Chapter 1 Summary. x2 +x y 1

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1 Chapter Summary. Distance Formula The distance between two points P = (x,y ) and P = (x,y ) in the xy-plane is Midpoint Formula P P = (x x ) +(y y ) The midpoint between two points P = (x,y ) and P = (x,y ) in the xy-plane is ( x +x (x,y) =, y ) +y. Intercepts To find x-intercepts, plug in y = 0 and solve for x. To find y-intercepts, plug in x = 0 and solve for y. Symmetry Given a point (x,y) on the graph of an equation: The graph is said to be symmetric about the x-axis if the point (x, y) is also on the graph. To test, replace y with y throughout the equation; if it simplifies back to the original equation, then it is symmetric about the x-axis. The graph is said to be symmetric about the y-axis if the point ( x,y) is also on the graph. To test, replace x with x throughout the equation; if it simplifies back to the original equation, then it is symmetric about the y-axis. The graph is said to be symmetric about the origin if the point ( x, y) is also on the graph. To test, replace x with x and y with y throughout the equation; if it simplifies back to the original equation, then it is symmetric about the origin. Equation of a Circle An equation of the circle with center (h,k) and radius r is (x h) +(y k) = r In particular, if the center is at the origin (0,0), this equation becomes x +y = r [Note: if you take the square root of both sides of this equation, you get the distance formula: the distance between the center (h,k) and a point (x,y) on the circle is r.]

2 . Slope of a Line The slope of a nonvertical line passing through the points P = (x,y ) and P = (x,y ) is m = rise run = y y x x = y y x x If we try using this formula with two points on a vertical line we get division by zero since x = x. Slope-Intercept Form of a Line The slope-intercept form of a [nonvertical] line with slope m and y-intercept b is y = mx+b Horizontal lines have slope m = 0, so in that case the equation becomes y = b Vertical lines can t be written in this form. The equation of a vertical line is Point-Slope Form of a Line x = a An equation of the line with slope m that passes through the point (x,y ) is y y = m(x x ) We ll often start with the point-slope form and then simplify to point-intercept. Note: To find the equation of a line, you need two things: Either the slope of the line and a point on the line or two points on the line (once you use them to find the slope you re back in the first case). Parallel and Perpendicular Lines. Two nonvertical lines are said to be parallel if and only if they have the same slope. (All vertical lines are parallel to each other.). Two lines with slopes m and m (both nonvertical) are said to be perpendicular if and only if m m =, i.e., if their slopes are negative reciprocals: m = m or, equivalently, m = m (A parallel line and a horizontal line will always be perpendicular to each other)..4 and.5 Functions A function f from a set A to a set B is a relation that assigns each element of A to exactly one element of the set B. The set A of inputs of is called the domain. The set of possible outputs of a function is called its range; the range will be a subset of B (but need not be the entire set).

3 Representing Functions We can represent functions in several ways:. Verbally.. Numerically (such as with a table of values or a set of input/output pairs, e.g., {(,), (,), (4,5)}).. Graphically. 4. Algebraically (with an equation). The variable that corresponds to the inputs is called the independent variable, and the variable that corresponds to the outputs is called the dependent variable. Testing for Functions. For a function described verbally: Does each input go to only one output?. For a function described numerically: Does each input appear with only one output?. For a function described graphically: Does the graph pass the vertical line test? Vertical Line Test: A graph represents a function if and only if we cannot draw a vertical line that touches/crosses the graph more than once. 4. For a function described algebraically: Can we show that each choice of the independent variable will yield only one corresponding value for the dependent variable? Domain of a Function The [implied] domain of a function is the set of all possible inputs (sometimes the domain is restricted to a smaller subset of these values). Difference Quotient The difference quotient of a function y = f(x) is Average Rate of Change f(x+h) f(x), h 0 h The average rate of change of a function y = f(x) over the interval [x,x ] (or between two points (x,f(x )) and (x,f(x )) ) is f(x ) f(x ) x x Graphically, the difference quotient can be thought of as representing the average rate of change (where h = x x ). Zeros/Roots The zeros or roots of a function f are all the x-values for which f(x) = 0. Even and Odd Functions A function f is even if and only if f( x) = f(x). A function f is odd if and only if f( x) = f(x).

4 .6 Library of Parent Functions (a) Constant, y = c (b) Identity, y = x (c) Absolute Value, y = x 4 (d) Square Root, y = x (e) Quadratic, y = x (f) Cubic, y = x (g) Reciprocal, y = x (h) Greatest Integer, y = x (i) Cube Root, y = x 4

5 .7 Rigid Transformations Given a function y = f(x) and a constant c > 0, we have the following: Shifting: y = f(x)+c shifts the graph up c units (add c to y-values). y = f(x) c shifts the graph down c units (subtract c from y-values). y = f(x+c) shifts the graph left c units (subtract c from x-values, replace x with x+c in the function). y = f(x c) shifts the graph right c units (add c to x-values, replace x with x c in the function). Reflecting: y = f(x) reflects graph about the x-axis (multiply all y values by, i.e., multiply the entire function by ). y = f( x) reflects graph about the y-axis (multiply all x values by, i.e., replace x with x in the function). Nonrigid Transformations Stretching and Shrinking: Given a function y = f(x) and a constant c > : y = c f(x) gives a vertical stretch by a factor of c (multiply y-values by c, i.e., multiply function by c). y = c f(x) gives a vertical shrink by a factor of c (divide y-values by c, i.e., divide function by c). y = f(c x) gives a horizontal shrink by a factor of c function). (divide x-values by c, i.e., replace x with c x in the y = f( c x) gives a horizontal stretch by a factor of c (multiply x-values by c, i.e., replace x with c x in the function)..8 Arithmetic Combinations of Functions If f(x) and g(x) are functions whose domains overlap, then for all x in the intersection of their domains we can define the following combinations of f and g: Sum: (f +g)(x) = f(x)+g(x) Difference: (f g)(x) = f(x) g(x) Product: (fg)(x) = f(x) g(x) Quotient: ( f g ) (x) = f(x) g(x), g(x) 0 Compositions of Functions Given functions f(x) and g(x) where the range of f is contained in the domain of g, the composition of f with g or f composed with g is (f g)(x) = f (g(x)) 5

6 .9 Inverse Functions Informally, the inverse of a function f(x) is a function that undoes whatever f(x) does. In order for a function to have an inverse, it must be one-to-one (often written - ): A function f(x) is - if and only if each output comes from only one input, i.e., if f(x ) = f(x ) then x = x. Graphically, to test to whether a function is - we use the Horizontal Line Test: The graph of a function is - if and only if we cannot draw a horizontal line that touches/crosses the graph more than once. Here is part of the book s definition of an inverse function: Let f and g be functions such that for every x in the domain of g and f (g(x)) = x g(f(x)) = x for every x in the domain of f. Under these conditions, the function g is the inverse function of the function f. The function g is denoted by f (read f-inverse ). How to find the inverse of a function:. Determine whether the function is (at least on the domain given). If not, then either the domain has to be restricted (usually told in the problem that this is the case) or you can t find the inverse.. Interchange the xs and ys, i.e., instead of y = f(x) now write x = f(y).. Solve for y in this new equation. The result will be f (x). 4. Check your work by composing the function and its inverse (both ways) and/or by comparing the domain and range. Facts About Inverse Functions (f f )(x) = x and (f f)(x) = x. The domain of f is the range of f, and the range of f is the domain of f. The graphs of f and f are symmetric about the line y = x. Note: f (x) is not the same as f(x)! If we wanted to denote f(x) we would have to write (f(x)). 6