Functions and Graphs (Chapter 1) – Key Concepts

Flashcards cover coordinates, fundamental theorem, solving and graphing equations, functions, notation, evaluation, vertical line test, difference quotient, and basic profit–cost–demand analysis with domain considerations.

What are the two axes in the Cartesian Coordinate System, and what is the origin?

The horizontal x-axis and the vertical y-axis; they intersect at the origin, the point (0,0).

How are the four quadrants labeled and ordered around the plane?

I (top-right), II (top-left), III (bottom-left), IV (bottom-right); ordered counterclockwise starting from the top-right.

What does the Fundamental Theorem of Analytic Geometry state about points and their coordinates?

Every point on the plane has a unique ordered pair (x,y), and every ordered pair (x,y) corresponds to exactly one point.

What makes an ordered pair (x,y) a solution to an equation in two variables?

Substituting x and y into the equation yields a true statement.

For the equation y = x + 3, is the ordered pair (2,5) a solution? Is (1,5)?

Yes, because 5 = 2 + 3; No, because 5 ≠ 1 + 3.

How do you graph an equation in two variables?

Find several solution pairs (x,y), plot them on the grid, then connect the points with a line or curve.

What is a function, in terms of inputs and outputs?

A relation where each input value from the domain corresponds to exactly one output value in the range.

Give an example of a function and a non-function.

Function: height for each person (Person → Height). Non-function: number of friends for each person (Person → many outputs).

In function notation, what is f(x) and what is x?

f(x) is the output (the dependent value) and x is the input (the independent variable).

What is function evaluation?

Finding the output for a specific input, e.g., f(3) for f(x) = 2x + 1 equals 7.

What does the Vertical Line Test determine?

If a vertical line intersects the graph in more than one point, the relation is not a function.

What is the difference quotient and what does it measure?

The expression [f(x+h) − f(x)]/h; it measures the average rate of change (slope) between two close points.

For f(x) = 2x + 1, what is the difference quotient?

2, since the slope is constant.

What are fixed costs and variable costs in the Profit-Loss model?

Fixed costs (a) do not depend on production; Variable costs (b x) increase with production.

How is the total cost function expressed?

C(x) = a + b x.

What is a price-demand function p(x)?

It shows how the price changes based on the quantity demanded; for example p(x) = 100 − 0.5x.

How is revenue defined?

R(x) = x p(x), the total revenue from selling x items at price p(x).

How is profit defined and when is there a profit, loss, or break-even?

P(x) = R(x) − C(x); profit if R(x) > C(x), loss if R(x) < C(x), break-even if R(x) = C(x).

Given R(x) = 100x − 0.5x^2 and C(x) = 1000 + 10x, what is P(x)?

P(x) = (100x − 0.5x^2) − (1000 + 10x) = −0.5x^2 + 90x − 1000.

What is the domain of f(x) = sqrt(x − 5)?

The radicand must be nonnegative, so x ≥ 5.