Find Real Zeros Multiplicity And Graph Behavior For F(x) = -7(x-3)(x+7)^2
Hey guys! Let's dive into finding the real zeros of the function f(x) = -7(x-3)(x+7)^2 and figure out what's going on with its graph. We'll break down each step so it's super easy to follow.
Identifying Real Zeros
First off, what exactly are real zeros? Simply put, they are the x-values that make the function f(x) equal to zero. These are also the points where the graph of the function intersects or touches the x-axis. To find these zeros, we set f(x) equal to zero and solve for x. So, we have:
-7(x-3)(x+7)^2 = 0
Now, remember the zero-product property: if the product of several factors is zero, then at least one of the factors must be zero. Applying this to our equation, we get:
x - 3 = 0 or (x + 7)^2 = 0
Solving these equations separately:
- x - 3 = 0 => x = 3
- (x + 7)^2 = 0 => x + 7 = 0 => x = -7
So, the real zeros of the function f(x) are x = 3 and x = -7. Easy peasy, right?
Determining Multiplicity
Now that we've found the zeros, let's talk about multiplicity. The multiplicity of a zero refers to the number of times a factor appears in the factored form of the function. This gives us valuable information about the behavior of the graph at that zero.
Looking back at our function:
f(x) = -7(x-3)(x+7)^2
We can identify the multiplicity of each zero:
- For x = 3: The factor (x - 3) appears once. So, the multiplicity of the zero x = 3 is 1.
- For x = -7: The factor (x + 7) appears squared, which means it appears twice. So, the multiplicity of the zero x = -7 is 2.
Understanding multiplicity is key to visualizing how the graph interacts with the x-axis at each zero. Stick around, and we'll get into that next!
Graph Behavior at Zeros: Crossing or Touching
Okay, guys, this is where it gets really interesting! The multiplicity of a zero tells us whether the graph of the function crosses or just touches the x-axis at that zero. Here's the rule of thumb:
- Odd Multiplicity: If a zero has an odd multiplicity (like 1, 3, 5, etc.), the graph crosses the x-axis at that zero. Think of it like the graph passing straight through the axis.
- Even Multiplicity: If a zero has an even multiplicity (like 2, 4, 6, etc.), the graph touches the x-axis at that zero and then turns around. It's like the graph bouncing off the axis.
Let's apply this to our zeros:
- For x = 3 (multiplicity 1, which is odd): The graph crosses the x-axis at x = 3. This means the function changes sign at this point—it goes from being negative to positive (or vice versa).
- For x = -7 (multiplicity 2, which is even): The graph touches the x-axis at x = -7. The function doesn't change sign here; it stays either positive or negative on both sides of x = -7.
So, to recap, the behavior of the graph at a zero is directly linked to its multiplicity. This is a super important concept in understanding polynomial functions.
Summarizing the Findings
Alright, let's put it all together and summarize what we've found. For the function f(x) = -7(x-3)(x+7)^2:
- Real Zeros: The real zeros are x = 3 and x = -7.
- Multiplicity:
- The zero x = 3 has a multiplicity of 1.
- The zero x = -7 has a multiplicity of 2.
- Graph Behavior:
- At x = 3, the graph crosses the x-axis.
- At x = -7, the graph touches the x-axis.
This information gives us a pretty clear picture of what the graph of f(x) looks like. We know where it intersects or touches the x-axis, and how it behaves at those points. That's some serious insight, guys!
Visualizing the Graph (Optional)
If you want to take it a step further, you can sketch a rough graph of the function. Knowing the zeros and their multiplicities is a huge help. You'll also want to consider the leading coefficient (which is -7 in this case). Since it's negative, the graph will generally go downwards as x goes to positive infinity.
Here’s a quick mental sketch:
- Mark the zeros x = 3 and x = -7 on the x-axis.
- At x = -7, the graph touches the axis and bounces back (because of the even multiplicity).
- At x = 3, the graph crosses the axis (because of the odd multiplicity).
- Since the leading coefficient is negative, the graph starts low on the right, crosses at x = 3, turns around somewhere, touches at x = -7, and then goes back down.
Of course, a precise graph would require more points and possibly calculus to find local maxima and minima, but this gives you a solid idea!
Why This Matters
You might be wondering, "Why do we care about all this stuff?" Well, understanding the zeros and behavior of polynomial functions is crucial in many areas of math and its applications. For example:
- Solving Equations: Finding zeros is the same as solving polynomial equations.
- Graphing Functions: Knowing zeros and multiplicities helps you sketch accurate graphs.
- Modeling Real-World Situations: Polynomial functions are used to model various phenomena, and understanding their behavior can provide valuable insights.
- Calculus: Zeros and their multiplicities play a significant role in calculus, especially when analyzing derivatives and integrals.
So, what you've learned here is not just an isolated concept; it's a building block for more advanced topics. Keep practicing, and you'll become a pro at this in no time!
Practice Makes Perfect
The best way to master this is to practice with different functions. Try finding the zeros and multiplicities for these:
- f(x) = (x - 2)^3(x + 1)
- f(x) = x^2(x - 5)^2
- f(x) = -2(x + 4)(x - 1)^3
Work through each one, and think about how the multiplicity affects the graph's behavior. You've got this!
Conclusion
And there you have it! We've successfully found the real zeros of f(x) = -7(x-3)(x+7)^2, determined their multiplicities, and discussed how the graph behaves at each zero. Remember, the key takeaways are:
- Real zeros are the x-values that make the function zero.
- Multiplicity tells you how many times a factor appears.
- Odd multiplicity means the graph crosses the x-axis; even multiplicity means it touches.
Keep up the great work, guys, and remember to have fun with math! You're doing awesome!
