Answered

Explore Westonci.ca, the top Q&A platform where your questions are answered by professionals and enthusiasts alike. Get immediate and reliable solutions to your questions from a community of experienced experts on our Q&A platform. Get precise and detailed answers to your questions from a knowledgeable community of experts on our Q&A platform.

How would you describe the difference between the graphs of [tex][tex]$f(x) = x^2 + 4$[/tex][/tex] and [tex][tex]$g(y) = y^2 + 4$[/tex][/tex]?

A. [tex][tex]$g(y)$[/tex][/tex] is a reflection of [tex][tex]$f(x)$[/tex][/tex] over the line [tex][tex]$y = x$[/tex][/tex].

B. [tex][tex]$g(y)$[/tex][/tex] is a reflection of [tex][tex]$f(x)$[/tex][/tex] over the [tex][tex]$y$[/tex][/tex]-axis.

C. [tex][tex]$g(y)$[/tex][/tex] is a reflection of [tex][tex]$f(x)$[/tex][/tex] over the [tex][tex]$x$[/tex][/tex]-axis.

D. [tex][tex]$g(y)$[/tex][/tex] is a reflection of [tex][tex]$f(x)$[/tex][/tex] over the line [tex][tex]$y = 1$[/tex][/tex].

Sagot :

GinnyAnswer
To understand the difference between the graphs of [tex]\( f(x) = x^2 + 4 \)[/tex] and [tex]\( g(y) = y^2 + 4 \)[/tex], we need to interpret the transformations that can be applied to these functions.

1. Graph of [tex]\( f(x) = x^2 + 4 \)[/tex]:
- This is a parabolic function.
- The vertex of this parabola is at the point (0, 4).
- The parabola opens upwards since the coefficient of [tex]\( x^2 \)[/tex] is positive.

2. Graph of [tex]\( g(y) = y^2 + 4 \)[/tex]:
- On the surface, it appears to be similarly structured to [tex]\( f(x) = x^2 + 4 \)[/tex], but written in terms of [tex]\( y \)[/tex] instead of [tex]\( x \)[/tex].
- When we set [tex]\( y = f(x) \)[/tex], we get [tex]\( y = x^2 + 4 \)[/tex].
- Reflecting [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex] means swapping the roles of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

Now let's determine which transformation appropriately describes [tex]\( g(y) \)[/tex]:

- Option A: [tex]\( g(y) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex].
- Reflecting over [tex]\( y = x \)[/tex] swaps [tex]\( x \)[/tex] and [tex]\( y \)[/tex].
- This corresponds exactly to how we've defined [tex]\( f(x) = x^2 + 4 \)[/tex] and [tex]\( g(y) = y^2 + 4 \)[/tex].

- Option B: [tex]\( g(y) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] over the [tex]\( y \)[/tex]-axis.
- Reflecting over the [tex]\( y \)[/tex]-axis changes [tex]\( x \)[/tex] to [tex]\(-x\)[/tex].
- [tex]\( f(x) \)[/tex] would then become [tex]\( f(-x) \)[/tex], or [tex]\((-x)^2 + 4 = x^2 + 4 \)[/tex], which is still [tex]\( f(x) \)[/tex], so this doesn't describe a change to [tex]\( g(y) \)[/tex].

- Option C: [tex]\( g(y) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] over the [tex]\( x \)[/tex]-axis.
- Reflecting over the [tex]\( x \)[/tex]-axis changes [tex]\( y \)[/tex] to [tex]\(-y\)[/tex].
- Thus, [tex]\( g(y) \)[/tex] becomes [tex]\( -f(x) \)[/tex], or [tex]\(-(x^2 + 4) = -x^2 - 4 \)[/tex], which doesn't match the form of [tex]\( g(y) \)[/tex].

- Option D: [tex]\( g(y) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = 1 \)[/tex].
- Reflecting over [tex]\( y = 1 \)[/tex] would translate the function up or down relative to [tex]\( y = 1 \)[/tex].
- This does not produce a simple [tex]\( y^2 + 4 \)[/tex] transformation.

Combining these interpretations, the correct difference between [tex]\( f(x) \)[/tex] and [tex]\( g(y) \)[/tex] is:

Option A: [tex]\( g(y) \)[/tex] is a reflection of [tex]\( f(x) \)[/tex] over the line [tex]\( y = x \)[/tex].
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. We're glad you chose Westonci.ca. Revisit us for updated answers from our knowledgeable team.