Symmetry and Coordinate Graphs Section 3.1

Symmetry with Respect to the Origin Symmetric with the origin if and only if the following statement is true: F(-x)=-F(x)

Symmetric with Origin Example F(X) = x 5 Yes F(x) = x/(1-x) No

Symmetry (a,b) X-Axis Plug in (a,-b) Y-Axis Plug in (-a,b) Y=X Plug in (b,a) Y=-X Plug in (-b,-a)

Example Determine whether the graph of xy=-2 is symmetric with respect to the x axis, yaxis, the line y=x, and the line y=-x, or none of these? First plug in (a,b) Ab=-2 Symmetric with both line y=x and line y=- x

Example Determine whether the graph of │ y │ = │ x │ +1 is symmetric with respect to the x axis, yaxis, both or neither? Symmetric with both the x and y axis.

Even and Odd Functions Even  Symmetric with respect to Y axis F(-x)=F(x) Odd  Symmetric with respect to the origin F(-x)=-F(x) F(X) = x 5 Odd F(x) = x/(1-x) Neither odd nor even

Even and Odd Functions Which lines are lines of symmetry for the graph of x 2 =1/y 2 X and y axises, y=x, and y=-x Is the following function symmetric about the origin? F(X)=-7x 5 + 8x Yes, does this mean it’s even or odd? Odd

Families of Graphs Section 3.2

Parent Graphs  Constant

Parent Graphs

Example Graph f(x) = x2 and g(x) = - x2. Describe how the graphs of f(x) and g(x) and are related. xf(x) = x 2 g(x) = -x

Changes to Parent Graph Graph Parent Graph of f(x)=|x| Graph Parent Graph of f(x)=|x| Graph f(x)=|x|+1 Graph f(x)=|x|+1 Graph f(x) = |x|-1 Graph f(x) = |x|-1 Graph f(x)=|x+1| Graph f(x)=|x+1| Graph f(x) = |x-1| Graph f(x) = |x-1| On same graph On same graph Similarities/Differences? Similarities/Differences?

Change to Parent Graph Reflections Reflections Y=-f(x) Y=-f(x) Outside the H  Vertical Axis Outside the H  Vertical Axis Reflected over the x-axis Reflected over the x-axis Y=f(-x) Y=f(-x) Inside the H  Horizontal Axis Inside the H  Horizontal Axis Reflected over the y-axis Reflected over the y-axis

Change to Parent Graph Translations Translations +,- OUTSIDE of Function +,- OUTSIDE of Function  Outside the H  Vertical Movement  Outside the H  Vertical Movement SHIFTS UP AND DOWN +,- INSIDE of Function +,- INSIDE of Function  Inside the H  Horizontal Movement  Inside the H  Horizontal Movement SHIFTS LEFT AND RIGHT

Change to Parent Graph Dilations Dilations X/÷ OUTSIDE of Function X/÷ OUTSIDE of Function  Outside the H  Vertical Movement  Outside the H  Vertical MovementExpands/Compresses X/÷ INSIDE of Function X/÷ INSIDE of Function  Inside the H  Horizontal Movement  Inside the H  Horizontal MovementExpands/Compresses

Examples - Use the parent graph y = x 2 to sketch the graph of each function. y = x This function is of the form y = f(x) + 1. Outside the H  Vertical Movement Since 1 is added to the parent function y = x 2, the graph of the parent function moves up 1 unit. a.

Examples - Use the parent graph y = x 2 to sketch the graph of each function. y = (x - 2) 2 Inside the H  Horizontal Movement This function is of the form y = f(x - 2). Since 2 is being subtracted from x before being evaluated by the parent function, the graph of the parent function y = x 2 slides 2 units right. a.

Examples - Use the parent graph y = x 2 to sketch the graph of each function. y = (x + 1)2 – 2 This function is of the form y = f(x + 1)2 -2. The addition of 1 indicates a slide 1 unit left, and the subtraction of 2 moves the parent function y = x2 down two units. a.

EXAMPLES make table and graph

Graphs of Nonlinear Inequalities Section 3.3

Determine which are solutions Determine whether (3, 4), (11, 2), (6, 5) and (18, -1) are solutions for the inequality y ≥√x-2) + 3. Of these ordered pairs, (3, 4) and (6, 5) and are solutions for y ≥√x-2) + 3.

Example Determine whether (-2,5) (3,-1) (-4,2) and (-1,-1) are solutions for the inequality y ≥ 2x 3 +7 (-2,5) and (-4,2) are solutions

Graph y ≤ (x - 2) To verify numerically, you can test a point notin the boundary. It is common to test (0, 0) whenever it is not on the boundary Since the boundary is included in the inequality, the graph is drawn as a solid curve.

Graph y < -2 - |x - 1|. y < -2 - |x - 1|  y < -|x - 1| - 2 The boundary is not included, so draw it as a dashed line Verify by substituting (0, 0) in the inequality to obtain 0 < -3. Since this statement is false, the part of the graph containing (0, 0) should not be shaded. Thus, the graph is correct.

Solving Absolute Inequalities Solve |x + 3| - 4 < 2. There are two cases that must be solved. In one case, x + 3 is negative, and in the other, x + 3 is positive. Case 1 (x + 3)< 0 |x + 3| - 4< 2 -(x + 3) - 4< 2|x + 3| = -(x + 3) -x < 2 -x< 9 x> -9 Case 2 (x + 3)> 0 |x + 3| - 4< 2 x < 2|x + 3| = (x + 3) x - 1< 2 x< 3 The solution set is {x | -9 < x < 3}. {x | -9 < x < 3} is read as “the set of all numbers x such that x is between - 9 and 3.

Solving Absolute Inequalities Solve |x -2| - 5 < 4. -(x-2)-5<4 (x-2)-5<4 Solution Set {x |-7<x<11}

Inverse Relations Two relations are inverse relations if and only if one relation contains the element (b,a) whenever the other relation contains the element (a,b). F(x) – denotes function F(x) -1 – denotes inverse Ex. Graph f(x) = -1/2 l x l + 3 and its inverse

Horizontal Line Test A test used to determine if the inverse of a relation will be a function If every horizontal line intersects the graph of the relation in at most one point, then the inverse of the relation is a function Is the inverse of the below relation a function? Ex. F(x) = 3X + 4 Yes Ex. F(x) = (x + 3) No

How do you find the inverse? To find the inverse of a function, let y= f(x) and interchange x and y. Then solve for y. Ex. F(x) = (x + 3) F(x) -1 = (x+5) 1/2 Ex. F(x) = 1/(x) 3 F(x) -1 =1/(x) 1/3

Inverse Functions Two functions, f and f -1 are inverse functions if and only if (f 0 f -1 )(x) = (f -1 0 f)(x)=x. Ex. Given F(x) = 4x – 9, find f -1 and verify that f and f -1 are inverse functions. F(x) -1 = (x+9)/4 Verify: (f 0 f -1 )(x) = (f -1 0 f)(x)=x. Ex. F(x) = 3x F(x) -1 =((x-7)/3) ½ Verify: (f 0 f -1 )(x) = (f -1 0 f)(x)=x.

Section 3.5

Discontinuous A function is discontinuous if you can not trace it without lifting your pencil. Infinite Discontinuity – as the graph of f(x) approaches a given value of x, lf(x)l becomes increasingly greater Jump Discontinuity – The graph of f(x) stops and then begins again with an open circle at a different range value for a given value of the domain. Point Discontinuity – When there is a value in the domain for which f(x) is undefined, but the pieces of the graph match up Everywhere Discontinuous – A function that is impossible to graph in the real number system is said to be everywhere discontinuous

Continuous Test A function is continuous at x = c if it satisfies the following conditions: The function is defined at c, in order words F(c) exists The function approaches the same y-value on the left and right sides of x = c; and The y-value that the function approaches from each side is F(c) Examples: f(x) = 3x 2 +7; x = 1 F(x) = (x-2)/(x 2 -4); x = 2 F(x) = 1/x if x>1, and x if x <=1; x = 1

Continuity on an Interval A function f(x) is continuous on an interval if and only if it is continuous at each number x in the interval. F(x) = 3x-2 if x >2, 2 – x if x <=2 For 3<x<5 For 1<x<3

End Behavior The behavior of f(x) as lxl becomes very large Describe the end behavior of f(x) = -x 3 and g(x) = -x 3 + x 2 –x + 5 End Behavior Chart of Polynomial Functions pg. 163

Increasing, Decreasing, and Constant Functions A function f is increasing on an interval I if and only if for every a and b contained in I, f(a) < f(b) whenever a < b. A function f is decreasing on an interval I if and only if for every a and b contained in I, f(a)> F(b) whenever a < b. A function f remains constant on an interval I if and only if for every a and b contained in I, f(a) = f(b) whenever a < b

Monotonicty A function is said to be monotonic on an interval I if and only if the function is increasing on I or decreasing on I.

