Answer :
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].
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].
