Graphing Functions And Their Inverses: Y=3x-1 & Y=x^2-4

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Hey guys! Today, we're diving into the fascinating world of functions and their inverses. Specifically, we're going to graph two functions: y = 3x - 1 and y = x² - 4 (only for x > 0), and then, plot their inverses all on the same coordinate plane. Buckle up; it's going to be a fun ride!

Graphing the Original Functions

1. Graphing y = 3x - 1

Let's kick things off with the linear function, y = 3x - 1. Graphing this is pretty straightforward because it's a straight line. To graph a line, all we need are two points. We can find these points by choosing two values for x and then calculating the corresponding y values.

  • Choosing x = 0:

    If x = 0, then y = 3(0) - 1 = -1. So, our first point is (0, -1). This point is also the y-intercept of the line.

  • Choosing x = 1:

    If x = 1, then y = 3(1) - 1 = 2. So, our second point is (1, 2).

Now that we have two points, (0, -1) and (1, 2), we can plot these on a coordinate plane and draw a straight line through them. Make sure to extend the line in both directions, as the function is defined for all real numbers.

Key characteristics of this line:

  • Slope: The slope of the line is 3, which means for every 1 unit we move to the right along the x-axis, we move 3 units up along the y-axis.
  • Y-intercept: The line crosses the y-axis at y = -1.
  • X-intercept: To find where the line crosses the x-axis, we set y = 0 and solve for x: 0 = 3x - 1, which gives x = 1/3. So the x-intercept is (1/3, 0).

Graphing this line accurately is crucial because it sets the stage for understanding its inverse later on. Take your time and make sure your line is straight and passes through the points you calculated.

2. Graphing y = x² - 4 for x > 0

Next up, we have the quadratic function y = x² - 4, but with a twist! We're only considering the part of the graph where x > 0. This means we're focusing on the right side of the y-axis.

To graph this, let's pick a few x values greater than 0 and find the corresponding y values:

  • Choosing x = 0:

    Although we're only considering x > 0, it's useful to see where the graph would start if it weren't restricted. If x = 0, then y = (0)² - 4 = -4. So, the point would be (0, -4). But remember, we're only graphing for x > 0, so this point will be the start (but not included) of our graph.

  • Choosing x = 1:

    If x = 1, then y = (1)² - 4 = -3. So, our point is (1, -3).

  • Choosing x = 2:

    If x = 2, then y = (2)² - 4 = 0. So, our point is (2, 0). This is the x-intercept of the restricted graph.

  • Choosing x = 3:

    If x = 3, then y = (3)² - 4 = 5. So, our point is (3, 5).

Now, plot these points on the coordinate plane: (1, -3), (2, 0), and (3, 5). Since x > 0, the graph starts just to the right of the y-axis at (0, -4) and curves upwards. It's a part of a parabola that opens upwards. Don't forget to only draw the part of the curve where x is strictly greater than 0.

Key characteristics of this restricted parabola:

  • Vertex (theoretical): The vertex of the full parabola is at (0, -4), but since we're restricting x > 0, we don't include this point in our graph. Instead, the graph starts approaching this point but never actually reaches it.
  • X-intercept: The restricted graph crosses the x-axis at (2, 0).
  • Shape: The graph is a right-side-up parabola, but only the right half is shown.

Graphing this restricted quadratic function correctly is also super important for visualizing its inverse, which we'll tackle next.

Graphing the Inverse Functions

Now comes the fun part: finding and graphing the inverse functions. Remember, to find the inverse of a function, we essentially swap the roles of x and y and then solve for y.

1. Finding and Graphing the Inverse of y = 3x - 1

To find the inverse of y = 3x - 1, we switch x and y to get x = 3y - 1. Now, we solve for y:

  • Add 1 to both sides: x + 1 = 3y
  • Divide by 3: y = (x + 1) / 3

So, the inverse function is y = (x + 1) / 3. This is another linear function, so we can graph it just like we did before. Let's find two points:

  • Choosing x = -1:

    If x = -1, then y = (-1 + 1) / 3 = 0. So, our first point is (-1, 0).

  • Choosing x = 2:

    If x = 2, then y = (2 + 1) / 3 = 1. So, our second point is (2, 1).

Plot these points, (-1, 0) and (2, 1), on the same coordinate plane as the original function and draw a straight line through them. You should notice that this line is a reflection of the original line, y = 3x - 1, across the line y = x. This is a key property of inverse functions: their graphs are reflections of each other across the line y = x.

Key characteristics of the inverse line:

  • Slope: The slope of the inverse line is 1/3, which is the reciprocal of the original line's slope (3).
  • Y-intercept: The inverse line crosses the y-axis at y = 1/3.
  • X-intercept: The inverse line crosses the x-axis at x = -1.

2. Finding and Graphing the Inverse of y = x² - 4 for x > 0

To find the inverse of y = x² - 4 (for x > 0), we switch x and y to get x = y² - 4. Now, we solve for y:

  • Add 4 to both sides: x + 4 = y²
  • Take the square root of both sides: y = ±√(x + 4)

Since we know that for the original function x > 0, the range will be y > -4. When finding the inverse we know that the domain of the inverse must be x > -4 and the range must be y > 0. Because we have a range of y > 0 we can say that the inverse function is y = √(x + 4). We only take the positive square root because we restricted the original function to x > 0.

Now, let's plot a few points:

  • Choosing x = -4:

    If x = -4, then y = √(-4 + 4) = 0. So, our point is (-4, 0). This is the start of our inverse function graph.

  • Choosing x = -3:

    If x = -3, then y = √(-3 + 4) = 1. So, our point is (-3, 1).

  • Choosing x = 0:

    If x = 0, then y = √(0 + 4) = 2. So, our point is (0, 2).

  • Choosing x = 5:

    If x = 5, then y = √(5 + 4) = 3. So, our point is (5, 3).

Plot these points on the coordinate plane. You'll notice that the graph is a reflection of the original restricted parabola across the line y = x. It's a sideways-opening square root function that starts at (-4, 0) and curves upwards to the right.

Key characteristics of the inverse square root function:

  • Starting Point: The graph starts at (-4, 0).
  • Shape: The graph is a sideways-opening square root function, curving upwards.

Putting It All Together

Okay, guys, you've now graphed the original functions and their inverses on the same coordinate plane. To recap:

  • You graphed the line y = 3x - 1 and its inverse y = (x + 1) / 3.
  • You graphed the restricted parabola y = x² - 4 (for x > 0) and its inverse y = √(x + 4).

Remember that the graphs of a function and its inverse are reflections of each other across the line y = x. Make sure your graphs demonstrate this property visually.

This exercise not only helps you understand how to graph functions and their inverses but also reinforces the concept of inverse functions as