Ch 1 Graphs y axis Quadrant II Quadrant

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Ch 1: Graphs y axis Quadrant II (-, +) Quadrant I (+, +) y

Ch 1: Graphs y axis Quadrant II (-, +) Quadrant I (+, +) y - $$ in thousands Origin (0, 0) -6 -2 (-6, -3) Quadrant III (-, -) 2 4 6 (6, 0) x axis (5, -2) y intercept (0, -3) x Yrs x intercept Quadrant IV (+, -) When distinct points are plotted as above the graph is called a scatter plot – ‘points that are scattered about’ A point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y) Graphs represent trends in data. For example: x – number of years in business y – thousands of dollars of profit Equation : y = ½ x – 3

1. 1 Distance & Midpoint y Things to know: 1. Find distance or midpoint

1. 1 Distance & Midpoint y Things to know: 1. Find distance or midpoint given 2 points 2. Given midpoint and 1 point, find the other point Origin (0, 0) -6 A point in the x/y coordinate plane is described by an ordered pair of coordinates (x, y) The Distance Formula To find the distance between 2 points (x 1, y 1) and (x 2, y 2) d = (x 2 – x 1)2 + (y 2 – y 1)2 -2 2 4 6 (-6, -3) x (5, -2) The Midpoint Formula To find the coordinates of the midpoint (M) of a segment given segment endpoints of (x 1, y 1) and (x 2, y 2) M x 1 + x 2, y 1 + y 2 2 2

1. 1 & 1. 2 Linear Equations The graph of a linear equation is

1. 1 & 1. 2 Linear Equations The graph of a linear equation is a line. A linear function is of the form y = mx + b, where m and b are constants. y y = 3 x + 2 y = 3 x + 5 x y = -2 x – 3 x y = (2/3)x -1 y=4 6 x + 3 y = 12 All of these equations are linear. Three of them are graphed above. x y=3 x+2 x y=2/3 x – 1 0 2 0 -1 1 5 3 1

X and Y intercepts Equation: y = ½ x – 3 y (6, 0)

X and Y intercepts Equation: y = ½ x – 3 y (6, 0) y intercept (0, -3) x x intercept -3 The y intercept happens where y is something & x = 0: (0, ____) Let x = 0 and solve for y: y = ½ (0) – 3 = -3 6 0) The x intercept happens where x is something & y = 0: (____, Let y = 0 and solve for x: 0 = ½ x – 3 => 3 = ½ x => x = 6

Slope is the ratio of RISE (How High) = RUN (How Far) y 2

Slope is the ratio of RISE (How High) = RUN (How Far) y 2 – y 1 y (Change in y) x 2 – x 1 x (Change in x) Slope = 5 – 2 1 -0 y =3 Slope = 1 – (-1) 3– 0 Things to know: 1. Find slope from graph 2. Find a point using slope 3. Find slope using 2 points 4. Understand slope between 2 points is always the same on the same line =2 3 x y = mx + b m = slope b = y intercept x y=3 x+2 x y=2/3 x – 1 0 2 0 -1 1 5 3 1

The Possibilities for a Line’s Slope (m) Negative Slope Positive Slope y y m>0

The Possibilities for a Line’s Slope (m) Negative Slope Positive Slope y y m>0 Line rises from left to right. Example: y=½x+2 y y m<0 x Undefined Slope Zero Slope m is undefined m=0 x x x Line falls from left to right. Line is horizontal. Example: y = -½ x + 1 Example: y=2 Line is vertical. Example: x=3 Question: If 2 lines are parallel do you know anything about their slopes? Things to know: 1. Identify the type of slope given a graph. 2. Given a slope, understand what the graph would look like and draw it. 3. Find the equation of a horizontal or vertical line given a graph. 4. Graph a horizontal or vertical line given an equation 5. Estimate the point of the y-intercept or x-intercept from a graph.

Linear Equation Forms (2 Vars) Standard Form Example: 6 x + 3 y =

Linear Equation Forms (2 Vars) Standard Form Example: 6 x + 3 y = 12 Slope Intercept Form Example: y = - ½ x - 2 Point Slope Form Ax + By = C A, B, C are real numbers. A & B are not both 0. Things to know: 1. Graph using x/y chart 2. Know this makes a line graph. y = mx + b m is the slope b is the y intercept Things to know: 1. Find Slope & y-intercept 2. Graph using slope & y-intercept 3. Application meaning of of slope & intercepts y – y 1 = m(x – x 1) Example: Write the linear equation through point P(-1, 4) with slope 3 y – y 1 = m(x – x 1) Things to know: y – 4 = 3(x - - 1) 1. Change from point slope to/from other forms. y – 4 = 3(x + 1) 2. Find the x or y-intercept of any linear equation

