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:
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
When we know a function's x-intercepts, we can plot them in a coordinate plane.
| Interval | x | (x+2)(x-3)(x+1)(x-1) | h |
|---|---|---|---|
| x<- 2 | - 3 | ( - 3+2)( - 3-3)( - 3+1)( - 3-1) | 48 |
| - 2 < x < - 1 | - 1.5 | ( - 1.5+2)( - 1.5-3)( - 1.5+1)( - 1.5-1) | ≈ - 2.8 |
| - 1 < x < 1 | 0 | ( 0+2)( 0-3)( 0+1)( 0-1) | 6 |
| 1 < x < 3 | 2 | ( 2+2)( 2-3)( 2+1)( 2-1) | - 12 |
| 3 < x | 4 | ( 4+2)( 4-3)( 4+1)( 4-1) | 90 |
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(- 3)&= 48 &&⇒ Abovethex-axis h(- 1.5)&≈ - 2.8 &&⇒ Belowthex-axis h(0)& = 6 &&⇒ Abovethex-axis h(2)& = - 12 &&⇒ Belowthex-axis h(4)& = 90 &&⇒ Abovethex-axis Going from above the x-axis to below the x-axis means the function is decreasing, and vice-versa. With this information, we can sketch the graph. We will draw it so that it intercepts the y-axis at (0,6). From our randomly chosen substitutions, we also know that it should pass through (2,- 12). Adding these 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.