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Honors Calculus I
Chapter P: Prerequisites
Section P.1: Lines in the Plane
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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.
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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.
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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.
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Points of Intersection
The points of intersection of the graphs of two equations is a point that satisfies both equations.
Think substitution or elimination.
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Honors Calculus I
Chapter P: Prerequisites
Section P.2: Linear Models and Rate of Changes
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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
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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
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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
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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)
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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
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Honors Calculus I
Section P.3: Functions and
Their Graphs
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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
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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
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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.
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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.
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Range of a Function
Think of the graph of the function and the intervals of y values related to the domain.
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The Graph of a Function
Identity Function Quadratic Function Cubic Function Square Root Function Absolute Value Function Rational Function Sine Function Cosine Function
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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 )
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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.
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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))
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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.