Introduction to Functions and Graphs in Calculus

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Understanding points and coordinates is fundamental in calculus, as it allows us to describe locations and relationships in the plane R2.

Coordinates: A point in R2 is described by an ordered pair (x, y), where x and y are real numbers.
x-axis (horizontal) and y-axis (vertical) are used to plot points and analyze graphs.
For example, (1,2), (-3,4), (0,0), (2,1) are all points in R2.

Equations are used to describe sets of points and curves in the plane. The graph of an equation is the set of all points that satisfy it.

Linear equations: For example, y = 1 describes a horizontal line through y = 1, while x = 2 describes a vertical line through x = 2.
Quadratic equations: The equation y = x^2 + 1 describes a curve with a vertex at (0,1) and a radius 1.

The graph of a function is a particularly important example of a curve in R2. Every curve is not necessarily the graph of a function.

Given a function f: D → R, its graph is the set of all points (x, f(x)) for x ∈ D.
Informally, we plot the values of f(x) vertically as y coordinates against the corresponding values of x along the horizontal axis.

The vertical line test is used to determine if a curve is the graph of a function.

If any vertical line cuts the curve more than once, it is not the graph of a function.

Intercepts are key features of graphs.

x-intercepts: Points where the graph cuts or touches the x-axis. Solve f(x) = 0 to find them.
y-intercept: Point where the graph cuts or touches the y-axis. Evaluate f(0) to find it.

Linear functions have the form f(x) = ax + b. Their graphs are straight lines.

If a > 0, the line has positive slope and moves from bottom left to top right.
If a < 0, the slope is negative and the line moves from top left to bottom right.
If a = 0, the graph is a horizontal line at y = b.

Quadratic functions are of the form f(x) = ax^2 + bx + c.

If a > 0, the graph is a parabola opening upwards; if a < 0, it opens downwards.
The discriminant determines the number of distinct x-intercepts.

Other important functions include the square root and absolute value functions.

The square root function is defined for D = [0, ∞) and grows more slowly than x^2.
The absolute value function f(x) = |x| has a 'V' shape and is defined for all real numbers.

Cubic functions have the form f(x) = ax^3 + bx^2 + cx + d. Their graphs can have inflection points and change direction.

If a > 0, f(x) will be large and positive when x is large and positive, and large and negative when x is large and negative.
If a < 0, f(x) will be large and negative when x is large and positive, and large and positive when x is large and negative.

Function Type	General Form	Graph Shape
Linear		Straight line
Quadratic		Parabola
Cubic		S-shaped curve
Square Root		Curve starting at (0,0), increasing slowly
Absolute Value		V-shaped graph

A function f: D → R with domain D and range R assigns each input x ∈ D a unique output f(x) ∈ R.

If there is some x ∈ D with f(x) = b, then b is in the range of f.
The graph of f is the set of points (x, f(x)).
The number of times the line y = b cuts the graph is the number of solutions to f(x) = b.

Shifting graphs is a common transformation in calculus, allowing us to analyze changes in position without altering the shape.

The graph of y = f(x + a) is obtained by shifting y = f(x) a units to the left.
The graph of y = f(x - a) is obtained by shifting y = f(x) a units to the right.
The graph of y = f(x) + a is obtained by shifting y = f(x) a units up.
The graph of y = f(x) - a is obtained by shifting y = f(x) a units down.

These notes provide foundational concepts for calculus, including function definition, graphing, and transformations, which are essential for understanding limits, derivatives, and integrals in later chapters.