In Exercises 43-46, describe a basic graph and a sequence of transformations that can be used to

In Exercises 43-46, describe a basic graph and a sequence of...

Key Concepts

Square Root Function Graph

The square root function, typically written as y = ?x, is the basic graph from which transformations are applied. It is defined only for non-negative x-values, starting at the origin (0,0) and increasing gradually to the right with a curve that flattens as x becomes larger.

Horizontal Translation

Horizontal translation involves shifting the graph left or right. In the function y = -3?(x+1), the term (x+1) indicates that the graph of the basic square root function is shifted horizontally to the left by 1 unit, as the input is increased by 1 before taking the square root.

Vertical Scaling

Vertical scaling refers to stretching or compressing the graph in the vertical direction. The multiplier 3 in the function y = -3?(x+1) scales the output values by a factor of 3, making the graph steeper or more elongated vertically compared to the basic function.

Reflection Across the X-Axis

Reflection across the x-axis occurs when the output of a function is multiplied by -1, causing the graph to flip upside down. In y = -3?(x+1), the negative sign causes the entire graph to be reflected over the x-axis, inverting its vertical orientation.

Transcript

00:01 Okay, so in this question, we'll give it this equation, why is equal to negative 3 times square root of x plus 1? and we're asked to identify the basic function and also asks how we can go about transforming this basic function into the function that we have.

00:16 So the basic function for this question, and we can see that we're really adding 1 to the x and we're times the square root by 3, and these are really constant, they're not really basic.

00:28 So the basic function for this, it's actually square root of x.

00:34 And it looks something like this.

00:37 It goes up like that, square of x.

00:42 So how do we transform square root of x into this y, our function we have today? so in order to do that, we need to be aware of three functional transformation.

00:52 One is adding a constant to x.

00:56 This is different from adding a function to the entire function.

01:01 So in this case, it would be different.

01:04 In this case, f of x is the square of x, right? so let's call it as f of x.

01:09 So in this case, we have square of x plus a.

01:13 And this thing would look like square of x, entire thing plus a.

01:17 So in this case, if a is greater than zero, meaning that less than zero, meaning it's negative, then it's going to shift the graph to the right by a units.

01:27 And if a is greater than zero, then it's going to shift the graph to the left by a units.

01:31 And two, second transformation we need to be aware of.

01:35 It's horizontally stretching, multiplying a constant to the entire function.

01:39 So in this case, it looks something like this.

01:43 Because we know f of x, it will look a times square of x.

01:46 It will look something like this.

01:48 So in this case, we're going to either vertically stretch it or vertically shrink the function.

01:53 If a is greater than one, then we're going to stretch vertically, vertically stretch by a.

02:00 And if a it's smaller than one, then we're going to vertically shrink by a.

02:08 And the third of functional transformation that we need to be aware of, actually we need to be, it's multiplying, have a negative sign in front of the entire function.

02:19 So in this case, it will look something like negative square of x...

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