Asked by zaynab
Find the zeros of the function. Then graph the function.
y=(x+3)(x-5)(x-6)
y=(x+3)(x-5)(x-6)
Answers
Answered by
GPT 3.5
To find the zeros of the function, we need to set y equal to zero and solve for x:
0 = (x + 3)(x - 5)(x - 6)
Using the zero product property, we know that one or more of the factors must equal zero. Therefore, we can set each factor equal to zero and solve for x:
x + 3 = 0 => x = -3
x - 5 = 0 => x = 5
x - 6 = 0 => x = 6
So the zeros of the function are x = -3, x = 5, and x = 6.
To graph the function, we can start by plotting these zeros on a coordinate plane.
The zero x = -3 is a root of the function, which means the graph will cross the x-axis at x = -3.
The zero x = 5 is also a root of the function, so the graph will cross the x-axis at x = 5.
The zero x = 6 is also a root of the function, so the graph will cross the x-axis at x = 6.
Next, we can find the behavior of the graph as x approaches positive infinity and negative infinity.
As x approaches negative infinity, the three factors of the function will all be negative, since (-)(-)(-) = - which means the function will be negative.
As x approaches positive infinity, the three factors of the function will all be positive, since (+)(+)(+) = + which means the function will be positive.
The graph will also be a cubic shape since the function has three factors, each with an x of degree 1.
Putting all this information together, we can sketch the graph of the function.
At x = -3, the graph will cross the x-axis.
At x = 5, the graph will also cross the x-axis.
At x = 6, the graph will also cross the x-axis.
Overall, the graph will start in the negative region, cross the x-axis at -3, rise up, then dip down to cross at 5, and rise up again to cross at 6. It will continue upwards without crossing the x-axis again.
The rough sketch of the graph is as follows:
_______________
/
_______________
/
_______________
-3 5 6
Note that this is a rough sketch and may not be perfectly accurate. A more accurate graph can be obtained by evaluating the function at additional x-values and more precisely plotting its shape.
0 = (x + 3)(x - 5)(x - 6)
Using the zero product property, we know that one or more of the factors must equal zero. Therefore, we can set each factor equal to zero and solve for x:
x + 3 = 0 => x = -3
x - 5 = 0 => x = 5
x - 6 = 0 => x = 6
So the zeros of the function are x = -3, x = 5, and x = 6.
To graph the function, we can start by plotting these zeros on a coordinate plane.
The zero x = -3 is a root of the function, which means the graph will cross the x-axis at x = -3.
The zero x = 5 is also a root of the function, so the graph will cross the x-axis at x = 5.
The zero x = 6 is also a root of the function, so the graph will cross the x-axis at x = 6.
Next, we can find the behavior of the graph as x approaches positive infinity and negative infinity.
As x approaches negative infinity, the three factors of the function will all be negative, since (-)(-)(-) = - which means the function will be negative.
As x approaches positive infinity, the three factors of the function will all be positive, since (+)(+)(+) = + which means the function will be positive.
The graph will also be a cubic shape since the function has three factors, each with an x of degree 1.
Putting all this information together, we can sketch the graph of the function.
At x = -3, the graph will cross the x-axis.
At x = 5, the graph will also cross the x-axis.
At x = 6, the graph will also cross the x-axis.
Overall, the graph will start in the negative region, cross the x-axis at -3, rise up, then dip down to cross at 5, and rise up again to cross at 6. It will continue upwards without crossing the x-axis again.
The rough sketch of the graph is as follows:
_______________
/
_______________
/
_______________
-3 5 6
Note that this is a rough sketch and may not be perfectly accurate. A more accurate graph can be obtained by evaluating the function at additional x-values and more precisely plotting its shape.