IA Functions, Equations, and Graphs Chapter 2

In this chapter, you will learn: What a function is. Review domain and range. Linear equations. Slope. Slope intercept form y = mx+b. Point-slope form y – y1 = m(x – x1). Linear regression.

What is a function? A function is a special type of relation in which each type of domain (x values) is paired of with exactly one range value (y value). FUNCTION NOT A FUNCTION FUNCTION NOT A FUNCTION

Relations and Functions Suppose we have the relation { (-3,1), (0,2), (2,4) } FUNCTION ONE – TO – ONE DOMAIN x - values RANGE y - values

Relations and Functions Suppose we have the relation { (-1,5), (1,3), (4,5) } FUNCTION NOT ONE – TO – ONE

Relations and Functions Suppose we have the relation { (5,6), (-3,0), (1,1), (-3,6) } NOT A FUNCTION

Domain and Range The set of all inputs, or x-values of a function. It is all the x – values that are allowed to be used. The set of all outputs, or y-values of a function. It is all the y – values that are represented.

Example 1 Domain = ________________ Range = _________________ All x – values or (- ∞, ∞) Just 4 or {4}

Example 2 Domain = ________________ Range = _________________ All y – values or (- ∞, ∞) Just -5 or {-5}

Example 3 Domain = ________________ Range = _________________ All x – values or (- ∞, ∞) From -6 on up or [- 6, ∞)

Example 4 Domain = ________________ Range = _________________ All y – values or (- ∞, ∞) From -6 on up or [- 6, ∞)

Example 5 Domain = ________________ Range = _________________ All y – values or (- ∞, ∞) All x – values or (- ∞, ∞)

Function Notation Function notation, f(x), is called “f of x” or “a function of x”. It is not f times x. Example: if y = x+2 then we say f(x) = x+2. If y = 5 when x = 3, then we say f(3) = 5 What is function notation?

Example 1 f(x) = 3x + 1 f( 13) = ____________________ f( 5) = ____________________ f( -11) = ____________________ 3 (5) + 1 = 16 3 (13) + 1 = 40 3 (-11) + 1 = -32

Example 2 f(x) = x² + 3x - 5 f( 0) = ____________________ f( 5) = ____________________ f( 4) = ____________________ 5² + 3 (5) – 5 = 35 0² + 3 (0) – 5 = -5 4² + 3 (4) – 5 = 23

SLOPE RISE RUN SLOPE

Slope Formula Given points (X 1,Y 1 ) and (X 2,Y 2 ) Is the same as ?

Example ( 4, 0 ) and ( 7, 6 ) (X 1,Y 1 ) (X 2,Y 2 )

Example ( -6, 5 ) and ( 2, 4 ) (X 1,Y 1 ) (X 2,Y 2 )

Example ( 5, 2 ) and ( -3, 2 ) (X 1,Y 1 ) (X 2,Y 2 )

Example ( 2, 7 ) and ( 2, -3 ) (X 1,Y 1 ) (X 2,Y 2 )

Recap of slope

Linear Forms of Linear Equations Standard Form (AX + BY = C) 1)A and B cannot be fractions. 2)A cannot be negative y = 3x – 5 y – 3x = – 5 – 3x + y = – 5 3x – y = 5 To find y-intercept, set x = 0 Y-intercept (0,-5) FINDING INTERCEPTS To find x-intercept, set y = 0 X-intercept (5/3,0)

Linear Forms of Linear Equations Slope Intercept Form (y = mx + b) 1)Y is isolated on one side. 2)Y is positive x – 2y = 6 –2y = – x + 6 y = ½ x + 6/-2 y = ½ x – 3 To find y-intercept, set x = 0 Y-intercept (0,-3) FINDING INTERCEPTS To find x-intercept, set y = 0 X-intercept (6,0)

Find a line || to 2x + 4y = -8 and passes thru (8,3) First find slope 2x + 4y = -8 4y = -2x – 8 y = - ½ x – 2 Slope Use y = mx + b y = - 1/2x + b Plug in point 3 = - ½ (8) + b 3 = -4 + b 7 = b FINAL EQUATION y = mx + b y = - ½ x + 7 Slope intercept

Find a line Perpendicular to 3x – 2y = 10 and passes thru (-6,2) First find slope 3x – 2y = y = -3x + 10 y = 3/2 x – 5 Slope Use y = mx + b y = -2/3 x + b Plug in point 2 = -2/3 (-6) + b 2 = 4+ b -2 = b FINAL EQUATION y = mx + b y = -2/3x - 2 Slope intercept

Find a line That has slope = 2 and passes thru (-4,7) Use y = mx + b y = 2x + b Plug in point 7 = 2 (-4) + b 7 = -8 + b 15 = b FINAL EQUATION y = mx + b y = 2 x + 15 Slope intercept

Find a line That has x-intercept = (5,0) and y-intercept (0,-3) Use y = mx + b y = 3/5 x + b Plug in point either one ! 0 = 3/5 (5) + b 0 = 3 + b -3 = b FINAL EQUATION y = mx + b y = 3/5 x - 3 Slope intercept First find slope m m = 3/5

Graph x = 3

Graph y = -3

Graph Y = -3/4 x + 2 Y = mx + b m = slope b = y-intercept 1)Plot the y-intercept first 2)Starting from the y-intercept, go up if + or down if – Then go right. 3) Plot point and draw your line. m = -3 / 4 b = 2

Graph 5x + 6y < 30 5x + 6y < 30 6y< -5x + 30 Y < -5/6 x +5

Graph 2x + 4y ≥ 16 2x + 4y ≥ 16 4y ≥ -2x + 16 Y ≥ -2/4 x + 4 Y ≥ -1/2 x + 4

Graph 5x - y < 8 5x – y < 8 – y < -5x + 8 y > 5x – 8

Graph y = | x | XY WHERE IS THE VERTEX? (0,0)

Graph y = | x - 3 | XY WHERE IS THE VERTEX? (3,0)

Graph y = | x + 3 | XY WHERE IS THE VERTEX? (-3,0)

Graph y = | x + 3 | – 2 XY WHERE IS THE VERTEX? (-3,-2)

Is there a formula for graphing absolute value equations??? Y = |x + 2| – 3 Y = |x + 5| + 8 Y = |x – 8| – 6 Y = |x – 7| + 4 Y = |x – 4| – 5 Y = |2x – 8| + 2 Y = |3x + 6| – 3 Y = |mx + b| + c  (-2, -3)  (-5, 8)  (8, -6)  (7, 4)  (4, -5)  (4, 2)  (-2, -3)  (- b/m, c)

Correlations positive As x increases Then y increases POSITIVE SLOPE “Trend line” or “regression line” Outlier

Correlations negative As x increases Then y decreases negativeSLOPE

Correlations none No real trend line