Parallel and Perpendicular Lines & Slopes Things to know: 1. Identify parallel/non-parallel lines. PARALLEL

Parallel and Perpendicular Lines & Slopes Things to know: 1. Identify parallel/non-parallel lines. PARALLEL • Vertical lines are parallel • Non-vertical lines are parallel if and only if they have the same slope y=¾x+2 y = ¾ x -8 PERPENDICULAR Same Slope Things to know: 1. Identify (non) perpendicular lines. 2. Find the equation of a line parallel or perpendicular to another line through a point or through a y-intercept. • Any horizontal line and vertical line are perpendicular • If the slopes of 2 lines have a product of – 1 and/or are negative reciprocals of each other then the lines are perpendicular. y= ¾ x+2 Negative reciprocal slopes y = - 4/3 x - 5 3 • -4 = 4 3 -12 = -1 12 Product is -1

Practice Problems 1. Find the slope of a line passing through (-1, 2) and

Practice Problems 1. Find the slope of a line passing through (-1, 2) and (3, 8) 2. Graph the line passing through (1, 2) with slope of - ½ 3. Is the point (2, -1) on the line specified by: y = -2(x-1) + 3 ? 4. Parallel, Perpendicular or Neither? 3 y = 9 x + 3 and 6 y + 2 x = 6 5. Find the equation of a line parallel to y = 4 x + 2 through the point (-1, 5) 6. Find the equation of a line perpendicular to y = - ¾ x – 8 through point (2, 7) 7. Find the equation of a line passing through the points (-2, 1) and (3, 7) 8. Graph (using an x/y chart – plotting points) and find intercepts of any equation such as: y = 2 x + 5 or y = x 2 – 4

Symmetry and Odd/Even Functions Y-Axis Symmetry even functions f (-x) = f (x) For

Symmetry and Odd/Even Functions Y-Axis Symmetry even functions f (-x) = f (x) For every point (x, y), the point (-x, y) is also on the graph. Test for symmetry: Replace x by –x in equation. Check for equivalent equation. Origin Symmetry odd functions f (-x) = -f (x) For every point (x, y), the point (-x, -y) is also on the graph. Test for symmetry: Replace x by –x , y by –y in equation. Check for equivalent equation. X-Axis Symmetry (For every point (x, y), the point (x, -y) is also on the graph. ) Test for symmetry: Replace y by –y in equation. Check for equivalent equation. y = x 3 Origin Symmetry Test -y = (-x)3 -y = -x 3 y = x 2 ODD) Y-axis Symmetry (EVEN) Symmetry Test y = (-x)2 y = x 2 x = y 2 X-axis Symmetry Test x = (-y)2 X = y 2 Try these without Using a graph: y = 3 x 2 – 2 y = x 2 + 2 x + 1

A Rational Function Graph & Symmetry y= 1 x x -2 -1 -1/2 0

A Rational Function Graph & Symmetry y= 1 x x -2 -1 -1/2 0 ½ 1 2 y -1/2 -1 -2 Undefined 2 1 ½ Intercepts: No intercepts exist If y = 0, there is no solution for x. If x = 0, y is undefined The line x = 0 is called a vertical asymptote. The line y = 0 is called a horizontal asymptote. Symmetry: y = 1/-x => No y-axis symmetry -y = 1/-x => y = 1/x => origin symmetry -y = 1/x => y = -1/x => no x-axis symmetry

1. 3 Functions and Graphs Year 1997 1998 1999 2000 $3111 $3247 $3356 $3510

1. 3 Functions and Graphs Year 1997 1998 1999 2000 $3111 $3247 $3356 $3510 Cost The cost depends on the year. independent variable (x) dependent variable (y) The table above establishes a relation between the year and the cost of tuition at a public college. For each year there is a cost, forming a set of ordered pairs. A relation is a set of ordered pairs (x, y). The relation above can be written as 4 ordered pairs as follows: S = {(1997, 3111), (1998, 3247), (1999, 3356), (2000, 3510)} x y x y Domain – the set of all x-values. D = {1977, 1998, 1999, 2000} Range – the set of all y-values. R = {3111, 3247, 3356, 3510} Thinking Exercise: Draw a ‘line’ in the x/y axes. What is the Domain & Range? Year(x) 1997 1998 1999 2000 Cost(y) 3111 3247 3356 3510

Input x Functions & Linear Data Modeling y – Profit in thousands of $$

Input x Functions & Linear Data Modeling y – Profit in thousands of $$ (Dependent Var) Function f Output y=f(x) (6, 0) y intercept (0, -3) x - Years in business (Independent Var) x intercept Equation: y = ½ x – 3 Function: f(x) = ½ x – 3 A function has exactly one output value (y) for each valid input (x). x 0 2 6 8 Use the vertical line test to see if an equation is a function. • If it touches 1 point at a time then FUNCTION • If it touches more than 1 point at a time then NOT A FUNCTION. y = f(x) -3 f(0) = ½(0)-3=-3 -2 f(2) = ½(2)-3=-2 0 f(6) = ½(6)-3=0 1 f(8) = ½(8)-3=1

Diagrams of Functions A function is a correspondence fro the domain to the range

Diagrams of Functions A function is a correspondence fro the domain to the range such that each element in the domain corresponds to exactly one element in the range. f Function: f(x) = ½ x – 3 x 0 2 6 8 y = f(x) -3 f(0) = ½(0)-3=-3 -2 f(2) = ½(2)-3=-2 0 f(6) = ½(6)-3=0 1 f(8) = ½(8)-3=1 -3 -2 0 1 0 2 6 8 f 1 2 4 4 5 5 6 3 A function NOT a function

How to Determine if an equation is a function Graphically: Use the vertical line

How to Determine if an equation is a function Graphically: Use the vertical line test Symbolically/Algebraically: Solve for y to see if there is only 1 y-value. Example 1: x 2 + y = 4 Example 2: x 2 + y 2 = 4 y = 4 – x 2 y 2 = 4 – x 2 For every value of x there Is exactly 1 value for y, so This equation IS A FUNCTION. y = 4 – x 2 or y = - 4 – x 2 For every value of x there are 2 possible values for y, so This equation IS NOT A FUNCTION.

Are these graphs functions? Use the vertical line test to tell if the following

Are these graphs functions? Use the vertical line test to tell if the following are functions: y = x 3 Origin Symmetry y = x 2 Y-axis Symmetry x = y 2 X-axis Symmetry

More on Evaluation of Functions f(x) = x 2 + 3 x + 5

More on Evaluation of Functions f(x) = x 2 + 3 x + 5 Evaluate: f(2) = (2)2 + 3(2) + 5 f(2) = 4 + 6 + 5 f(2) = 15 Evaluate: f(x + 3) = (x + 3)2 + 3 (x + 3) + 5 f(x + 3) = (x + 3) + 3 x + 9 + 5 f(x + 3) = (x 2 + 3 x + 9) + 3 x + 14 f(x + 3) = (x 2 + 6 x + 9) + 3 x + 14 f(x + 3) = x 2 + 9 x + 23 Evaluate: f(-x) f (-x) = ( -x)2 + 3( -x) + 5 f (-x) = x 2 - 3 x + 5

More on Domain of Functions A function’s domain is the largest set of real

More on Domain of Functions A function’s domain is the largest set of real numbers for which the value f(x) is a real number. So, a function’s domain is the set of all real numbers MINUS the following conditions: • specific conditions/restrictions placed on the function • Bounds relating to real-life data modeling (Example: y = 7 x, where y is dog years and x is dog’s age) • values that cause division by zero • values that result in an even root of a negative number What is the domain the following functions: 1. f(x) = 6 x x 2 – 9 2. g(x) = 3 x + 12 3. h(x) = 2 x + 1

Slope & Average Rate of Change y - $$ in thousands y=½x– 3 (6,

Slope & Average Rate of Change y - $$ in thousands y=½x– 3 (6, 0) y = x 2 - 4 x + 4 x Yrs (0, -3) The slope of a line may be interpreted as the rate of change. The rate of change for a line is constant (the same for any 2 points) y 2 – y 1 x 2 – x 1 Non-linear equations do not have a constant rate of change. But you can Find the average rate of change from x 1 to x 2 along a secant to the graph. f(x 2) – f(x 1) x 2 – x 1 See Page 38 -39 for more examples.

Definition of a Difference Quotient The average rate of change for f(x) is called

Definition of a Difference Quotient The average rate of change for f(x) is called the “difference quotient” and is defined below. (This is an important concept in calculus – it becomes the mathematical definition of the derivative you will learn about next semester. Example: Find the difference quotient for : f(x) = 2 x 2 -3 f(x + h) = 2(x + h )2 - 3 = 2(x + h) -3 = 2(x 2 + 2 xh + h 2 ) -3 = 2 x 2 + 4 xh + 2 h 2 -3 = = = So, the difference quotient is: 2 x 2 + 4 xh + 2 h 2 -3 – (2 x 2 -3) h 2 x 2 + 4 xh + 2 h 2 -3 – 2 x 2 + 3 h 4 xh + 2 h 2 h 4 x + 2 h

