c. describe the transformation from the solid function to the dashed function and write the horizontal asymptote of the dashed function.

transformation:
horizontal asymptote:

transformation:
horizontal asymptote:

transformation:
horizontal asymptote:

Question

c. describe the transformation from the solid function to the dashed function and write the horizontal asymptote of the dashed function.

transformation:
horizontal asymptote:

Accepted Answer

### First Graph (Left)
# Explanation:
## Step1: Analyze Vertical Shift
The solid function (exponential decay, horizontal asymptote $ y = 1 $) and dashed function (exponential, horizontal asymptote $ y = -2 $): The dashed function is shifted down. Calculate the vertical shift: $ 1 - (-2) = 3 $ units down? Wait, no—solid’s asymptote is $ y = 1 $, dashed’s is $ y = -2 $. So vertical shift: $ 1 - 3 = -2 $, so shift down 3 units? Wait, solid is $ y = a^x + 1 $, dashed is $ y = a^x - 2 $? Wait, no, the dashed is below. Wait, the solid curve is above the x - axis, dashed is below. Wait, maybe reflection? No, solid is decay (decreasing), dashed is increasing? Wait, no, the solid is a decay curve (like $ y = (\frac{1}{2})^x + 0 $? No, the solid has horizontal asymptote $ y = 1 $ (since it approaches $ y = 1 $ as $ x \to \infty $). The dashed curve is an exponential growth? Wait, no, the dashed is a curve that starts low and increases, while solid decreases. Wait, maybe vertical shift and reflection? Wait, no, let's check the horizontal asymptote. Solid’s asymptote: $ y = 1 $. Dashed’s asymptote: $ y = -2 $? Wait, no, looking at the grid, the solid curve approaches $ y = 1 $ (since it’s near $ y = 1 $ on the right). The dashed curve approaches $ y = -2 $? Wait, no, the dashed curve (dotted) is below, maybe vertical shift down by 3 units? Wait, no, maybe reflection over x - axis and shift? Wait, maybe better to see: the solid function is $ y = b^x + 1 $ (decay, so $ 0 < b < 1 $), dashed is $ y = -b^{-x} - 2 $? No, this is getting complicated. Wait, the first graph: solid is a decay curve (decreasing, horizontal asymptote $ y = 1 $), dashed is a growth curve (increasing, horizontal asymptote $ y = -2 $)? No, maybe vertical shift down by 3 units and reflection over x - axis? Wait, no, let's focus on transformation: from solid to dashed. The solid is above the x - axis, dashed is below. Maybe vertical shift down by 3 units? Wait, the horizontal asymptote of solid: $ y = 1 $, dashed: $ y = -2 $. So the transformation: vertical shift down by 3 units? Wait, no, $ 1 - 3 = -2 $, so vertical shift down 3 units. And also, reflection over x - axis? Wait, no, the solid is decreasing, dashed is increasing. So maybe reflection over x - axis (which would flip the direction) and vertical shift down? Wait, this is confusing. Alternatively, maybe the solid function is $ y = (\frac{1}{2})^x + 1 $, and the dashed is $ y = -(\frac{1}{2})^{-x} - 2 $? No, maybe simpler: the transformation is a vertical shift down by 3 units and a reflection over the x - axis? Wait, no, let's check the first graph's transformation:

## Step1: Identify Horizontal Asymptote of Solid
The solid curve approaches $ y = 1 $ (horizontal asymptote $ y = 1 $).

## Step2: Identify Horizontal Asymptote of Dashed
The dashed curve approaches $ y = -2 $? Wait, no, looking at the grid, the solid curve is near $ y = 1 $ on the right. The dashed curve (dotted) is near $ y = -2 $ on the right? No, the dashed curve is a curve that starts at the bottom left and increases, while solid decreases. Maybe the transformation is a vertical shift down by 3 units and a reflection over the x - axis? Wait, maybe the correct transformation is: vertical shift down by 3 units and reflection over the x - axis? No, perhaps the solid function is $ y = b^x + 1 $ (decay), dashed is $ y = -b^{-x} - 2 $, but this is too complex. Alternatively, let's look at the second graph.

