Solve Y=3^x: Find Missing Coordinates

by Admin 38 views
Solve y=3^x: Find Missing Coordinates

Hey guys! Let's dive into solving some equations and finding those missing coordinates. We're going to tackle the equation y = 3x, and our mission is to fill in the blanks in ordered pairs so that each pair becomes a valid solution. This means that when we plug in the x and y values, the equation holds true. Sound like fun? Let's get started!

Understanding the Equation

Before we jump into the problems, let's make sure we're all on the same page about what the equation y = 3x means. This is an exponential equation where 3 is the base and x is the exponent. Basically, for any value of x, y is going to be 3 raised to the power of x. Understanding this relationship is key to finding the missing coordinates.

For example:

  • If x = 0, then y = 30 = 1
  • If x = 1, then y = 31 = 3
  • If x = 2, then y = 32 = 9
  • If x = -1, then y = 3-1 = 1/3

And so on. This pattern will guide us as we solve for the missing coordinates. Remember, a coordinate pair is written as (x, y), so we need to find either the x or the y value that makes the equation true.

Solving for Missing Coordinates

Now, let's get to the heart of the matter: finding the missing coordinates. We'll look at a few examples to illustrate the process. For each example, we'll be given an ordered pair with one coordinate missing, and we'll use the equation y = 3x to find the missing value. Get your thinking caps on!

Example 1: Finding y when x is given

Suppose we have the ordered pair (2, ?). Here, we know that x = 2, and we need to find the corresponding y value. To do this, we simply plug x = 2 into our equation:

y = 32 = 9

So, the missing coordinate is 9, and the complete ordered pair is (2, 9). This means that when x is 2, y is 9, and the equation y = 3x holds true.

Example 2: Finding x when y is given

Now let's try a slightly different scenario. Suppose we have the ordered pair (?, 27). In this case, we know that y = 27, and we need to find the corresponding x value. This requires a bit more thinking. We need to find the exponent x such that 3 raised to that power equals 27.

We can rewrite the equation as:

27 = 3x

We know that 27 is 3 * 3 * 3, which is 33. So:

33 = 3x

Therefore, x = 3. The complete ordered pair is (3, 27). When x is 3, y is 27, and the equation y = 3x is satisfied.

Example 3: Dealing with Fractional Values

Let's make things a little more interesting with fractional values. Suppose we have the ordered pair (?, 1/3). Here, y = 1/3, and we need to find x. We need to remember our exponent rules. A negative exponent means we take the reciprocal of the base raised to the positive exponent.

So, we have:

1/3 = 3x

We know that 1/3 is the same as 3-1. Therefore:

3-1 = 3x

So, x = -1. The complete ordered pair is (-1, 1/3). When x is -1, y is 1/3, and the equation y = 3x works perfectly.

Example 4: When y = 1

What if we have the ordered pair (?, 1)? In this case, y = 1, and we need to find x. Remember that any number (except 0) raised to the power of 0 is 1. So:

1 = 3x

Since 30 = 1, x = 0. The complete ordered pair is (0, 1). This is a crucial point to remember for exponential functions: when x is 0, y is always 1, regardless of the base.

Tips and Tricks for Solving

Here are some handy tips and tricks to help you solve these types of problems:

  1. Know Your Powers: Familiarize yourself with the powers of common numbers like 2, 3, 4, and 5. This will make it easier to recognize patterns and find the missing coordinates quickly.
  2. Use Exponent Rules: Remember the rules of exponents, especially when dealing with fractions and negative numbers. For example, a-n = 1/an and a0 = 1.
  3. Rewrite the Equation: Sometimes, rewriting the equation can make it easier to solve. For example, if you have y = 27, rewrite it as 3x = 27 and then express 27 as a power of 3.
  4. Use Logarithms: If you're comfortable with logarithms, they can be a powerful tool for solving for x when y is given. The equation y = 3x can be rewritten as x = log3(y).
  5. Check Your Work: Always double-check your answers by plugging the completed ordered pair back into the original equation to make sure it holds true.

Practice Problems

Alright, guys, now it's your turn to practice! Here are a few problems for you to try on your own:

  1. (4, ?)
  2. (?, 81)
  3. (-2, ?)
  4. (?, 1/9)
  5. (1/2, ?)

Solve these problems using the techniques we've discussed. Remember to show your work and double-check your answers. Practice makes perfect!

Conclusion

Finding missing coordinates in ordered pairs for the equation y = 3x might seem tricky at first, but with a solid understanding of exponential equations and some practice, you'll become a pro in no time! Remember the key principles, use the tips and tricks, and always double-check your work. Now go out there and conquer those equations! You've got this!