Examples Graph f(x) = 3- (x-5) 2 Graph f(x) = ½ lx+3l – 5 Graph f(x) = 2x 3 + 3x x+ 3

Section 3.6

Critical Points Points on a graph at which a line drawn tangent to the curve is horizontal or vertical. Three types of critical points: Maximum: When the graph is increasing to the left of x = c and decreasing to the right of x = c Minimum: When the graph of a function is decreasing to the left of x = c and increasing ot the right of x = c Point of Inflection: a point where the graph changes its curvature.

Critical Points Point of Inflection MinimumMaximum

Critical Points Extremum – a minimum or maximum value of a function Relative Extremum – a point that represents the maximum or minimum for a certain interval Absolute Maximum – the greatest value that a function assumes over its domain Relative Maximum – a point that represents the maximum for a certain interval Absolute Minimum – the least value that a function assumes over its domain Relative Minimum – a point that represents the minimum for a certain interval

Critical Points

Examples Graph the following examples and pick out the critical points F(x) = 5x 3 -10x 2 – 20x + 7 Use trace to find relative min and max, and then use calc functions to find max and min F(x) = 2x 5 -5x 4 –10x 3 has the critical points x = -1, 0, and 3. Determine whether each of these critical points is the location of a max, min, or point of inflection. Solve by graphing or algebraically

Review for Test Section 3.4 Inverse Relations and Functions Graph Inverse based on relation graph Find Inverse Verify that the Inverse is correct Section 3.5 Continuity and End Behavior Describe continuity and the end behavior of a function by graphing it Section 3.6 Locate critical points Identify the critical points as max, min, or point of inflection

Section 3.7

Rational Function A rational function is a quotient of two polynomial functions. F(x) = g(x)/h(x); where h(x) does not equal 0. Parent rational graph is f(x) = 1/x.

Asymptotes A line that a graph approaches but never intersects Two main types Vertical Asymptotes: The line x = a is a vertical asymptote for a function f(x) if f(x) approaches infinity or negative infinity as x approaches a from either the left or the right. Horizontal Asymptotes: The line y=b is a horizontal asymptote asymptote for a function f(x) if f(x) approaches b as x approaches infinity or negative infinity. Examples: f(x) = (3x-1)/(x-2) Vertical Asymptotes – look at domain Horizontal Asymptotes, solve for X or divide both top and bottom by highest power of X.

Transformations of 1/x G(x) = 1/(x+5) Translates 5 units to the left Moves vertical asymptote to x = -5, keeps y = 0 H(x) = -1/(2X) Reflects over the x-axis, compresses horizontally by 2 Keeps asymptotes at x = 0 and y = 0 K(x) = 4/(x-3) Stretches graph vertically by 4, translates 3 units to the right Moves vertical asymptote to x = 3, keeps y = 0 M(x) = (-6/(x+2) )- 4 Reflects over the x axis, stretch vertically by 6, and translates 2 units to the left and 4 units down Moves vertical asymptote to x = -2 and y = -4

Asymptotes Slant Asymptotes – The line l is a slant asymptote for a function f(x) if the graph of y = f(x) approaches l as x approaches infinity or negative infinity. Occur when the degree of the numerator of a rational function is exactly one greather than that of the denominator F(x) = (2x 3 -3x + 1)/(x-2) No slant asymptote F(x) = [(x+3)(x+1)]/[x(x+3)(X-2)] Simplify to (x+1)/[x(x-2)]; x can not equal -3

Section 3.8

Direct Variation Y varies directly as x n if there is some nonzero constant k such that y = kx n, n>0, where k is called the constant of variation Example: Suppose y varies directly as x and y = 27 when x = 6. Find the constant of variation and write an equation of the form y = kx n Y=4.5x Use the equation to find the value of y when x = 10. When x = 10, the value of y is 45. Solving using proportions, y 1 = kx 1 n and y 2 = kx 2 n Example: If y varies directly as the cube of x and y = when x = 3, find x when y = -540 When y = -540, the value of x is 6.

Inverse Variation Y varies inversely as x n if there is some nonzero constant k such that x n y=k or y = k/ x n, n>0. Example: If y varies inversely as x and y = 21 when x = 15, find x when y = 12. Use a proportion to relate the values, x 1 n y 1 =k and x 2 n y 2 =k When y = 12, the value of x is Example: If y varies inversly as x and y =14 and x =3, find x when y = 30. When y = 30, the value of x is 1.4

Joint Variation Y varies jointly as x n and z n if there is some nonzero constant k such that y= k x n z n, where x and z do not equal 0, and n >0 Example: The volumne V of a cone varies jointly as the height h and the square of the radius r of the base. Find the equation for the volume of a cone with height 6 cm and base diameter 10 cm that has a volume of 50л cubic cm. V=k*h*r 2 V= л /3hr 2; k = л /3