### Second Graph (Middle)
Solid curve: horizontal asymptote $ y = 1 $ (approaches $ y = 1 $ as $ x \to \infty $). Dashed curve: horizontal asymptote $ y = -3 $ (approaches $ y = -3 $ as $ x \to \infty $)? Wait, no, the dashed curve is below, and the solid is above. The solid curve is a decay curve (decreasing), dashed is a curve that decreases (like a decay) but below the x - axis. Wait, solid’s asymptote: $ y = 1 $, dashed’s asymptote: $ y = -3 $. So vertical shift down by 4 units? No, $ 1 - 4 = -3 $. Or reflection over x - axis and shift? Wait, solid is $ y = (\frac{1}{2})^x + 1 $, dashed is $ y = -(\frac{1}{2})^x - 3 $? No, this is not right. Wait, the middle graph: solid is a curve with horizontal asymptote $ y = 1 $ (approaching from above), dashed is a curve with horizontal asymptote $ y = -3 $ (approaching from below). So transformation: vertical shift down by 4 units? No, $ 1 - 4 = -3 $. Or reflection over x - axis (which would make the asymptote $ y = -1 $) and then shift down by 2 units? $ -1 - 2 = -3 $. So reflection over x - axis and vertical shift down by 2 units.

### Third Graph (Right)
Solid curve: a parabola? Wait, no, it's a curve that looks like a parabola, opening upwards, with horizontal asymptote? No, parabolas don't have horizontal asymptotes, but maybe it's an exponential? Wait, no, the solid curve is a parabola - like curve, opening upwards, with vertex at $ (0, -1) $? Wait, no, the grid: x - axis and y - axis. The solid curve (black) and dashed (dotted) curve. The dashed curve is above, solid is below. The horizontal asymptote: for exponential functions, but this looks like a parabola. Wait, maybe it's a quadratic function. The solid curve: $ y = x^2 - 1 $, dashed: $ y = (x - 0)^2 + 2 $? So vertical shift up by 3 units? The horizontal asymptote for quadratics: none, but maybe it's an exponential. Wait, no, the third graph: solid is a curve opening upwards, dashed is also opening upwards, dashed is above solid. So transformation: vertical shift up by 3 units? The horizontal asymptote: if they are exponential, but they look like parabolas. Wait, maybe the third graph is a quadratic, so no horizontal asymptote, but the question says "horizontal asymptote", so they must be exponential functions.

Wait, maybe I made a mistake. Let's start over.

#### First Graph (Left)
- Solid function: horizontal asymptote $ y = 1 $ (approaches $ y = 1 $ as $ x \to \infty $), decreasing (exponential decay: $ y = a^x + 1 $, $ 0 < a < 1 $).
- Dashed function: horizontal asymptote $ y = -2 $ (approaches $ y = -2 $ as $ x \to \infty $)? No, as $ x \to \infty $, dashed is increasing, so maybe as $ x \to -\infty $ it approaches $ y = -2 $. Wait, exponential functions: $ y = a^x + k $, horizontal asymptote $ y = k $. So solid: $ y = (\frac{1}{2})^x + 1 $ (as $ x \to \infty $, $ (\frac{1}{2})^x \to 0 $, so $ y \to 1 $; as $ x \to -\infty $, $ (\frac{1}{2})^x \to \infty $, so $ y \to \infty $). Dashed: $ y = -2^x - 2 $ (as $ x \to \infty $, $ -2^x \to -\infty $, so $ y \to -\infty $; as $ x \to -\infty $, $ -2^x \to 0 $, so $ y \to -2 $). So from solid ($ y = (\frac{1}{2})^x + 1 $) to dashed ($ y = -2^x - 2 $): reflection over y - axis (since $ (\frac{1}{2})^x = 2^{-x} $), reflection over x - axis (multiply by - 1: $ -2^{-x} $), and vertical shift down by 1 unit? $ -2^{-x} - 1 $, no, this is not matching.

Alternative approach for transformation:

#### First Graph:
- Transformation: Vertical shift down by 3 units and reflection over the x - axis (or reflection over x - axis and vertical shift down by 2 units). The horizontal asymptote of the dashed function: $ y = -2 $.

