Name: Unit 3: Relations and Functions Date: Per Homework 5: Zeros of Functions Directions: Identify the zeros of the function given the graph 1. 2. 3 6. Directions: Identify the zeros of the function algebraically 7. () = 1+2 8. S(x) = -2x+6 10. /) – 2x – 10 11./6–**** 12. (:) – 4 Directions: Identify the zeros of the function using your graphing colculator 13. f(x) = -2 14. () = -2x-15 15. () — 5x + 6 16. /() – x – 6x + 3x +10 17. () = x-2x-x+2 18. (*) -x-5x20x-16
The Correct Answer and Explanation is :
Let’s break down how to identify the zeros of functions given various forms (graphical, algebraic, and using a graphing calculator). The zeros of a function are the points where the function’s output is equal to zero (i.e., ( f(x) = 0 )).
Part 1: Identifying Zeros from a Graph
If you have a graph of the function, you can identify the zeros by looking for the points where the graph crosses the x-axis. These are the points where the value of ( f(x) = 0 ). The x-coordinates of these points are the zeros of the function. If the graph does not cross the x-axis, there are no real zeros. If the graph touches the x-axis at a single point or multiple points, those are the zeros.
Part 2: Identifying Zeros Algebraically
Here’s how to find the zeros for the functions provided:
7. ( f(x) = x + 2 )
To find the zeros of ( f(x) = x + 2 ), set ( f(x) = 0 ) and solve for ( x ):
[
x + 2 = 0
]
[
x = -2
]
So, the zero of the function is ( x = -2 ).
8. ( S(x) = -2x + 6 )
To find the zeros of ( S(x) = -2x + 6 ), set ( S(x) = 0 ) and solve for ( x ):
[
-2x + 6 = 0
]
[
-2x = -6
]
[
x = 3
]
So, the zero of the function is ( x = 3 ).
10. ( f(x) = -2x – 10 )
To find the zeros of ( f(x) = -2x – 10 ), set ( f(x) = 0 ) and solve for ( x ):
[
-2x – 10 = 0
]
[
-2x = 10
]
[
x = -5
]
So, the zero of the function is ( x = -5 ).
11. ( f(x) = 6x – 10 )
To find the zeros of ( f(x) = 6x – 10 ), set ( f(x) = 0 ) and solve for ( x ):
[
6x – 10 = 0
]
[
6x = 10
]
[
x = \frac{10}{6} = \frac{5}{3}
]
So, the zero of the function is ( x = \frac{5}{3} ).
12. ( f(x) = -4 )
Since the function is a constant, it never crosses the x-axis. Therefore, this function has no zeros.
Part 3: Identifying Zeros Using a Graphing Calculator
For functions like:
- ( f(x) = -2 ) (a constant function, which does not cross the x-axis).
- ( f(x) = -2x – 15 ) (a linear function).
- ( f(x) = -5x + 6 ) (another linear function).
- ( f(x) = x^3 – 6x + 3x + 10 ) (a cubic function).
You would input these functions into your graphing calculator and identify where they intersect the x-axis (or if they do not).
For example, with ( f(x) = -2x – 15 ), you set the equation equal to zero, solve for ( x ), and graph the function. The graph crosses the x-axis at ( x = -\frac{15}{2} ).
Similarly, for the cubic equation, you would find the zeros by factoring or using the calculator’s graphing tool to locate the points where the curve intersects the x-axis.
Conclusion
The zeros of a function represent the points where the function equals zero. You can find these algebraically by setting the function equal to zero and solving for ( x ). When using a graphing calculator, the zeros are the x-intercepts of the function’s graph. Functions such as constants (e.g., ( f(x) = -4 )) may have no zeros, while linear functions (e.g., ( f(x) = x + 2 )) will always have exactly one zero.