Concept Byte: Graphing Polynomials Using Zeros
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How can you find the x-intercepts of a function written in factored form?
Sketch:
Graphing calculator:
g(x)= 0
Use the Zero Product Property
If we know how the function moves around these points, we get an idea of what it looks like. In order to do that, we can calculate the g-values of the following intervals. Note that g represents the function's values for different values of x. x&<- 1 - 1< x& < 3 3< x& < 5 5< x& Let's choose some arbitrary x-values in these intervals and find their corresponding g-values.
| Interval | x | (x+1)(x-3)(x-5) | g |
|---|---|---|---|
| x<- 1 | - 2 | ( - 2+1)( - 2-3)( - 2-5) | - 35 |
| - 1 < x < 3 | 0 | ( 0+1)( 0-3)( 0-5) | 15 |
| 3 < x < 5 | 4 | ( 4+1)( 4-3)( 4-5) | - 5 |
| 5 < x | 6 | ( 6+1)( 6-3)( 6-5) | 21 |
With the possible exception of the y-intercept, the actual g-values for the given x-values we used are not important. Instead, we are more interested if the function is above or below the x-axis in the given intervals. This will tell us how the function grows, which we can use to sketch the graph. g(- 2)&= - 35 &&⇒ Belowthex-axis g(0)& = 15 &&⇒ Abovethex-axis g(4)& = - 5 &&⇒ Belowthex-axis g(6)& = 21 &&⇒ Abovethex-axis Going from above to below the x-axis means that the function is decreasing, and vice-versa. We will sketch the graph so that it intercepts the y-axis at (0,15). Adding the obtained points to the diagram will help to make a better sketch.
Now let's graph the function on our graphing calculator and compare. Notice that we can use the same window-setting as in our sketch to make sure the proportions are the same.
The graphing calculator image is very similar to our graph.