1. 4 Increasing, Decreasing, and Constant Functions A function is increasing on an interval

1. 4 Increasing, Decreasing, and Constant Functions A function is increasing on an interval if for any x 1, and x 2 in the interval, where x 1 < x 2, then f (x 1) < f (x 2). A function is decreasing on an interval if for any x 1, and x 2 in the interval, where x 1 < x 2, then f (x 1) > f (x 2). A function is constant on an interval if for any x 1, and x 2 in the interval, where x 1 < x 2, then f (x 1) = f (x 2). (x 2, f (x 2)) (x 1, f (x 1)) Increasing f (x 1) < f (x 2) (x 1, f (x 1)) (x 2, f (x 2)) Decreasing f (x 1) > f (x 2) Constant f (x 1) = f (x 2)

More Examples a. b. 5 5 4 4 3 3 2 1 1 -5

More Examples a. b. 5 5 4 4 3 3 2 1 1 -5 -4 -3 -2 -1 -1 -2 1 2 3 4 5 -5 -4 -3 -2 -1 -1 -2 -3 -3 -4 -5 Observations • Decreasing on the interval (-oo, 0) 1 2 3 4 5 Observations a. Two pieces (a piecewise function) b. Constant on the interval (-oo, 0). • • Increasing on the interval (0, 2) Decreasing on the interval (2, oo). c. Increasing on the interval (0, oo). Challenge Yourself: What might be the definition of the piecewise function for this graph? (You will learn about these Later. Can you guess what it might be? )

f(x) = sin (x) x 0 /2 The point at which a function changes

f(x) = sin (x) x 0 /2 The point at which a function changes its increasing or decreasing behavior is called a relative minimum or relative maximum. 3 /2 y 2 2 Relative (local) Min & Max (90, f(90)) 1 y 0 1 0 -1 0 f(90), or 1, is a local max x 0 90 -1 -2 A function value f(a) is a relative maximum of f if there exists an open interval about a such that f(a) > f(x) for all x in the open interval. 180 270 360 (270, f(270)) f(270), or -1, is a local min A function value f(b) is a relative minimum of f if there exists an open interval about b such that f(b) < f(x) for all x in the open interval.

1. 4 Library of Functions/Common Graphs y=c y=x y = x 2 x x

1. 4 Library of Functions/Common Graphs y=c y=x y = x 2 x x y = x 3 y= x y = |x| x y = 1/x x x y = x 1/3 x x

Step Function Application Example y = int(x) f(x) = int(x) or y = [[x]]

Step Function Application Example y = int(x) f(x) = int(x) or y = [[x]] (Greatest Integer Function) y – Tax (+) or Refund (-) in thousands of $$ x – Income in $10, 000’s Find: 1) f (1. 06) 2) f (1/3) 3) f (-2. 3) • What other applications of the step function can you think of?

Piecewise Functions A function that is defined by two (or more) equations over a

Piecewise Functions A function that is defined by two (or more) equations over a specified domain is called a piecewise function. f(x) = x 2 + 3 5 x + 3 if x < 0 if x>=0 f(-5) = (-5)2 + 3 = 25 + 3 = 28 f(6) = 5(6) + 3 = 33 See Page 247 for more examples

1. 5 Transformation of Functions A transformation of a graph is a change in

1. 5 Transformation of Functions A transformation of a graph is a change in its position, shape or size. Example function: y = x 2 For a given function, y = f(x) +c [shift up c] y = f(x) – c [shift down c] y = f(x + c) [shift left c] y = f(x – c) [shift right c] y = -f(x) [flip over x-axis] y = f(-x) [flip over y-axis] y = cf(x) [multiply y value by c] [if c > 1, stretch vertically] [if 0 < c < 1, shrink vertically] Graph: y = x 2 + 4 y = x 2 - 4 y = (x+4)2 y = (x – 4)2 y = -x 2 y = (-x)2 y = ½ x 2 Can you graph : y = ½ (x + 2)3 + 2

More Transformation Practice Suppose that the x-intercepts of the graph of y = f(x)

More Transformation Practice Suppose that the x-intercepts of the graph of y = f(x) are -5 and 3 (a)What are the x-intercepts of the graph of y = f(x + 2) (b)What are the x-intercepts of the graph of y = f(x – 2) (c)What are the x-intercepts of the graph y = 4 f(x) (d)What are the x-intercepts of the graph of y = f(-x)

