Concept Byte: Graphing Polynomials Using Zeros
Sign In
How can you find the x-intercepts of a function written in factored form?
Sketch:
Graphing calculator:
To sketch the graph of a function, we must find its zeros and determine how the function changes around these points. Since the function is already in factored form, we can determine the zeros by substituting h(x)=0 and then using the Zero Product Property.
h(x)= 0
Use the Zero Product Property
Now that we know the function's x-intercepts, we can plot them on a coordinate plane.
| Interval | x | (x+6)(x-7) | h |
|---|---|---|---|
| x<- 6 | - 7 | ( - 7+6)( - 7-7) | 14 |
| - 6 < x < 7 | 0 | ( 0+6)( 0-7) | - 42 |
| 7 < x | 8 | ( 8+6)( 8-7) | 14 |
With the possible exception of the y-intercept, the actual h-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. h(- 7)&= 14 &&⇒ Abovethex-axis h(0)&= - 42 &&⇒ Belowthex-axis h(8)& = 14 &&⇒ 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,- 42). 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.