Honors Calculus I Chapter P: Prerequisites Section P.1: Lines in the Plane
Intercepts of a Graph The x-intercept is the point at which the graph crosses the x-axis. (a, 0) Let y = 0, and solve for x. The y-intercept is the point at which the graph crosses the y-axis. (0, b) Let x = 0, and solve for y.
Symmetry of a Graph A graph is symmetric with respect to the y-axis if, whenever (x, y) is a point on the graph, (-x, y) is also a point on the graph. A graph is symmetric with respect to the x-axis if, whenever (x, -y) is a point on the graph, (-x, y) is also a point on the graph. A graph is symmetric with respect to the origin if, whenever (x, y) is a point on the graph, (-x, -y) is also a point on the graph.
Tests for Symmetry The graph of an equation in x and y is symmetric with respect to the y-axis if replacing x by -x yields an equivalent equation. The graph of an equation in x and y is symmetric with respect to the x-axis if replacing y by -y yields an equivalent equation. The graph of an equation in x and y is symmetric with respect to the origin if replacing x by -x AND y by -y yields an equivalent equation.
Points of Intersection The points of intersection of the graphs of two equations is a point that satisfies both equations. Think substitution or elimination.
Honors Calculus I Chapter P: Prerequisites Section P.2: Linear Models and Rate of Changes
Slope of a Line Slope = Delta,, means change in Given two points in the plane:
Point Slope Form of a Linear Equation Given two points in the plane: Point-Slope Form: Find Slope & Cross-multiply Now, switch sides
Slope-Intercept from of a Linear Equation Slope-intercept form: m is the slope of the given line b is the y -intercept of the given line the point (0, b) is on the graph
Equations of special lines Vertical lines intersect the x -axis, therefore the equation of a vertical line is x = a Where a is the x -intercept x = 3 intersects the x -axis at 3 Horizontal lines intersect the y -axis, therefore the equation of a horizontal line is y = b Where b is the y -intercept y = 3 intersects the y -axis at 3 (& has a slope of 0)
Parallel and Perpendicular Lines Parallel lines never intersect, therefore they have the SAME SLOPE Perpendicular lines intersect at right angles, therefore they have OPPOSITE INVERSE SLOPES
Honors Calculus I Section P.3: Functions and Their Graphs
Function and Function Notation A relation is a set of ordered pairs ( x, y ). A function is a relation in which each x value is paired with exactly one y value. A function f(x) is read f of x The independent variable: x The domain is the set of all x The dependent variable: y The range is the set of all y
Equations An explicit form of an equation is solved for y or f(x) An implicit form of an equation is when the equation is not solved for (or cannot be solved for) y. It is implied.
Domain of Function The implied domain is the set of all real numbers for which the function is defined. Two considerations: The expression under an even root must be non-negative (positive or zero). The expression in the denominator cannot equal zero.
Domain of a Function For those two considerations: Set the expression under an even root 0. Set the expression in the denominator equal to zero to find out what the variable CANNOT be. The domain is everything else. Use interval notation to designate domain.
Range of a Function Think of the graph of the function and the intervals of y values related to the domain.
The Graph of a Function Identity Function Quadratic Function Cubic Function Square Root Function Absolute Value Function Rational Function Sine Function Cosine Function
Transformations of Functions Horizontal Shift to the Right: y = f(x c) Horizontal Shift to the Left: y = f(x + c) Vertical Shift Up: y = f(x ) + c Vertical Shift Down: y = f(x ) c Reflection about the x -axis: y = f(x ) Reflection about the y -axis: y = f( x ) Reflection about the origin : y = f( x )
Classifications of Functions Algebraic Functions: Polynomial Functions: expressed as a finite number of operations of x n Rational Functions: expressed as a fraction Radical Functions: expressed with a root Transcendental Functions: Trigonometric Functions: sine, cosine, tangent, etc.
Composite Functions Combination of functions such that the range of one function is the domain of the other. (f g)(x) = f(g(x)) (g f)(x) = g(f(x))
Even and Odd Functions An even function is symmetric with respect to the y- axis Test: substitute - x for x and get back the original function. An odd function is symmetric with respect to the origin. Test: substitute - x for x AND - y for y and get back the original function.