Honors Calculus I Chapter P: Prerequisites Section P.1: Lines in the Plane.
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Transcript of Honors Calculus I Chapter P: Prerequisites Section P.1: Lines in the Plane.
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:
rise
run
y
x
m y2 y1
x2 x1
x1,y1 x2, y2
Point Slope Form of a Linear Equation
Given two points in the plane:
Point-Slope Form:
m y y1
x x1
m x x1 y y1
y y1 m x x1
x1,y1 x,y 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
y mx b
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
m1 m2
m1 1
m2
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.
y 2
3x 5
x 2 y 2 16
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 xn
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.