Finding The X-Intercept: F(x) = (x-8)(x+9) Explained

by Andrew McMorgan 53 views

Hey guys! Today, we're diving into a common math problem that might seem tricky at first, but I promise, it's super manageable once you get the hang of it. We're talking about finding the x-intercept of a quadratic function. Specifically, we'll be tackling the function f(x) = (x-8)(x+9). So, let's break it down step by step, making it easy for everyone to understand. Ready? Let's jump in!

Understanding X-Intercepts

Let's start with the basics. What exactly is an x-intercept? Think of it as the point where your function's graph crosses the x-axis. In other words, it's where the value of y (or f(x) in our case) is zero. This is a crucial concept because it gives us a direct method for finding these points. When f(x) equals zero, we know that (x-8)(x+9) must also equal zero. This is because any value multiplied by zero results in zero, and that's exactly what we need to find the x-intercepts. So, to nail this down, remember the x-intercept is where the graph intersects the x-axis, and at this point, the y-value is always zero. Keeping this definition in mind makes solving these types of problems much more intuitive, so let’s keep going!

The Zero Product Property

The next key concept we need in our toolkit is the Zero Product Property. This might sound super formal, but it's actually quite simple. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. In mathematical terms, if A * B = 0, then either A = 0 or B = 0 (or both!). This property is incredibly useful when we're dealing with factored quadratic equations, like the one we have, f(x) = (x-8)(x+9). This is where we use this property to actually solve for our x-intercepts. So how does this help us? Well, our function is already factored into two parts: (x-8) and (x+9). If the entire function f(x) equals zero, then one or both of these factors must equal zero. This is the magic key that unlocks our solution, because it allows us to turn one equation into two simpler ones, each of which we can solve individually.

Solving for X-Intercepts: Step-by-Step

Okay, let’s put everything together and find those x-intercepts. Remember, we're working with the function f(x) = (x-8)(x+9), and we want to find the values of x when f(x) = 0. Here’s the breakdown:

  1. Set f(x) to Zero: We start by setting the function equal to zero: (x-8)(x+9) = 0.
  2. Apply the Zero Product Property: Now we use the Zero Product Property, which tells us that either (x-8) = 0 or (x+9) = 0.
  3. Solve the First Equation: Let's solve (x-8) = 0. To do this, we add 8 to both sides of the equation: x = 8. So, one of our x-intercepts occurs when x is 8.
  4. Solve the Second Equation: Next, we solve (x+9) = 0. We subtract 9 from both sides of the equation: x = -9. This gives us another x-intercept when x is -9.

So, we've found two values for x that make f(x) equal to zero: x = 8 and x = -9. But remember, x-intercepts are points, not just x-values. To write them as points, we need to include the y-value, which we know is 0 at the x-intercept. Therefore, our x-intercepts are (8, 0) and (-9, 0).

Identifying the Correct Option

Now that we've done the math and found our x-intercepts, let’s look at the options provided in the question:

A. (0, 8) B. (0, -8) C. (9, 0) D. (-9, 0)

Comparing our calculated x-intercepts (8, 0) and (-9, 0) with the options, we see that option D, (-9, 0), matches one of our solutions. Therefore, the correct answer is D. (-9, 0). Remember, we found that x can be either 8 or -9 when f(x) is zero, and we’ve confirmed that (-9, 0) is indeed one of the x-intercepts.

Why Other Options Are Incorrect

It's always a good idea to understand why the other options are incorrect. This helps solidify your understanding and prevents you from making similar mistakes in the future. Let's quickly go through why options A, B, and C are not the x-intercepts of the given function.

  • Option A: (0, 8) This point represents the y-intercept, not the x-intercept. The y-intercept is where the graph crosses the y-axis, which occurs when x = 0. If we plug x = 0 into the function f(x) = (x-8)(x+9), we get f(0) = (0-8)(0+9) = -72, not 8. So, (0, 8) is not on the graph of this function.
  • Option B: (0, -8) Similar to option A, this point also represents a y-intercept, but it's not the correct one for this function. As we calculated above, the actual y-intercept is (0, -72), not (0, -8).
  • Option C: (9, 0) This point looks like an x-intercept because the y-coordinate is 0, but it's incorrect. We found that the x-intercepts occur when x = 8 and x = -9. While 9 is close to 8, it doesn't make the function equal to zero. If we plug x = 9 into the function, we get f(9) = (9-8)(9+9) = 1 * 18 = 18, which is not zero. Therefore, (9, 0) is not an x-intercept.

By understanding why these options are incorrect, you reinforce the process of finding x-intercepts and ensure you can confidently tackle similar problems in the future.

Key Takeaways for Finding X-Intercepts

Before we wrap up, let's quickly recap the key takeaways from this discussion. Knowing these points will make finding x-intercepts a breeze:

  • Definition of X-Intercept: Remember, the x-intercept is the point where the graph of a function crosses the x-axis. At this point, the y-value (or f(x)) is always zero.
  • The Zero Product Property: This property is your best friend when dealing with factored equations. It states that if A * B = 0, then either A = 0 or B = 0 (or both).
  • Steps to Find X-Intercepts:
    1. Set the function equal to zero.
    2. Apply the Zero Product Property to set each factor equal to zero.
    3. Solve each resulting equation to find the x-values.
    4. Write the x-intercepts as points (x, 0).
  • Double-Check Your Answers: Always verify your solutions by plugging them back into the original function to ensure they make the function equal to zero.

Practice Makes Perfect

Alright, guys, that’s it for today! Finding the x-intercept of a quadratic function doesn't have to be intimidating. With a solid understanding of the concepts and a bit of practice, you’ll be solving these problems like a pro in no time. So, next time you encounter a question like this, remember our step-by-step guide and don't forget the Zero Product Property. Keep practicing, and you'll ace it! Happy solving!