Connective tissue

Price Definition

• Price is the value assigned, representing what a consumer exchanges for an offering which includes money, time, and effort.
• Opportunity cost must also be considered when determining the overall price.

Cost-Plus Pricing Strategy

• Markup is added to the total costs for the product.
• Easy to implement, but risks overpricing the product.

Competition-Based Pricing Strategy

• Prices are set based on competitors' charges.
• This strategy can lead to price wars.

Demand-Based Pricing Strategy

• Prices are set based on consumer's willingness to pay.
• This requires accurate estimation of demand.

Value-Based Pricing Strategy

• Prices are set based on the perceived value to the customer.
• Understanding customer needs and willingness to pay is essential for its implementation.

Skimming Pricing Strategy

• A high initial price is set to capture early adopters, then the price is lowered later.
• Most effective for innovative products with limited competition.

Penetration Pricing Strategy

• A low initial price is set to quickly gain market share.
• Best suits price-sensitive markets with high competition.

Reference Pricing

• Consumers compare a product's price to their internal reference price.
• Marketing and store placement can influence this comparison.

Price-Quality Inferences

• Consumers assume that higher-priced products have higher quality.
• Useful where quality is difficult to assess directly.

Odd-Even Pricing

• Prices ending in odd numbers (like $19.99) are used to create the perception of a lower price.
• This exploits how consumers perceive prices.

Price Bundling

• Two or more products are sold together at a single price.
• Increases sales and reduces marketing costs, but requires a lower price than purchasing items individually.

Price Fixing

• Competitors agree to set prices collectively.
• Illegal under antitrust laws.

Price Discrimination

• Different prices are charged to different customers for the same product.
• Illegal if it harms competition.

Deceptive Pricing

• Consumers are misled about prices using tactics such as bait-and-switch.

Modeling Relationships with Functions

• Functions can express relationships between quantities and model real-world behavior.
• Independent variable: input.
• Dependent variable: output.

Example: AC Unit Sales

• Temperature is the independent variable in determining number of AC units sold.
• AC unit sales represents the dependent variable.

Verbal Descriptions and Mathematical Models

• "y is four more than x" translates to: $y = x + 4$
• "y is four less than x" translates to: $y = x - 4$
• "y is four times x" translates to: $y = 4x$
• "y is four more than three times x" translates to: $y = 3x + 4$
• "y is four less than three times x" translates to: $y = 3x - 4$
• "y is four more than three times the square of x" translates to: $y = 3x^2 + 4$
• "y is four less than three times the square root of x" translates to: $y = 3\sqrt{x} - 4$
• "y varies directly as x" translates to: $y = kx$
• "y is directly proportional to x" translates to: $y = kx$
• "y varies inversely as x" translates to: $y = \frac{k}{x}$
• "y is inversely proportional to x" translates to: $y = \frac{k}{x}$
• "y varies jointly as x and z" translates to: $y = kxz$
• "y is jointly proportional to x and z" translates to: $y = kxz$
• "y varies directly as x and inversely as z" translates to: $y = k\frac{x}{z}$
• "y is directly proportional to x and inversely to z" translates to: $y = k\frac{x}{z}$
• "y varies jointly as x and the square of z" translates to: $y = kxz^2$
• "y varies inversely as the square root of x" translates to: $y = \frac{k}{\sqrt{x}}$
• "y varies jointly as x and z and inversely as w" translates to: $y = k\frac{xz}{w}$
• "y varies jointly as x and z and inversely to the sum of w and r" translates to: $y = k\frac{xz}{w + r}$

Scatter Plots

• Scatter plots visually represent the relationship between two variables and help to quickly identify correlations.

Linear Modeling

• A line of best fit is used to represent a linear trend in data with the equation $y = mx + b$.
• $m$ is the slope, and b is the y-intercept.
• Correlation coefficient, $r$, measures the strength and direction of correlation.
  • $r$ close to 1 is a strong positive correlation.
  • $r$ close to -1 is a strong negative correlation.
  • $r$ close to 0 is a weak or no linear correlation.

Linear Modeling Process

• Create a scatter plot of the data.
• Determine if a linear relationship exists.
• Find the line of best fit using a calculator.
• Determine the the correlation coefficient r.
• Graph the line of best fit to assess the fit.
• Use the equation for predictions.

Exponential Models

• Exponential functions grow faster than linear functions with the general form $y = ab^x$.
• $a$ is the initial value.
• $b$ is the growth/decay factor.
  • $b > 1$ is exponential growth.
  • $0 < b < 1$ is exponential decay.

Process for Exponential Models

• Create a scatter plot of the data.
• Determine if an exponential relationship exists.
• Find the exponential curve of best fit using a calculator ($y = ab^x$).
• Graph the curve of best fit to assess the fit.
• Use the equation to make predictions.

Logarithmic Models

• Logarithmic functions are the inverse of exponential functions.
• Has the general form $y = a + b \ln(x)$.
• Used to model relationships where the rate of change decreases as the independent variable increases.

Process for Logarithmic Models

• Create a scatter plot of the data.
• Determine if a logarithmic relationship exists.
• Find the logarithmic curve of best fit using a calculator ($y = a + b \ln(x)$).
• Graph the curve of best fit to assess the fit.
• Use the equation to make predictions.

Sinusoidal Models

• Sinusoidal functions are periodic functions oscillating between max and min values.
• Has the general form $y = A\sin(Bx - C) + D$
  • $A$ is the amplitude.
  • $B$ relates to the period $P = \frac{2\pi}{B}$.
  • $C$ is the phase shift.
  • $D$ is the vertical shift (midline).

Process for Sinusoidal Models

• Create a scatter plot of the data.
• Determine if a Sinusoidal relationship exists.
• Determine equation using high and low values.
  • $D = \frac{High + Low}{2}$
  • $A = High - D$
• Determine the period P from the graph.
• Calculate $B = \frac{2\pi}{P}$

Example: Average Daily Temperature

• Average daily temperature throughout year follows a sinusoidal pattern.
• Model with: $y = A\sin(Bx - C) + D$

Model Selection

• Begin with a scatter plot of the data.
• Consider the nature of variables and potential relationships.
• Linear models suit constant rates of change.
• Exponential models suit constant percentage change growth or decay.
• Logarithmic models suit decreasing rates of change.
• Sinusoidal models suit periodic patterns.
• Graph the model to assess the fit; accurate predictions indicate a good fit.

Key Questions for Model Selection

• Is there a pattern?
• Is the pattern increasing or decreasing?
• Is the rate of change constant, increasing, or decreasing?
• Is the function periodic?