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 Dependent Variable

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# Graphs of Equations and Functions

## 1 1.1 Functions and Mathematical Modeling p.2

Functions
• A real-valued function f defined on a set D of real numbers is a rule that assigns to
each number x in D exactly one real number, denoted f(x).
• The set D of all numbers for which f(x) is defined is called the domain.
• The number f(x) is called the value of the function f at the point x.
• The set of all values y = f(x) is called the range of f.
• When a function f is described by writing a formula y = f(x), we call x the inde-
pendent variable and y the dependent variables because the value of y depends
on the choice of x.

Idea. A function is discrete if it only takes on certain isolated values. A function is contin-
uous if it can take on any numbers.

Domains and Intervals

• The domain of the function f is the set of all real numbers x for which the expression
f(x) makes sense and produces a real number y.

Often, a domain is R, or all real numbers. There are two restrictions that we know. you
can not take the square root of a negative number and you can not divide by zero.
• A closed interval contains both its endpoints x = a and x = b and is written [a, b].
• A open interval contains neither of its endpoints, written (a, b).
• A half open interval contains exactly one of its endpoints, like (a, b] or [a, b).
• An unbounded interval has positive or negative infinity as an `endpoint', written ## 2 1.2 Graphs of Equations and Functions p.12

• The slope-intercept equation is y = mx + b for the straight line with slope m = angle of inclination and y-intercept b.

• We also define the slope m as "rise over run." • The point-slope equation for a line is y - y0 = m(x - x0), with slope m passing
through the point (x0, y0).

Horizontal lines have slope zero.
Vertical lines have no defined slope.
Parallel lines have the same slope.

Graphs of More General Equations

• The graph of an equation in two variables x and y is the set of all points (x, y) in the
plane that satisfy the equation.

• The Pythagorean theorem implies the distance formula • x2 + y2 = r2 is the equation for a circle of radius r centered at (0, 0).

• More generally, (x-h)2+(y-k)2 = r2 is the equation for a circle of radius r centered
at (h, k).

Translates of Graphs

Translation Principle. When the graph of an equation is translated h units to the right
and k units up, the equation of the translated curve is obtained from the original equation
by replacing x with x - h and y with y - k.
Example of completing the square to find the center of a circle, page 14.

Graphs of Functions
• The graph of a function is a special case of the graph of an equation.
• The graph of the function f is the graph of the equation y = f(x), so the set of all
points in the plane (x, f(x)).
The Vertical Line Test. Each vertical line through a point in the domain of a function
meets its graph in exactly one point.
• The values of x where the value of f(x) makes a jump are called points of discon-
tinuity.

Parabolas
• The graph of a quadratic function of the form f(x) = ax2 + bx + c for a ≠ 0, is a
parabola.
• To draw, make a table of a few points and sketch.
• The size of the coefficient a determines the width of the parabola.
• The sign on a determines if the parabola opens up or down.
• Any quadratic equation can be manipulated to be of the form y - k = a(x - h)2 by
completing the square. It is now easy to see the vertex of the parabola is at (h, k).