: The point $(h, k)$ is the vertex of the parabola.

• Axis of Symmetry: The vertical line $x = h$ is the axis of symmetry.

• Direction: The graph opens upward if $a > 0$ and downward if $a < 0$.

• Width: The graph is wider than $y = x^2$ if $0 < |a| < 1$ and narrower if $|a| > 1$.

Graphing a Quadratic Function: Example

Given $P(x) = -x^2 - 6x - 8 = -(x + 3)^2 + 1$:

• Vertex: $(-3, 1)$

• y-intercept: $P(0) = -(0 + 3)^2 + 1 = -9 + 1 = -8$; so the y-intercept is $(0, -8)$.

• Another point: $P(-1) = -(-1 + 3)^2 + 1 = -4 + 1 = -3$; so $(-1, -3)$ is on the graph.

• Axis of Symmetry: $x = -3$

Graphical Features:

• The parabola opens downward (since $a = -1 < 0$).

• The vertex is the highest point (maximum).

• The axis of symmetry divides the parabola into two mirror-image halves.

Domain and Range

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

• Range: If the parabola opens downward, the range is $(-\infty, k]$; if upward, $[k, \infty)$, where $k$ is the y-coordinate of the vertex.

• Intervals of Increase/Decrease:

  • The function increases on $(-\infty, h)$ and decreases on $(h, \infty)$ if $a < 0$ (downward opening).

  • If $a > 0$, the function decreases on $(-\infty, h)$ and increases on $(h, \infty)$.

Vertex Formula for a Parabola

The vertex $(h, k)$ of the quadratic function $P(x) = ax^2 + bx + c$ can be found using:

• Vertex (h, k):

  • $h = -\frac{b}{2a}$

  • $k = P(h) = a(h)^2 + b(h) + c$

Key Features to Include When Graphing a Quadratic Function

• Vertex

• Axis of symmetry

• y-intercept

• x-intercepts (if any)

• Extreme values (maximum or minimum)

Extreme Values (Maximum/Minimum)

• If $a > 0$, the vertex is the minimum point and $k$ is the minimum value.

• If $a < 0$, the vertex is the maximum point and $k$ is the maximum value.

Example: Identifying Extreme Points and Values

• For $P(x) = 2x^2 + 4x - 16$:

  • $a = 2$, $b = 4$, $c = -16$

  • $h = -\frac{4}{2 \times 2} = -1$

  • $k = 2(-1)^2 + 4(-1) - 16 = 2 - 4 - 16 = -18$

  • Vertex: $(-1, -18)$ (minimum point, since $a > 0$)

• For $P(x) = -x^2 - 6x - 8$:

  • Vertex: $(-3, 1)$ (maximum point, since $a < 0$)

Applications: Projectile Motion

Quadratic functions model the height of a projected object under gravity (neglecting air resistance):

• Height function: $s(t) = -16t^2 + vt + s_0$

• $s(t)$: height at time $t$ (in feet)

• $v$: initial velocity (ft/s)

• $s_0$: initial height (ft)

Example: A ball is projected upward from 100 ft with initial velocity 80 ft/s.

• $s(t) = -16t^2 + 80t + 100$

• Maximum height occurs at $t = -\frac{b}{2a} = -\frac{80}{2 \times -16} = 2.5$ seconds

• Maximum height: $s(2.5) = -16(2.5)^2 + 80(2.5) + 100 = 200$ ft

Summary Table: Key Properties of Quadratic Functions

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