Advanced Algebra/Trig Chapter2 Notes Analysis of Graphs of Functios

2-1: Graphs of Basic Functions and Relations; Symmetry Continuity- a function that can be drawn over an interval of it’s domain without lifting the pencil If a function is not continuous, then it may have a point of discontinuity or it may a vertical asymptote (we will talk about asymptotes in ch 4) point of discontinuity at:nu continuous over intervals:

2-1: Graphs of Basic Functions and Relations; Symmetry

Increasing: Decreasing: Continuous:

2-1: Graphs of Basic Functions and Relations; Symmetry

Symmetry with respect to the y-axis If a function f is defined so that f(-x) = f(x) For all x in its domain, then the graph of f is symmetric with respect to the y- axis. If (a, b) is on the graph, so is (-a, b) If a function is symmetric with respect to the y-axis, we can fold the graph in half along the y-axis.

2-1: Graphs of Basic Functions and Relations; Symmetry

Symmetry with Respect to the Origin If a function f is defined so that f(-x) = -f(x) For all x in its domain, then the graph is symmetric with respect to the origin You can also rotate the graph 180 degrees and find the same graph If (a, b) is on the graph, then so is (-a, -b)

2-1: Graphs of Basic Functions and Relations; Symmetry

Even and Odd Functions A function f is called an even function if f(-x) = f(x) for all x in the domain. (it is symmetric with respect to the y-axis) A function f is called an odd function if f(-x) = -f(x) for all x in the domain. (its graph is symmetric with respect to the origin) x f(x) x f(x)

2-1: Graphs of Basic Functions and Relations; Symmetry

2-2 Vertical and Horizontal Shifts of Graphs

Each of those examples show a vertical shift or a vertical translation. Vertical Shifting occurs by shifting the graph upward a distance of c when c>0 or downward a distance of c when c<0 y = f(x) + c

2-2 Vertical and Horizontal Shifts of Graphs

Each of those show horizontal shift or a horizontal translation Horizontal Shifting occurs by shifting the graph right a distance of c when c>0 or left a distance of c when c<0 y = f(x - c) *Horizontal shifting goes in the opposite direction of what is with the x!!! (look for what would make it zero)

2-2 Vertical and Horizontal Shifts of Graphs

Domains and Ranges of shifted graphs Vertical translations can affect the RANGE Horizontal translations can affect the DOMAIN DOMAIN: RANGE: shift graph left 2 DOMAIN: RANGE: shift graph down 2 DOMAIN: RANGE:

2-2 Vertical and Horizontal Shifts of Graphs

2-3 Stretching, Shrinking, and Reflecting Graphs

These are all examples of vertical stretching or shrinking Vertical Stretching- If a point (x, y) lies on the graph of y = f(x), then (x, cy) lies on the graph of y = c f(x). If c > 1, then this is vertical stretching Vertical Shrinking- If 0 < x < 1, then this is vertical shrinking

2-3 Stretching, Shrinking, and Reflecting Graphs Horizontal stretching and skrinking If a point (x, y) lies on the graph of y = f(x), then the point (x/c, y) lies on the graph of y = f(x). For the graph y = f(cx), If 0 < c < 1, then this is an example of horizontal stretching If c > 1, then this is an example of horizontal shrinking

2-3 Stretching, Shrinking, and Reflecting Graphs

Reflecting across an axis The graph y = -f(x) is a reflection across the x-axis The graph y = f(-x) is a reflection across the y-axis

2-3 Stretching, Shrinking, and Reflecting Graphs

2-4: Absolute Value Functions Absolute value function y = |f(x)| Domain: (-∞, ∞) Range: [0, ∞) y = |x – 4| + 3  shifted right 4 and up 3 Domain: Range:

2-4: Absolute Value Functions

Graph y = |ax + b| by hand Graph: y = 2x – 1 y = |2x – 1|

2-4: Absolute Value Functions Solving absolute value equations Solve |2x + 1| = 7|2x + 1| must be 7 away from zero 2x + 1 = 7OR2x + 1 = -7 2x = 62x = -8 x = 3x = -4 So, x = -4 or 3 The solution set is {-4, 3} Plug each back in to the original equation to check

2-4: Absolute Value Functions Inequalities Example 1) with > signs: |3x + 1| > 6 3x + 1 > 6OR3x + 1 < -6 flip the sign on the negative answer! 3x > 53x < -7 X > 5/3x < -7/3 Example 2) with < signs |2x + 4| < 8 -8 < 2x + 4 < < 2x < 4 -6 < x < 2

2-4: Absolute Value Functions What about problems like: |3x + 1| = -3no solution absolute value can’t = negative |3x + 1| < -3no solution absolute value can’t ever be < negative |3x + 1| > -3 all solutions possible : (-∞, ∞) absolute value will ALWAYS be > negative

2-4: Absolute Value Functions |ax + b| = |cx + d| ax + b must either equal cx + d or equal the opposite Solve: ax + b = cx + dORax + b = -(cx + d) |x + 6| = |2x – 3| x + 6 = 2x – 3ORx + 6 = -(2x – 3) 6 = x – 3x + 6 = -2x + 3 x = 93x + 6 = 3 3x = -3 x = -1

2-4: Absolute Value Functions |2x – 10| = 4 2x – 10 = 4or2x – 10 = -4 2x = 142x = 6 X = 7x=3 |2x – 10| > 4 2x – 10 > 4or2x – 10 < -4 2x > 142x < 6 X > 7x < 3 |2x – 10| < 4 -4 < 2x – 10 < 4 6 < 2x < 14 3 < x < 7

2-4: Absolute Value Functions 5 | x – 4 | = 20 |x – 4| = 4 x – 4 = 4or x – 4 = -4 X = 8x = 0 |-5x + 1| = |3x – 4| -5x + 1 = 3x – 4or-5x + 1 = -(3x – 4) 1 = 8x – 4-5x + 1 = -3x + 4 8x = 51 = 2x + 4 X = 5/82x = -3 x = -3/2

2-5: Piecewise-Defined Functions Piecewise Function- a function defined by different rules (formulas) over different subsets of its domain. Domain: Range: Is it continuous on its Domain? f(-3)= f(0)= f(5)=

2-5: Piecewise-Defined Functions f(-8) f(-3) f(-1) f(10)

2-5: Piecewise-Defined Functions Find the formula from the graph:

2-6: Operations and Composition

Domains of f + g, f – g, and fg include all real numbers in the intersection of the domains of f and g The domain of f/g include those in the intersection of the domains of f and g for which g(x) ≠ 0 ∩ means AND ᶸ means OR

2-6: Operations and Composition