Additional Graphing Exercises

[Technology Exe...

Question

Additional Graphing Exercises

[Technology Exercise] Graph the curves in Exercises 109–112. Explain the relationship between the curve’s formula and what you see.

y = −1 / √(4 − x²)

Accepted Answer

The graph of Y equals -1, divided by the square root of 9 minus X2 is shown. Explain how the equation of the curve relates to this graph. Now, we're given the graph of this function here, and we're asked to prove that this is the real graph. So let's detail some of the attributes of this function. Well, first, find the horizontal extent. In other words, we'll find our domain. We have 1 divided by the square root of 9 minus X2. Our domain will be 9 minus X2 is greater than 0. This is because the square root. Has to have all positive values underneath the radical, and because this is a reciprocal function, it cannot equal 0. So we have 9 minus X2 squared than 0. Let's go ahead and solve this. We get X2 equals 9. We take the square root, we get X equals plus or minus 3. That means our horizontal extent will be from -3 to 3. Now, let's look at the Y intercept. We look at our graph, we see that the Y intercept. Occurs at 0. 1/3. This is because if we plug in our valued function. -1, divided by the square root of 9 minus 02. You would get -1 divided by the square root of 9, which is just 1 1/3. Now, we also noticed that this is all below the X-axis. Because this is all below the x-axis, we have a negative sign. In front of the fraction Normally, the function would be above the X-axis if the negative were there, but because the negatives there, It makes everything below the x-axis. Now, let's check its symmetry. We noticed that this has symmetry over the y axis, and we can prove this by checking. If we replace X with negative X, we'll see if those two functions are still the same. We have 1 divided by the square root of 9 minus X2, which just gives us -1 divided by the square root of 9 minus X2. That's the same as the original function, so we can confirm. That it is symmetric across the y axis. Finally, we can check its vertical asymptotes. We can do this by checking the limits. We have the limit. As X approaches 3. And we will check right so And the limit as X approaches 3 on the left side. We have 1 divided by the square root. Of 9 minus 3 squad. Which is -1 divided by 0, or negative infinity. Similarly, We can check the left side by doing the same thing. To get negative infinity. We can tell then our function goes to negative infinity, which means our vertical asymptotes. Occur At X equals 3 and X equals -3 because of symmetry. Based on these 5 attributes, we can explain how this equation relates to the graph shown here. OK, I hope to help you solve the problem. Thanks for watching. Goodbye.

Additional Graphing Exercises

[Technology Exercise] Graph the curves in Exercises 109–112. Explain the relationship between the curve’s formula and what you see.

y = −1 / √(4 − x²)

Key Concepts

Graphing Functions

Understanding Square Roots

Behavior of Rational Functions

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