College Algebra: Quadratic Functions, Equations, and Applications Study Guide

Quadratic Functions and Their Properties

Vertex, Maximum/Minimum, and Concavity

Quadratic functions are polynomial functions of degree two, typically written in the form f(x) = ax^2 + bx + c. The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the sign of the leading coefficient a.

• Vertex: The vertex is the highest or lowest point on the graph of a quadratic function. For f(x) = a(x-h)^2 + k, the vertex is at (h, k).

• Maximum or Minimum: If a > 0, the parabola opens upward and the vertex is a minimum. If a < 0, the parabola opens downward and the vertex is a maximum.

• Concavity: The direction the parabola opens is called its concavity. Upward means concave up; downward means concave down.

Example:

• For f(x) = -4(x-7)^2 + 11:

  • Vertex: (7, 11)

  • Maximum or Minimum: Maximum (since a = -4 < 0)

  • Concave up or down: Down

• For f(x) = x^2 + 3x - 2:

  • Vertex: $x = -\frac{b}{2a} = -\frac{3}{2}$, $y = f(-\frac{3}{2})$

  • Maximum or Minimum: Minimum (since a = 1 > 0)

  • Concave up or down: Up

Graphing Quadratic Functions

Axis of Symmetry, Range, and Intervals of Increase/Decrease

The graph of a quadratic function is symmetric about a vertical line called the axis of symmetry. The range and intervals of increase/decrease depend on the vertex and the direction the parabola opens.

• Axis of Symmetry: $x = h$ for f(x) = a(x-h)^2 + k, or $x = -\frac{b}{2a}$ for f(x) = ax^2 + bx + c.

• Range: If the parabola opens up, the range is $[k, \infty)$; if it opens down, the range is $(-\infty, k]$.

• Intervals: The function decreases to the vertex and increases after (if opening up), or vice versa (if opening down).

Example: For a parabola with vertex at (2, -3) opening upward:

• Axis of symmetry: $x = 2$

• Range: $[-3, \infty)$

• Increasing on $(2, \infty)$, decreasing on $(-\infty, 2)$

Quadratic Equations and the Discriminant

Real Solutions and the Discriminant

The discriminant of a quadratic equation $ax^2 + bx + c = 0$ is $D = b^2 - 4ac$.

• If $D > 0$, there are two distinct real solutions.

• If $D = 0$, there is one real solution (a repeated root).

• If $D < 0$, there are no real solutions (the solutions are complex).

Example: If the graph of a quadratic function does not intersect the x-axis, then $D < 0$ and there are no real solutions.

Solving Exponential and Radical Equations

Exponential Equations

To solve equations like $4^x = 108$, take the logarithm of both sides:

• $4^x = 108$

• $x \ln 4 = \ln 108$

• $x = \frac{\ln 108}{\ln 4}$

Radical Equations

For equations like $2x^{1/3} = \frac{1}{3}$:

• Isolate the radical: $x^{1/3} = \frac{1}{6}$

• Cube both sides: $x = (\frac{1}{6})^3 = \frac{1}{216}$

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions over different intervals of the domain.

• Example: $f(x) = \begin{cases} 3x-2 & \text{if } x < -1 \\ 4-x^2 & \text{if } x \geq -1 \end{cases}$

• To graph, plot each piece on its specified interval, paying attention to endpoints.

Applications: Revenue, Cost, and Profit Functions

Profit Function and Optimization

In business applications, the profit function is defined as revenue minus cost:

• $P(x) = R(x) - C(x)$, where $x$ is the number of units produced and sold.

• To find the domain, consider the context (e.g., $x \geq 0$).

• To maximize profit, find the vertex of the quadratic profit function.

Example: If $R(x) = 4x - 20$ and $C(x) = 0.002x^2 + 0.8x + 50$, then:

• $P(x) = (4x - 20) - (0.002x^2 + 0.8x + 50) = -0.002x^2 + 3.2x - 70$

• Maximum profit at $x = -\frac{b}{2a} = -\frac{3.2}{2 \times -0.002} = 800$ units

Quadratic Applications: Projectile Motion

Height Functions and Maximum Height

The height of a projectile is often modeled by a quadratic function: $h(t) = -16t^2 + v_0 t + h_0$.

• When does the ball hit the ground? Set $h(t) = 0$ and solve for $t$.

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

• Time to reach maximum height: Same as above.

Modeling with Exponential Functions

Growth and Decay

Exponential functions model situations where quantities grow or decay by a constant percentage rate. The general form is $y = ab^{x}$.

• Concavity: Exponential growth is always concave up.

• Applications: Used to model populations, finance, and health statistics.

Example: $y = 4.97x^{0.076}$ models the percent of the U.S. adult population with diabetes, where $x$ is years after 2000.

Piecewise Cost Functions

Definition and Interpretation

Piecewise cost functions are used when the cost structure changes after a certain threshold.

• Example: $C(x) = \begin{cases} 300 & \text{if } 0 < x \leq 30 \\ 300 + 12(x-30) & \text{if } x > 30 \end{cases}$

• For $x \leq 30$, the cost is fixed at $300$.

• For $x > 30$, each additional meal costs $12$.

Example: $C(60) = 300 + 12(60-30) = 300 + 360 = 660$