Difference between revisions of "Graph of a function"
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=== Solution === | === Solution === | ||
| − | + | The quadratic equation can be written as <math>(2x+3)(x-1)</math> making the roots <math>x=-\frac{3}{2}</math> and <math>x=1</math>. Since the coefficient of the term with the highest power (in this case <math>x^2</math>) is <math>2>0</math>, the graph is above the <math>x</math>-axis for <math>(-\infty, -\frac{3}{2})</math> and <math>(1, +\infty)</math> and below the <math>x</math>-axis for <math>(-\frac{3}{2}, 1)</math>. This allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline: | |
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<center>[[Image:Parabola1.PNG]]</center> | <center>[[Image:Parabola1.PNG]]</center> | ||
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=== Solution === | === Solution === | ||
| − | First, find the zeros of the function. | + | First, we need to find the zeros of the function. Notice that if <math>x=1</math> or <math>x=2</math>, <math>y=0</math>. Hence, the polynomial reduces to <math>y=(x-1)(x-2)(x^2+x-6)</math>. Factoring the quadratic gives <math>(x-1)(x-2)^2(x+3)</math>. So the roots are <math>1</math> and <math>-3</math> and a double root at <math>2</math>. The final graph looks like: |
<center>[[Image:Quartic1.PNG]]</center> | <center>[[Image:Quartic1.PNG]]</center> | ||
Latest revision as of 04:11, 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
The quadratic equation can be written as 
making the roots 
and 
. Since the coefficient of the term with the highest power (in this case 
) is 
, the graph is above the -axis for 
and 
and below the -axis for 
. This allows the graph to be drawn as a smooth curve curve through the zeros using this information as a guideline:
Problem
Graph 
.
Solution
First, we need to find the zeros of the function. Notice that if or 
, . Hence, the polynomial reduces to 
. Factoring the quadratic gives 
. So the roots are 
and 
and a double root at 
. The final graph looks like:
