Intersection Points & Odd Functions: A Tricky Algebra Problem

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Hey guys! Let's dive into a super interesting algebra problem that combines the concepts of intersection points, odd functions, and exponential equations. We're tasked with finding the product of the abscissas (that's the x-coordinates, for those not in the know!) of the intersection points between a line and the graph of a special kind of function – an odd function. This function is defined across all real numbers except zero and has a specific formula for positive x-values. To top it off, we need to multiply our final product by 9. Sounds like a fun challenge, right? Buckle up, because we're about to break it down step by step.

Understanding the Problem

Before we even think about calculations, let's make sure we really understand what the problem is asking. We need to find where the line y = 12 intersects the graph of an odd function. Remember, an odd function has a special symmetry: it's symmetrical about the origin. This means that if a point (x, y) is on the graph, then the point (-x, -y) is also on the graph. This property is super important for solving this problem. We're given the formula for the function when x is greater than 0: y = 2^(3x-8) - 20. This is an exponential function, and it's going to play a key role in finding our intersection points. The line y = 12 is a horizontal line, so we're essentially looking for the x-values where our exponential function (and its odd function counterpart) equals 12. Once we find those x-values (the abscissas), we'll multiply them together and then multiply the result by 9. So, the keywords here are odd function, intersection points, and exponential equation. Keeping these in mind will guide our solution.

Cracking the Exponential Equation

Let's focus on the part of the function we know: y = 2^(3x-8) - 20 for x > 0. We need to find the x-value where this function equals 12, because that's where it intersects the line y = 12. So, we set up the equation:

12 = 2^(3x-8) - 20

Now we need to solve for x. First, let's isolate the exponential term by adding 20 to both sides:

32 = 2^(3x-8)

Hey, 32 is a power of 2! Specifically, 32 = 2^5. This is awesome because it allows us to rewrite our equation as:

2^5 = 2^(3x-8)

When the bases are the same (both 2 in this case), we can equate the exponents:

5 = 3x - 8

Now we have a simple linear equation! Add 8 to both sides:

13 = 3x

And finally, divide by 3:

x = 13/3

So, we've found one of our intersection points! When x = 13/3, the function y = 2^(3x-8) - 20 equals 12. But remember, this is only for x > 0. We need to use the property of odd functions to find the other intersection point.

Leveraging the Odd Function Property

This is where the odd function property comes in clutch! We know that if a point (x, y) is on the graph of an odd function, then the point (-x, -y) is also on the graph. We found that when x = 13/3, y = 12. So, the point (13/3, 12) is on the graph of our function. Because the function is odd, the point (-13/3, -12) must also be on the graph. However, this isn't quite the intersection point we're looking for. We need the point where y = 12, not y = -12. But, it gives us a crucial piece of the puzzle.

Here's the key insight: the odd function property reflects the graph across both the x-axis and the y-axis. We already found the intersection point for x > 0. To find the other intersection point where y = 12, we need to consider the "reflected" part of the function for x < 0. Since the function is odd, the reflection across the origin means that if y = 2^(3x-8) - 20 for x > 0, then for x < 0, the function will essentially be the negative of the reflection of this. To find where the odd function equals 12 when x < 0, we need to find where the original function (before the odd reflection) would equal -12. Think of it like this: the odd function "flips" the part of the graph where x > 0 over both axes. So, to find the intersection on the negative side, we need to consider what x value on the positive side would give us a y-value of -12, before the flip.

So, we set up the equation:

-12 = 2^(3x-8) - 20

Add 20 to both sides:

8 = 2^(3x-8)

And again, we can express 8 as a power of 2: 8 = 2^3, so

2^3 = 2^(3x-8)

Equate the exponents:

3 = 3x - 8

Add 8 to both sides:

11 = 3x

Divide by 3:

x = 11/3

This means that before the odd function reflection, when x = 11/3, y would have been -12. After the odd function reflection, the point (-11/3, 12) is on the graph. So, our second intersection point has an x-coordinate of -11/3.

Final Calculation: Multiplying and Scaling

We've found the x-coordinates (abscissas) of the two intersection points: 13/3 and -11/3. Now, we just need to multiply them together and then multiply by 9, as the problem instructed. The product of the abscissas is:

(13/3) * (-11/3) = -143/9

Finally, we multiply by 9:

(-143/9) * 9 = -143

So, the final answer is -143! Woohoo! We tackled a complex problem by breaking it down into smaller, manageable steps. We used our understanding of odd functions, exponential equations, and a bit of algebraic manipulation to arrive at the solution. This is the beauty of math – connecting different concepts to solve challenging problems. You guys nailed it!

Key Takeaways

  • Odd functions have symmetry about the origin, meaning f(-x) = -f(x). This property is crucial for solving problems involving odd functions.
  • Solving exponential equations often involves expressing both sides of the equation with the same base.
  • Carefully consider the implications of function transformations, especially when dealing with symmetry.
  • Breaking down complex problems into smaller steps makes them easier to solve.

Remember, practice makes perfect! Keep challenging yourselves with these types of problems, and you'll become algebra masters in no time.