Difference between revisions of "Graph of a function"
Smileapple (talk | contribs) m (→See also) |
(→Solution) |
||
| Line 30: | Line 30: | ||
=== Solution === | === Solution === | ||
| − | Luckily the quadratic factors as <math>(2x+3)(x-1)</math> making the roots <math>x=-\frac 32</math> and <math>x=1</math>. The quadratic can only switch signs | + | Luckily the quadratic factors as <math>(2x+3)(x-1)</math> making the roots <math>x=-\frac 32</math> and <math>x=1</math>. The quadratic can only switch signs at its zeros. So picking one point less than <math>-\frac 32</math> and plugging it in will determine whether the graph is above or below the <math>x</math>-axis for all <math>x</math> on the interval <math>\left(-\infty, -\frac 32\right).</math> Since <math>f(-2)=3</math> is positive, the graph is above the <math>x</math>-axis. |
Likewise, a sign analysis on the intervals <math>\left(-\frac 32, 1\right)</math> and <math>(1, \infty)</math> allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline: | Likewise, a sign analysis on the intervals <math>\left(-\frac 32, 1\right)</math> and <math>(1, \infty)</math> allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline: | ||
Revision as of 04:06, 8 June 2023
A graph is a visual representation of a function. If 
then the point 
lies on the graph of 
.
Contents
Graphing Points
A single point is the simplest thing to graph. The graph of 
would be a dot 2 units to the right of 
-axis and 5 units above the 
-axis.
.PNG)
Graphing Lines
Given two distinct points on a line, one can construct the whole line. So one way to graph a line given its equation is to just find two points on it and to draw a straight line through them.
Problem
Graph the line 
.
Solution
To graph a line, it is necesasry to find two points that satisfy . Letting 
gives 
. So 
is one point on the graph.
Find another point by letting 
. Plugging this in and solving gives 
. So 
is our other point.
Now plot these in the coordinate plane and draw a line through them:
The arrowheads on the ends of the line segment indicate that the line goes on infinitely in both directions.
Graphing Polynomials
The first step in graphing a polynomial, 
, is to find the zeros of 
. Then a smooth curve should be drawn through the zeros accounting for multiple roots and making sure the signs match up (i.e. the graph is above the -axis when the polynomial is positive and below it when the polynomial is negative). This process is best understood through examples.
Problem
Graph the parabola 
.
Solution
Luckily the quadratic factors as 
making the roots 
and 
. The quadratic can only switch signs at its zeros. So picking one point less than 
and plugging it in will determine whether the graph is above or below the -axis for all on the interval 
Since 
is positive, the graph is above the -axis.
Likewise, a sign analysis on the intervals 
and 
allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline:
Problem
Graph 
.
Solution
First, find the zeros of the function. Note that if or 
, . After synthetic division, the polynomial reduces to 
. Factor the quadratic gives 
. So the roots are 1 and -3 and a double root at 2. The final graph looks like:
