Slope, Intercepts, and Economic Models Review

Flashcards covering slope, intercepts, line equations from points, depreciation context, inequalities, equilibrium in demand/supply, correlation, revenue/cost/profit, domain considerations, and the difference quotient.

What is the slope formula?

m = (y2 - y1) / (x2 - x1)

How do you find the y-intercept b in y = m x + b using a known point (x0, y0) on the line?

b = y0 - m x0

If a line has slope m = 225 and passes through (20, 5000), what is the y-intercept b?

b = 500

What is the general method to find the equation of a line through two points?

Compute m = (y2 - y1)/(x2 - x1); then use y = m x + b with one point to solve for b; the equation is y = m x + b

What is the slope for the points (-2, 4) and (2, -2)?

-3/2

Using the line y = (-3/2)x + b, what is b if it passes through (-2, 4)?

b = 1

What does a negative slope indicate in a depreciation context?

Values decrease over time (per year)

In solving inequalities, what happens to the inequality sign when you multiply or divide by a negative number?

The inequality sign flips

How do you find the equilibrium quantity in a price-demand and price-supply model?

Set the price-demand equation equal to the price-supply equation (Pd = Ps) and solve for Q

How do you obtain the equilibrium price once you have the equilibrium quantity Q?

Plug Q into either the demand or the supply equation to get P

What does the correlation coefficient measure and what is its range?

Measures linear relationship; ranges from -1 to 1 (negative to positive correlation)

Given p(x) and costs F and V, what is the profit function P(x)?

P(x) = R(x) - C(x) = p(x) x - (F + V x)

With p(x) = 2000 - 60x, F = 400, V = 200, what is the profit function P(x)?

P(x) = (2000 - 60x)x - (400 + 200x) = -60x^2 + 1800x - 400

If profit is 11,600, what x-values satisfy P(x) = 11,600 and what are the corresponding prices?

x = 10 or x = 20; corresponding prices are p(10) = 1400 and p(20) = 800

What is the domain for x when the price must be nonnegative for p(x) = 2000 - 60x?

0 ≤ x ≤ 2000/60 ≈ 33.33

What is the domain for sqrt(x + 2)?

x ≥ -2

What is the difference quotient and what does it represent?

Difference quotient: (f(x + h) - f(x)) / h; represents the average rate of change over an interval h

What is true about the difference quotient for linear functions?

For a linear function, the difference quotient is constant and equals the slope, regardless of h