College Algebra: Quadratic Functions, Graphs, and Problem Solving

Quadratic Functions and Their Properties

Definition and Standard Form

A quadratic function is a polynomial function of degree 2, typically written in the form:

• Standard form: $f(x) = ax^2 + bx + c$, where $a \neq 0$.

• Vertex form: $f(x) = a(x-h)^2 + k$, where $(h, k)$ is the vertex.

Quadratic functions graph as parabolas, which may open upward ($a > 0$) or downward ($a < 0$).

Key Features of Quadratic Functions

• Vertex: The highest or lowest point of the parabola. The vertex $(h, k)$ can be found using:

  • $h = -\frac{b}{2a}$

  • $k = f(h)$

• Axis of Symmetry: The vertical line $x = h$ that divides the parabola into two mirror images.

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

• X-intercepts (Roots): The points where the graph crosses the x-axis, found by solving $ax^2 + bx + c = 0$.

Methods for Solving Quadratic Equations

• Factoring: Express the quadratic as a product of two binomials and set each factor to zero.

• Quadratic Formula: For $ax^2 + bx + c = 0$, the solutions are: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• Completing the Square: Rearrange the equation to form a perfect square trinomial.

• Square Root Method: Used when the quadratic is in the form $a(x-h)^2 = k$.

Discriminant and Nature of Roots

The discriminant $D = b^2 - 4ac$ determines the nature of the roots:

• If $D > 0$: Two distinct real roots.

• If $D = 0$: One real root (a repeated root).

• If $D < 0$: No real roots (two complex roots).

Examples

• Example 1: Find the vertex of $f(x) = 2x^2 + 12x - 1$.

  • $h = -\frac{12}{2 \times 2} = -3$

  • $k = f(-3) = 2(-3)^2 + 12(-3) - 1 = 18 - 36 - 1 = -19$

  • Vertex: $(-3, -19)$

• Example 2: Solve $x^2 - 5x - 7 = 0$ using the quadratic formula.

  • $a = 1$, $b = -5$, $c = -7$

  • $x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4 \cdot 1 \cdot (-7)}}{2 \cdot 1}$

  • $x = \frac{5 \pm \sqrt{25 + 28}}{2} = \frac{5 \pm \sqrt{53}}{2}$

Transformations of Functions

Shifting Graphs

Transformations change the position or shape of a graph:

• Vertical Shifts: $f(x) + k$ shifts the graph up ($k > 0$) or down ($k < 0$).

• Horizontal Shifts: $f(x - h)$ shifts the graph right ($h > 0$) or left ($h < 0$).

Example: $y = f(x - 3)$ shifts the graph of $y = f(x)$ three units to the right.

Function Composition and Inverses

Composition of Functions

The composition $f(g(x))$ means applying $g$ first, then $f$ to the result.

• Example: If $f(x) = 3x + 6$ and $g(x) = x - 5$, then $f(g(x)) = 3(x - 5) + 6 = 3x - 15 + 6 = 3x - 9$.

Inverse Functions

Two functions $f$ and $g$ are inverses if $f(g(x)) = x$ and $g(f(x)) = x$ for all $x$ in their domains.

• Example: $f(x) = x^2$ and $g(x) = \sqrt{x}$ are inverses on $x \geq 0$.

Applications: Quadratic Word Problems

Projectile Motion

Quadratic functions model the height of an object in projectile motion:

• General form: $h(t) = -16t^2 + vt + h_0$, where $h_0$ is initial height, $v$ is initial velocity, and $t$ is time.

• Finding when the object hits the ground: Set $h(t) = 0$ and solve for $t$.

• Maximum height: Occurs at the vertex $t = -\frac{b}{2a}$.

Example: A coin is thrown from a height of 150 ft. The height after $t$ seconds is $h(t) = -16t^2 + 16t + 150$.

• To find when it hits the ground, solve $-16t^2 + 16t + 150 = 0$ using the quadratic formula.

• Maximum height occurs at $t = -\frac{16}{2 \times -16} = 0.5$ seconds.

Factoring Quadratic Expressions

Factoring Techniques

• Find two numbers that multiply to $ac$ and add to $b$.

• Split the middle term and factor by grouping.

• Check for special forms: difference of squares, perfect square trinomials.

Example: Factor $3x^2 - 11x + 6$.

• Find factors of $3 \times 6 = 18$ that add to $-11$ ($-9$ and $-2$).

• Rewrite: $3x^2 - 9x - 2x + 6 = (3x^2 - 9x) + (-2x + 6) = 3x(x - 3) -2(x - 3) = (x - 3)(3x - 2)$.

Summary Table: Methods for Solving Quadratic Equations

Method	When to Use	Example
Factoring	When the quadratic can be factored easily	$x^2 - 5x + 6 = (x - 2)(x - 3)$
Quadratic Formula	Always works; use when factoring is difficult	$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
Completing the Square	Useful for converting to vertex form	$x^2 + 6x + 5 = (x + 3)^2 - 4$
Square Root Method	When quadratic is in form $a(x-h)^2 = k$	$x - 3 = \pm \sqrt{2}$

Additional info:

• Some content inferred from context and standard College Algebra curriculum, such as the general forms and methods for solving quadratics.

• Graph sketches and matching problems reinforce understanding of transformations and function properties.

• Projectile motion problems are classic applications of quadratic functions in algebra.