1. 6 Sum, Difference, Product, and Quotient of Functions Let f and g be

1. 6 Sum, Difference, Product, and Quotient of Functions Let f and g be two functions. The sum of f + g, the difference f – g, the product fg, and the quotient f /g are functions whose domains are the set of all real numbers common to the domains of f and g, defined as follows: Sum: Difference: Product: Quotient: (f + g)(x) = f (x)+g(x) (f – g)(x) = f (x) – g(x) (f • g)(x) = f (x) • g(x) (f / g)(x) = f (x)/g(x), provided g(x) does not equal 0 Example: Let f(x) = 2 x+1 and g(x) = x 2 -2. f+g = 2 x+1 + x 2 -2 = x 2+2 x-1 f-g = (2 x+1) - (x 2 -2)= -x 2+2 x+3 fg = (2 x+1)(x 2 -2) = 2 x 3+x 2 -4 x-2 f/g = (2 x+1)/(x 2 -2)

Adding & Subtracting Functions If f(x) and g(x) are functions, then: (f + g)(x)

Adding & Subtracting Functions If f(x) and g(x) are functions, then: (f + g)(x) = f(x) + g(x) (f – g)(x) = f(x) – g(x) Examples: f(x) = 2 x + 1 and Method 1 (f + g)(4) = 2(4) + 1 + -3(4) – 7 = 8+1 + -12 – 7 = 9 + -19 = -10 Method 2 (f + g)(4) = 2 x + 1 + -3 x – 7 = -x – 6 = -4 – 6 = - 10 Adding/subtracting also extends to non-linear functions you will see in a subsequent chapter. g(x) = -3 x – 7 Method 1 (f – g)(6) = 2(6) + 1 – [-3(6) – 7] = 12 + 1 - [-18 – 7] = 13 - [-25] = 13 + 25 Method 2 = 38 (f - g)(6) = 2 x + 1 - [-3 x – 7] = 2 x + 1 + 3 x + 7 = 5 x + 8 = 5(6) + 8 = 30 + 8 = 38

The Composition of Functions f o g - composition of the function f with

The Composition of Functions f o g - composition of the function f with g is is defined by the equation (f o g)(x) = f (g(x)). f (x) = 3 x – 4 and g(x) = x 2 + 6 (f o g)(x) = f (g(x)) = 3 g(x) – 4 = 3(x 2 + 6) – 4 = 3 x 2 + 18 – 4 = 3 x 2 + 14 (g o f)(x) = g(f (x)) = (f (x))2 + 6 = (3 x – 4)2 + 6 = 9 x 2 – 24 x + 16 + 6 = 9 x 2 – 24 x + 22

1. 7 Inverse Function If f (g(x)) = x for every x in the

1. 7 Inverse Function If f (g(x)) = x for every x in the domain of g g(f (x)) = x for every x in the domain of f. and Then the function g is the inverse of the function f denoted by f -1 and the function f is the inverse of the function g denoted by g -1 The domain of f is equal to the range of f -1, and vice versa. (x, y) in f => (y, x) in f--1 Examples: Verifying inverses (Are f & g inverses? ) f (x) = 5 x and g(x) = x/5. f(x)= 3 x + 2 g(x) = x - 2 3 f (g(x)) = 5 x = x f (g(x)) = 3(g(x))+2 5 = 3 x-2 + 2 g( f (x)) = f(x) = 5 x = x 3 5 5 =x– 2+2=x f (g(x)) = x and g( f (x)) = x Thus they are inverses. f(x) = 5 x f-1(x)=x/5 g(f(x) = f(x) – 2 = 3 x + 2 – 2 = x 3 3 Thus they are inverses f(x ) = 3 x + 2 f-1(x) = (x-2)/3

How to Find the Inverse of a Function Example: Find the inverse of f

How to Find the Inverse of a Function Example: Find the inverse of f (x) = 7 x – 5. y = 7 x – 5 Step 1: Replace f (x) by y : Step 2: Interchange x and y : x = 7 y – 5. Step 3: Solve for y. : x + 5 = 7 y x+5=y 7 Step 4 Replace y by f -1(x) = x+5 7

The Horizontal Line Test For Inverse Functions A function f has an inverse that

The Horizontal Line Test For Inverse Functions A function f has an inverse that is a function, f – 1, if there is no horizontal line that intersects the graph of the function f at more than one point y x f(x) = x 2+3 x-1 NO Inverse Function f(x) = x + 4 YES this has an inverse Function

Graphing the Inverse of a Function y x Create your own Example: 1. Draw

Graphing the Inverse of a Function y x Create your own Example: 1. Draw any function (f) that passes the horizontal line test. 2. To graph the inverse (f--1) reverse each (x, y) point on f, graphing (y, x).