From e0116782ee332bd0a9b106614b8d34bc562cc74d Mon Sep 17 00:00:00 2001 From: Nick Chura Date: Fri, 26 May 2017 11:43:08 -0700 Subject: [PATCH] changed some wording --- src/set3B-Polynomial_Functions.mbx | 38 +++++++++++++++--------------- 1 file changed, 19 insertions(+), 19 deletions(-) diff --git a/src/set3B-Polynomial_Functions.mbx b/src/set3B-Polynomial_Functions.mbx index db8f724..8d25b86 100644 --- a/src/set3B-Polynomial_Functions.mbx +++ b/src/set3B-Polynomial_Functions.mbx @@ -222,7 +222,7 @@ Saylor.org, pages 192-193-->
-
i -
-
- \end{tikzpicture}]]>
- + -

Each of these graphs look quite different from each other, but share some characteristics. +

Each of these graphs look quite different from each other, but they do share some characteristics.

  • The graphs are all connected with smooth bends. They do not look like piece-wise functions.
  • The domain of each consists of all real numbers. (Why?)
    • -
  • The ends of the graph eventually settles into an upward or downward curve. (Note: The graph doesn't really end, it continues indefinitely, but if you go out far enough, eventually it will consistently move in one direction, either up or down.)
    • +
  • The ends of the graph eventually settles into an upward or downward curve. (Note: The graph doesn't really end, but it continues indefinitely. If you go out far enough, eventually it will consistently move in one direction, either up or down.)
  • The graphs all have one vertical intercept, but the number of horizontal intercepts varies from none to many.

@@ -330,10 +330,10 @@ Saylor.org, pages 192-193-->
- ] (-5.9,0) -- (5.9,0); - \draw[line width=0.55mm,black,<->] (0,-1.9) -- (0,5.9); + \draw[line width=0.55mm,black,<->] (0,-1.9) -- (0,5.9); \draw[line width=0.55mm,black] (-4 cm,3pt) -- (-4 cm,-3pt); \draw[line width=0.55mm,black] (4 cm,3pt) -- (4 cm,-3pt); \draw[line width=0.55mm,black] (3pt,4 cm) -- (-3pt,4 cm); @@ -347,7 +347,7 @@ Saylor.org, pages 192-193--> \fill (-1,2) circle (.1cm); \fill (0,1) circle (.1cm); \fill (1,2) circle (.1cm); - \fill (2,5) circle (.1cm); + \fill (2,5) circle (.1cm); \end{tikzpicture}]]>
@@ -401,7 +401,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH

Plotting a few more points gives the graph:

- ] (-5.9,0) -- (5.9,0); \draw[line width=0.55mm,black,<->] (0,-4.9) -- (0,4.9); @@ -418,7 +418,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH \node at (2.2,2.5) {\Large $g(x)=-x^2-x+2$}; \fill (-2,0) circle (.1cm); \fill (0,2) circle (.1cm); - \fill (1,0) circle (.1cm); + \fill (1,0) circle (.1cm); \end{tikzpicture}]]>
@@ -432,7 +432,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH 0 \amp = \frac{2}{7}x^3-\frac{1}{7}x^2-3x \amp = \frac{1}{7}x(2x^2-x-21) \amp = \frac{1}{7}x(2x- 7)(x+3) - So, + So, \frac{1}{7}x\amp = 0\amp\amp\text{or}\amp 2x - 7 \amp = 0\amp\amp\text{or}\amp x + 3 \amp = 0 x\amp = 0\amp\amp\text{or}\amp x \amp = \frac{7}{2} \amp\amp\text{or}\amp x \amp = -3 @@ -442,7 +442,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH

Plotting a few more points gives the graph:

- ] (-5.9,0) -- (5.9,0); \draw[line width=0.55mm,black,<->] (0,-4.9) -- (0,4.9); @@ -459,7 +459,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH \node at (-3.5,3.5) {\Large $h(x)=\frac{2}{7}x^3-\frac{1}{7}x^2-3x$}; \fill (-3,0) circle (.1cm); \fill (0,2) circle (.1cm); - \fill (3.5,0) circle (.1cm); + \fill (3.5,0) circle (.1cm); \end{tikzpicture}]]>
@@ -499,7 +499,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH

Plot a few more points and we are done:

- ] (-5.9,0) -- (5.9,0); \draw[line width=0.55mm,black,<->] (0,-8.9) -- (0,1.9); @@ -516,7 +516,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH \node at (-3.5,-6.5) {\Large $k(x)=x^4-2x^3-2x^2$}; \fill (-.732,0) circle (.1cm); \fill (0,0) circle (.1cm); - \fill (2.732,0) circle (.1cm); + \fill (2.732,0) circle (.1cm); \end{tikzpicture}]]>
@@ -566,7 +566,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH

Note: Why study polynomials? What are they good for? Many functions are difficult to calculate without a calculator or computer, for example, trigonometric functions. How DO calculators or computers calculate values for these functions? The answer is based on polynomials which are easier to calculate since they only involve multiplying and addition/subtraction. The graph below shows how the polynomial f(x) = x - \frac{1}{6}x^3 + \frac{1}{120}x^5 approximates the function g(x) = \sin(x) for -3 \leq x \leq3.

- ] (-6,0) -- (6,0); \draw[line width=0.55mm,black,<->] (0,-4) -- (0,4); \foreach [evaluate=\x as \z using int(2*\x)] \x in {-5,...,5} @@ -576,7 +576,7 @@ on the left side of NOT SURE WHETHER TO GO INTO THIS HERE OR IF THIS IS THE RIGH \draw[line width=0.75mm, domain=-2.4:2.3,smooth,variable=\x,red] plot ({\x},{\x-2*\x*\x*\x/3+2*\x*\x*\x*\x*\x/15}); \draw[dashed, line width=0.75mm, blue](-5.5, .5) cos (-4.71,0) sin (-3.93,-.5) cos (-3.14,0) sin (-2.36,.5) cos (-1.57,0) sin (-.785,-.5) cos (0,0) sin (.785,.5) cos (1.57,0) sin (2.36,-.5) cos (3.14,0) sin (3.93,.5) cos (4.71,0) sin (5.5, -.5); \node at (4,3.2) {\Large $f(x)=x - \frac{1}{6}x^3 + \frac{1}{120}x^5$}; - \node at (4, -1) {\Large $g(x)=\sin(x)$}; + \node at (4, -1) {\Large $g(x)=\sin(x)$}; \end{tikzpicture}]]>