Function: f(x) = x²
Analysis of End Behavior:
1. Testing Positive Values:
Let’s plug in x = 10:
f(10) = (10)² = 100
As x gets larger and approaches positive infinity, the value of x² will continue to grow without bound. This shows that the right end behavior of the graph goes to positive infinity.
2. Testing Negative Values:
Let’s plug in x = -10:
f(-10) = (-10)² = 100
As x gets smaller and approaches negative infinity, the value of x² will also continue to grow without bound. Since squaring a negative number results in a positive value, the left end behavior of the graph also goes to positive infinity.
Conclusion: The end behavior of the function f(x) = x² is such that as x approaches positive or negative infinity, the function values approach positive infinity.
Function: f(x) = -x²
Analysis of End Behavior:
1. Testing Positive Values:
Let’s plug in x = 10:
f(10) = -(10)² = -100
As x gets larger and approaches positive infinity, the value of -x² will continue to decrease without bound. This shows that the right end behavior of the graph goes to negative infinity.
2. Testing Negative Values:
Let’s plug in x = -10:
f(-10) = -(-10)² = -100
As x gets smaller and approaches negative infinity, the value of -x² will also continue to decrease without bound. Since squaring a negative number results in a positive value, and we have a negative sign in front of the expression, the left end behavior of the graph also goes to negative infinity.
Conclusion: The end behavior of the function f(x) = -x² is such that as x approaches positive or negative infinity, the function values approach negative infinity.
Function: f(x) = x² + 2x
Analysis of End Behavior:
1. Testing Positive Values:
Let’s plug in x = 10:
f(10) = (10)² + 2(10) = 100 + 20 = 120
As x gets larger and approaches positive infinity, both the x² and 2x terms will increase without bound. This shows that the right end behavior of the graph goes to positive infinity.
2. Testing Negative Values:
Let’s plug in x = -10:
f(-10) = (-10)² + 2(-10) = 100 – 20 = 80
As x gets smaller and approaches negative infinity, the x² term will increase, but the 2x term will decrease. Since the x² term grows faster than the linear term, the left end behavior of the graph goes to positive infinity as well.
Conclusion: The end behavior of the function f(x) = x² + 2x is such that as x approaches positive or negative infinity, the function values approach positive infinity.
Function: f(x) = -x³ + 2x² + x
Analysis of End Behavior:
1. Testing Positive Values:
Let’s plug in x = 10:
f(10) = -(10)³ + 2(10)² + 10 = -1000 + 200 + 10 = -790
As x gets larger and approaches positive infinity, the -x³ term will dominate the other terms and decrease without bound. This shows that the right end behavior of the graph goes to negative infinity.
2. Testing Negative Values:
Let’s plug in x = -10:
f(-10) = -(-10)³ + 2(-10)² + (-10) = 1000 + 200 – 10 = 1190
As x gets smaller and approaches negative infinity, the -x³ term will dominate the other terms and increase without bound. This shows that the left end behavior of the graph goes to positive infinity.
Conclusion: The end behavior of the function f(x) = -x³ + 2x² + x is such that as x approaches positive infinity, the function values approach negative infinity, and as x approaches negative infinity, the function values approach positive infinity.
Function: f(x) = (x-1)(x+2) = x² + x – 2
Analysis of End Behavior:
1. Testing Positive Values:
Let’s plug in x = 10:
f(10) = (10-1)(10+2) = 9 × 12 = 108
As x gets larger and approaches positive infinity, the x² term will dominate the other terms and increase without bound. This shows that the right end behavior of the graph goes to positive infinity.
2. Testing Negative Values:
Let’s plug in x = -10:
f(-10) = (-10-1)(-10+2) = -11 × -8 = 88
As x gets smaller and approaches negative infinity, the x² term will still dominate the other terms and increase without bound. This shows that the left end behavior of the graph also goes to positive infinity.
Conclusion: The end behavior of the function f(x) = (x-1)(x+2) is such that as x approaches both positive and negative infinity, the function values approach positive infinity.
Function: f(x) = (x-1)(x+3)(x-5) = x³ – 3x² – 13x + 15
Analysis of End Behavior:
1. Testing Positive Values:
Let’s plug in x = 10:
f(10) = (10-1)(10+3)(10-5) = 9 × 13 × 5 = 585
As x gets larger and approaches positive infinity, the x³ term will dominate the other terms and increase without bound. Since the coefficient of x³ is positive, the right end behavior of the graph goes to positive infinity.
2. Testing Negative Values:
Let’s plug in x = -10:
f(-10) = (-10-1)(-10+3)(-10-5) = -11 × -7 × -15 = -1155
As x gets smaller and approaches negative infinity, the x³ term will still dominate the other terms, but since the coefficient of x³ is positive, the left end behavior of the graph goes to negative infinity.
Conclusion: The end behavior of the function f(x) = (x-1)(x+3)(x-5) is such that as x approaches positive infinity, the function values approach positive infinity, and as x approaches negative infinity, the function values approach negative infinity.
Function: f(x) = (x+1)(x-12) = x² – 11x – 12
Concern with Using x = 10:
If one of the zeros of the function is x = 12, then using x = 10 to analyze the end behavior is too close to this zero. The value x = 10 is only 2 units away from the zero, and it may not provide a clear indication of the trend as x approaches positive infinity.
Recommended Value:
Instead of using x = 10, a value further from the zero, such as x = 20 or even larger, might be more appropriate. For example:
f(20) = (20+1)(20-12) = 21 × 8 = 168
This value is further from the zero at x = 12 and may provide a better indication of the end behavior of the function as x approaches positive infinity.
Conclusion: When analyzing the end behavior of a polynomial function, especially near its zeros, it’s important to choose values that are sufficiently far from those zeros to understand the overall trend of the function. In this case, using a value like x = 20 or larger would be more informative than x = 10.
Function: f(x) = (x+1)²(x-2)³
Analysis:
The function has zeros at x = -1 (with multiplicity 2) and x = 2 (with multiplicity 3). To analyze the end behavior, we’ll choose values that are sufficiently far from these zeros.
Using x = -10:
f(-10) = (-10+1)²(-10-2)³ = (-9)²(-12)³ = 81 × (-1728) = -139968
This value indicates the behavior of the function as x approaches negative infinity.
Using x = 10:
f(10) = (10+1)²(10-2)³ = 11² × 8³ = 121 × 512 = 61952
This value indicates the behavior of the function as x approaches positive infinity.
Why x = 10 is Safe: The choice of x = 10 is safe because it is far enough from the zeros at x = -1 and x = 2. Since the zeros are the points where the function changes behavior, choosing a value far from them ensures that we capture the overall trend of the function as x approaches positive infinity.
Conclusion: The function f(x) = (x+1)²(x-2)³ exhibits different behaviors as x approaches negative and positive infinity. The chosen values, x = -10 and x = 10, are sufficiently far from the zeros to provide a clear indication of these trends. As x approaches negative infinity, the function decreases, and as x approaches positive infinity, the function increases.
