Graph the functions in Exercises 37-56 y=√(x+4)

Name: Problem 37, Chapter 1: Thomas Calculus
Uploaded: 2024-10-15T22:02:37-08:00
Duration: 2 min 56 s
Channel: Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsohdsno3/5Wc5Bsgjhyqjgxswzij15Z06 Bcrypt_Sha256$$2B$12$Koudzt7Vugfesdqzt.Btsoec1F7Nikxndin/Owbntbjji9Jzcznki
Description: Graph the functions in Exercises 37-56 y=√(x+4)

Key Concepts

Domain Considerations

Determining the domain of a function is crucial when graphing, especially for functions with radicals. For the function y = ?(x + 4), the expression under the square root must be non-negative. This restriction leads to the inequality x + 4 ? 0, which means x must be greater than or equal to -4. This domain informs where on the x-axis the graph will exist.

Square Root Function

A square root function is a type of function that involves taking the square root of the input variable. Its graph typically starts at the point where the radicand (the expression inside the square root) is zero, and the output values are non-negative. This function represents half of a parabola, as it produces only one branch of the quadratic relationship.

Function Transformation - Horizontal Translation

Function transformations involve shifting, stretching, or reflecting the graph of a basic function. In the case of y = ?(x + 4), the expression inside the square root indicates a horizontal shift. By rewriting the function as y = ?[x - (-4)], we see that the graph of the base square root function, y = ?x, is translated 4 units to the left.

Transcript

00:01 We want to graph the function y equals the square root of x plus four.

00:05 So first we need to realize that that is the graph of the square root of x, y equals the square root of x shifted, shifted to the left for units.

00:20 If we add four to the argument of the function, which is what's inside of the function, in this case the square root symbol, it's going to shift to the left that amount.

00:33 So we first need to know what does the graph of y equals square of x look y? we do a quick table of values for x and y, and if our x is zero, y would be the square root of zero, which is zero.

00:50 And if x is one, y would be the square root of one, which is one.

00:54 And then i'm going to skip to a perfect square since that'll be easier to take the square root of.

00:58 If x is four, then y would be the square root of four, which is two, and we'll do one more.

01:05 If x is nine, the square root of nine would give us three.

01:08 So if we give it a sketch of those four points, you have zero, zero, and i'll label our graph here.

01:18 So one, two, three, four, five, six, seven, eight, nine.

01:25 Let me just label those by twos real quick and then the y -axis will go by ones here so that'll be two and that will be four there and then if we plot those points we have zero zero here and then we have the point one one and then we have the point four two which would be right here and then the point nine three should be about right here we can connect those points and we get actually half of a sideways parabola, and it's going to continue on in that direction.

02:00 Now, to graph that function shifted left four, now this is just y -equal squared of x...

Please give Ace some feedback