#### Second Graph:
- Transformation: Reflection over the x - axis and vertical shift down by 2 units (or vertical shift down by 4 units? No, solid’s asymptote $ y = 1 $, dashed’s $ y = -3 $, so $ 1 - 4 = -3 $, so vertical shift down by 4 units? No, $ 1 - 3 = -2 $, no).

#### Third Graph:
- Transformation: Vertical shift up by 3 units (solid curve is below, dashed above). Horizontal asymptote: for exponential, but it looks like a parabola. Wait, maybe the third graph is a quadratic, so no horizontal asymptote, but the question says "horizontal asymptote", so it must be exponential. Maybe the solid is $ y = 2^x - 1 $, dashed is $ y = 2^x + 2 $, so vertical shift up by 3 units. Horizontal asymptote: none for exponential? No, exponential functions have horizontal asymptote $ y = 0 $ if in form $ y = a^x $, but with shifts, $ y = a^x + k $ has $ y = k $. So if solid is $ y = 2^x - 1 $ (horizontal asymptote $ y = -1 $), dashed is $ y = 2^x + 2 $ (horizontal asymptote $ y = 2 $)? No, the third graph's solid curve has horizontal asymptote $ y = -1 $, dashed $ y = 2 $, so vertical shift up by 3 units.

This is getting too confusing. Let's provide the most probable transformations and horizontal asymptotes:

### First Graph:
- Transformation: Vertical shift down by 3 units and reflection over the x - axis (or reflection over x - axis and vertical shift down by 2 units).
- Horizontal Asymptote: $ y = -2 $

### Second Graph:
- Transformation: Reflection over the x - axis and vertical shift down by 2 units (or vertical shift down by 4 units).
- Horizontal Asymptote: $ y = -3 $

### Third Graph:
- Transformation: Vertical shift up by 3 units.
- Horizontal Asymptote: $ y = 2 $ (if solid has $ y = -1 $, dashed $ y = 2 $)

But maybe a better way:

#### First Graph:
- Solid function: horizontal asymptote $ y = 1 $ (exponential decay: $ y = (\frac{1}{2})^x + 1 $)
- Dashed function: horizontal asymptote $ y = -2 $
- Transformation: Vertical shift down by 3 units (since $ 1 - 3 = -2 $) and reflection over the x - axis (to flip the direction from decay to growth)
- Horizontal Asymptote: $ y = -2 $

#### Second Graph:
- Solid function: horizontal asymptote $ y = 1 $ (exponential decay: $ y = (\frac{1}{2})^x + 1 $)
- Dashed function: horizontal asymptote $ y = -3 $
- Transformation: Reflection over the x - axis (asymptote $ y = -1 $) and vertical shift down by 2 units ($ -1 - 2 = -3 $)
- Horizontal Asymptote: $ y = -3 $

#### Third Graph:
- Solid function: horizontal asymptote $ y = -1 $ (exponential growth: $ y = 2^x - 1 $)
- Dashed function: horizontal asymptote $ y = 2 $
- Transformation: Vertical shift up by 3 units ($ -1 + 3 = 2 $)
- Horizontal Asymptote: $ y = 2 $

# Answers:

### First Graph:
- Transformation: Vertical shift down by 3 units and reflection over the x - axis (or reflection over x - axis and vertical shift down by 2 units)
- Horizontal Asymptote: $ y = -2 $

### Second Graph:
- Transformation: Reflection over the x - axis and vertical shift down by 2 units (or vertical shift down by 4 units)
- Horizontal Asymptote: $ y = -3 $

### Third Graph:
- Transformation: Vertical shift up by 3 units
- Horizontal Asymptote: $ y = 2 $

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QUESTION IMAGE

Question

Explanation:

First Graph (Left)

Step1: Analyze Vertical Shift

Step1: Identify Horizontal Asymptote of Solid

Step2: Identify Horizontal Asymptote of Dashed

Second Graph (Middle)

Answer:

First Graph:

Second Graph:

Third Graph: