Lesson 1 Functions and Linear Equations

1.1 Lecture Content

Video: Functions

Video: Common Functions

Video: Two Equations in Two Unknowns

Video: Supply and Demand Models

Video: Linear Functions and Models

1.2 Lecture Notes

1.2.1 Common Functions and Their Graphs

Horizontal Line

Equation: \[ y = c \]

Graph:
A flat, horizontal line that crosses the y-axis at \(y = c\).

Rule:
The output is constant for every value of \(x\).

General Linear Equation

Equation: \[ y = mx + b \]

Graph:
A straight line with slope \(m\) and y-intercept \(b\).

Rule:
Linear functions increase or decrease at a constant rate.

Quadratic Function (Parabola)

Equation: \[ y = ax^2 + bx + c \]

Graph:
A U-shaped curve: - Opens upward if \(a > 0\) - Opens downward if \(a < 0\)

Vertex:
The turning point of the parabola.

Axis of Symmetry:
A vertical line through the vertex.

Cubic Function

Equation: \[ y = ax^3 + bx^2 + cx + d \]

Graph:
An S-shaped curve that may have one or two turning points depending on coefficients.

Rule:
Cubic functions change curvature and direction.

Reciprocal Function

Equation: \[ y = \frac{1}{x} \]

Graph: - Two curved branches. - Undefined at \(x = 0\).

Asymptotes: - Vertical: \(x = 0\) - Horizontal: \(y = 0\)

Absolute Value Function

Equation: \[ y = |x| \]

Graph:
A V-shaped graph with the vertex at the origin \((0, 0)\).

Rule:
Negative inputs are reflected to positive outputs.

1.2.2 Finding the Domain of a Function

Rational Functions

Example: \[ f(x) = \frac{1}{x - 3} \]

Rule:
Exclude any value of \(x\) that makes the denominator zero.

Domain:
All real numbers except \(x = 3\).

Square Root Functions

Example: \[ f(x) = \sqrt{x - 2} \]

Rule:
The expression under the square root must be non-negative.

Domain:
All real numbers \(x \geq 2\).

1.2.3 Solving Systems Using the Elimination Method

Steps:

  1. Multiply one or both equations so that one variable has the same (or opposite) coefficient.
  2. Add or subtract the equations to eliminate one variable.
  3. Solve for the remaining variable.
  4. Substitute back to find the other variable.

Example: \[ \begin{align*} 2x + 3y &= 12 \\ 4x - 3y &= 6 \end{align*} \]

Add the two equations: \[ (2x + 3y) + (4x - 3y) = 12 + 6 \\ 6x = 18 \Rightarrow x = 3 \]

Substitute \(x = 3\) into the first equation: \[ 2(3) + 3y = 12 \Rightarrow 6 + 3y = 12 \Rightarrow y = 2 \]

Solution:
\(x = 3, \quad y = 2\)

1.2.4 Revenue, Cost, and Profit Functions

Revenue Function

Formula: \[ R(x) = p(x) \cdot x \]

Where: - \(R(x)\): Revenue - \(p(x)\): Price per unit - \(x\): Quantity sold

Cost Function

Formula: \[ C(x) = \text{Fixed Cost} + \text{Variable Cost} \cdot x \]

  • Fixed Costs: Constant (e.g., rent, salaries)
  • Variable Costs: Depend on production level

Profit Function

Formula: \[ P(x) = R(x) - C(x) \]

Profit is the difference between revenue and cost.

Demand Function

A typical linear demand function: \[ p(x) = a - bx \]

  • As quantity \(x\) increases, price \(p\) usually decreases.
  • This represents an inverse relationship between price and demand.

1.3 Examples

1.3.1 Applications of Linear Equations

Video: Solutions

  1. Find the final amount of money in an account if $2700is deposited at 5.5% interest compounded quarterly (every 3 months) and the money is left for 5 years.

  2. What present value amounts to $12,448.29 if it is invested for 2 years at 11% compounded monthly?

  3. Suppose a calculator manufacturer has the total cost function \[C(x) = 43x + 11,000,\] and the total revenue function \[R(x) = 55x.\]

    1. What is the equation of the profit function
    2. What is the profit on 2100 units?

  4. Suppose a stereo receiver manufacturer has the total cost function \[C(x) = 400x + 2490,\] and the total revenue function \[R(x) = 830x.\]

    1. What is the equation of the profit function for this commodity?
    2. What is the profit on 310 units?

  5. Given: (q is number of items)

    • Demand function: \(P = 1040.4 - 0.4q^2\)
    • Supply function: $p = 0.5q^2 Find the equilibrium price and quantity.

  6. If the demand for a pair of designer high heel shoes is given by \[4p + 5q = 825,\] and the supply function for the shoes is \[p - 6q = -35,\]

    • The quantity demanded at $175 is?
    • The quantity supplied at $175 is?
    • At $175, do we have a surplus or a shortfall?

1.3.2 Systems of Linear Equations

Video: Solutions

  1. Solve the system of equations by graphing: \[\begin{align*} y - 26 &= -6x\\ y - 5x &= -29 \end{align*}\]

  2. Solve the system of equations: \[\begin{align*} -4x - y &=14\\ -2x + y &= 4 \end{align*}\]

  3. Solve the system of equations by graphing: \[\begin{align*} 3x - 9y &= -15\\ -6x + 18y &= 30 \end{align*}\]

  4. Solve the system of equations by graphing: \[\begin{align*} 4x + 4y &= -4\\ 2x + y &=2 \end{align*}\]

  5. The admission fee at an amusement park is 1.5 dollars for children and 4 dollars for adults. On a certain day, 292 people entered the park, and the admission fees collected totaled 788 dollars. How many children and how many adults were admitted?

  6. What quantity of 70% acid solution must be mixed with a 20% solution to produce 900 mL of a 50% solution?

  7. Convert the equations in the system below into slope-intercept form and then classify the system. \[\begin{align*} -x - y &= -2\\ 2x + 2y &= 4 \end{align*}\]

1.4 Practice Problems

  1. Find the equation of the straight line through the points (3,4) and (5,2).
  2. Find the equation of the straight line through the point (3,4) and parallel to the line \(2x - 3y = 4\).
  3. If \(f(x) = -3x + 7\), evaluate \(f(-3)\) and determine the domain of \(1/f(x)\) using interval notation.

  1. Solve the system:
    \[ \begin{cases} 2x + y = 2 \\ -2x + y = 2 \end{cases} \]

  2. Solve the system:
    \[ \begin{cases} 2x - 3y = 2 \\ 6x - 9y = 3 \end{cases} \]

  1. (APPLIED) A piano manufacturer has a daily fixed cost of $1,200 and a marginal cost of $1,500 per piano. Find the cost function \(C(x)\) for manufacturing \(x\) pianos in one day. Answer the following:
    1. What is the cost of manufacturing 3 pianos?
    2. What is the cost of manufacturing the 3rd piano that day?
    3. What is the cost of manufacturing the 11th piano that day?
    4. Graph \(C\) as a function of \(x\).
  2. (APPLIED) Anthony Altino is mixing food for his young daughter and wants the meal to supply 1 gram of protein and 5 milligrams of iron. He mixes cereal (0.5 g protein and 1 mg iron per ounce) and fruit (0.2 g protein and 2 mg iron per ounce). What mixture will provide the desired nutrition?

1.5 Self-Assessment

Time yourself and try to solve the following within 20 minutes:

  1. Find the equation of the straight line through (3,1) and parallel to \(6x - 2y = 11\).
  2. Solve:
    \[ \begin{cases} \frac{-2x}{3} + \frac{y}{2} = \frac{-1}{6} \\ \frac{x}{4} - y = \frac{-3}{4} \end{cases} \]
  3. If the addition or subtraction of two linear equations results in \(0 = 3\), what does this say about their graphs?
  4. Determine the domain of \(f(x) = \sqrt{x-10}\). Express your answer in interval notation.
  5. In 2005, the Las Vegas monorail charged $3 per ride with an average ridership of 28,000 per day. After raising the fare to $5, ridership dropped to 19,000. Determine a linear function that relates ridership \(q\) to fare \(p\).

1.6 Lesson Checklist

This checklist is designed to help you keep track of what you need to work on. The main goal is to be aware of what you need to focus more attention on. Place an \(X\) in the appropriate box beside the skill below.

\[\begin{align*} &\textbf{Developing (D):} &&\textrm{You still need to work on this skill.}\\ &\textbf{Consistent (CON):} &&\textrm{You use the skill correctly most of the time.}\\ &\textbf{Competent (COM):} &&\textrm{You show mastery of the skill.} \end{align*}\]

Skill D CON COM
Find the domain of a rational function in interval notation
Find the domain of a square root function in interval notation
Find the equation of a line given various information
Use the elimination method to solve a 2x2 linear system
Solve applied problems involving linear equations