7. the cube root parent function is vertically stretched by a factor of 6, reflected over the x-axis, horizontal translation -5 (left) units and vertical translation by 2 units up. write the new cube root function.
8. the cube root parent function is reflected across the y-axis, vertically compressed by a factor of $\frac{1}{8}$, and horizontal translation -2 (left) units. write the new cube root function.
9. determine the transformations applied to the following cubic function.
10. determine the transformations applied to the following cube root function.

Question

Accepted Answer

### Question 7
# Explanation:
## Step1: Recall the parent function
The parent cube root function is $ y = \sqrt[3]{x} $.

## Step2: Apply vertical stretch
A vertical stretch by a factor of $ a $ is given by multiplying the function by $ a $. Here, $ a = 6 $, so we have $ y = 6\sqrt[3]{x} $.

## Step3: Reflect over the x - axis
Reflecting over the x - axis changes the sign of the function. So now we have $ y=- 6\sqrt[3]{x} $.

## Step4: Apply horizontal translation
A horizontal translation of $ h $ units is given by replacing $ x $ with $ x - h $. A horizontal translation of - 5 (left 5 units) means $ h=-5 $, so we replace $ x $ with $ x-(-5)=x + 5 $. The function becomes $ y=-6\sqrt[3]{x + 5} $.

## Step5: Apply vertical translation
A vertical translation of $ k $ units up is given by adding $ k $ to the function. Here, $ k = 2 $, so we add 2 to the function. The final function is $ y=-6\sqrt[3]{x + 5}+2 $.

# Answer:
$ y=-6\sqrt[3]{x + 5}+2 $

### Question 8
# Explanation:
## Step1: Recall the parent function
The parent cube root function is $ y=\sqrt[3]{x} $.

## Step2: Reflect over the y - axis
Reflecting over the y - axis means replacing $ x $ with $ -x $. So the function becomes $ y = \sqrt[3]{-x}=-\sqrt[3]{x} $ (since $ \sqrt[3]{-x}=-\sqrt[3]{x} $).

## Step3: Apply vertical compression
A vertical compression by a factor of $ a=\frac{1}{8} $ is given by multiplying the function by $ a $. So we have $ y=\frac{1}{8}\times(-\sqrt[3]{x})=-\frac{1}{8}\sqrt[3]{x} $.

## Step4: Apply horizontal translation
A horizontal translation of - 2 (left 2 units) means we replace $ x $ with $ x-(-2)=x + 2 $. The final function is $ y =-\frac{1}{8}\sqrt[3]{x + 2} $.

# Answer:
$ y=-\frac{1}{8}\sqrt[3]{x + 2} $

### Question 9
# Explanation:
1. **Identify the parent function**: The parent cubic function is $ y = x^{3} $.
2. **Horizontal translation**: The graph of the cubic function seems to be shifted horizontally. The vertex (or the point of inflection) of the parent cubic function $ y=x^{3} $ is at $ (0,0) $. For the given graph, the point of inflection seems to be at $ (0,-3) $? Wait, no, looking at the graph, the key point (the point where the function changes its curvature) is at $ x = 0,y=-3 $? Wait, actually, let's re - examine. The parent cubic function $ y = x^{3} $ passes through $ (0,0) $, $ (1,1) $, $ (- 1,-1) $. The given graph: Let's check the x - intercept or the point of inflection. The graph is shifted horizontally? Wait, no, the graph of the cubic function in the picture: Let's see the transformation. The parent function $ y=x^{3} $. If we look at the graph, it seems to be reflected over the x - axis (since the cubic function is decreasing from left to right, while the parent $ y = x^{3} $ is increasing from left to right for $ x>0 $ and decreasing for $ x < 0 $, but the given graph has a different behavior). Wait, actually, the cubic function $ y=-x^{3}-3 $? Wait, no, let's do it step by step.
   - **Reflection**: The parent cubic function $ y = x^{3} $ has a positive leading coefficient (the coefficient of $ x^{3} $ is 1). The given graph is decreasing as $ x $ increases, so it is reflected over the x - axis (multiply by - 1), so we have $ y=-x^{3} $.
   - **Vertical translation**: The graph of $ y=-x^{3} $ has its point of inflection at $ (0,0) $. The given graph has its point of inflection at $ (0,-3) $, so it is shifted down by 3 units (vertical translation of - 3 units). So the transformations are: reflection over the x - axis, vertical translation 3 units down, and maybe no horizontal translation (or horizontal translation 0 units). Wait, also, looking at the graph, the shape seems to be vertically stretched or compressed? Wait, the parent $ y=-x^{3} $ at $ x = 1 $ is $ y=-1 $, at $ x = 2 $ is $ y=-8 $. In the given graph, at $ x = 1 $, what's the y - value? The graph at $ x = 1 $ seems to be at $ y=-4 $ or so? Wait, maybe there is a vertical stretch. Wait, the parent cubic function $ y = x^{3} $. Let's assume the function is $ y=-2x^{3}-3 $? Wait, no, maybe I made a mistake. Let's start over.
   - The general form of a cubic function is $ y=a(x - h)^{3}+k $, where $ (h,k) $ is the point of inflection.
   - For the parent function $ y = x^{3} $, $ h = 0,k = 0,a = 1 $.
   - Looking at the given graph, the point of inflection is at $ (0,-3) $, so $ h = 0,k=-3 $.
   - The function is decreasing, so $ a<0 $. Let's take two points. Let's say when $ x = 1 $, the y - value: from the graph, when $ x = 1 $, $ y\approx - 5 $ (since at $ x = 1 $, the graph is below $ y=-4 $). For the function $ y=a(x - 0)^{3}-3=ax^{3}-3 $. When $ x = 1 $, $ y=a - 3 $. If $ y=-5 $, then $ a=-2 $. So the function is $ y=-2x^{3}-3 $. So the transformations are: vertical stretch by a factor of 2, reflection over the x - axis (since $ a=-2 $, the negative sign indicates reflection over x - axis and the absolute value 2 indicates vertical stretch), and vertical translation 3 units down.

# Brief Explanations:
The parent cubic function $ y = x^{3} $ undergoes the following transformations:
1. **Reflection over the x - axis**: The function changes from increasing to decreasing (the sign of the leading coefficient changes from positive to negative).
2. **Vertical stretch by a factor of 2**: The coefficient of $ x^{3} $ is - 2 (the absolute value 2 indicates the vertical stretch).
3. **Vertical translation 3 units down**: The function is shifted down by 3 units (the constant term is - 3).

# Answer:
The cubic function $ y = x^{3} $ is vertically stretched by a factor of 2, reflected over the x - axis, and vertically translated 3 units down to get the function (e.g., $ y=-2x^{3}-3 $).

### Question 10
# Explanation:
1. **Identify the parent function**: The parent cube root function is $ y=\sqrt[3]{x} $.
2. **Horizontal translation**: The parent cube root function $ y = \sqrt[3]{x} $ has its "corner" (the point where the function changes its rate of increase) at $ (0,0) $. The given cube root function has its "corner" at $ (2,1) $? Wait, no, looking at the graph, the key point (the point where the function starts to increase more rapidly) is at $ (2,2) $? Wait, no, let's look at the coordinates. The parent cube root function $ y=\sqrt[3]{x} $ passes through $ (0,0) $, $ (1,1) $, $ (8,2) $. The given graph passes through $ (0,1) $, $ (2,2) $, etc.
   - The general form of a transformed cube root function is $ y=a\sqrt[3]{x - h}+k $.
   - The parent function $ y=\sqrt[3]{x} $ has $ a = 1,h = 0,k = 0 $.
   - Looking at the graph, the point $ (0,1) $: If we assume $ h = 2 $ (horizontal translation), $ k = 1 $ (vertical translation), and $ a = 1 $. Let's check: $ y=\sqrt[3]{x - 2}+1 $. When $ x = 2 $, $ y=\sqrt[3]{0}+1 = 1 $? No, the graph has a "corner" at $ (2,2) $. Wait, when $ x = 2 $, $ y = 2 $. For $ y=\sqrt[3]{x - 2}+1 $, when $ x = 2 $, $ y=0 + 1=1 $, which is not correct. Wait, when $ x = 10 $, $ \sqrt[3]{10-2}=\sqrt[3]{8}=2 $, then $ y=2 + 1=3 $, but the graph at $ x = 8 $ has $ y = 4 $? No, maybe my approach is wrong.
   - Let's take two points. The parent function $ y=\sqrt[3]{x} $ has points $ (0,0) $, $ (1,1) $, $ (8,2) $. The given graph has points: Let's see, when $ x = 0 $, $ y = 1 $; when $ x = 2 $, $ y = 2 $; when $ x = 10 $, $ y = 4 $. Wait, the difference between the y - values: from $ x = 0,y = 1 $ to $ x = 2,y = 2 $, the change in $ x $ is 2, change in $ y $ is 1. From $ x = 2,y = 2 $ to $ x = 10,y = 4 $, change in $ x $ is 8, change in $ y $ is 2. Which is similar to the parent function $ y=\sqrt[3]{x} $ (change in $ x $ from 0 to 8 is 8, change in $ y $ from 0 to 2 is 2). So the transformation:
     - **Horizontal translation**: The parent function's point $ (8,2) $ corresponds to the given function's point $ (10,4) $? No, wait the given graph: Let's look at the "corner" point. The parent cube root function $ y=\sqrt[3]{x} $ has its key point at $ (0,0) $. The given graph has its key point at $ (2,1) $? No, the graph in the picture: the curve starts to rise more steeply at $ x = 2 $, $ y = 2 $. Wait, maybe the function is $ y=\sqrt[3]{x - 2}+1 $. Let's check:
       - When $ x = 2 $, $ y=\sqrt[3]{0}+1 = 1 $, no, that's not matching. Wait, maybe the vertical translation is 1 unit up and horizontal translation 2 units right? No, the graph is shifted left? Wait, the parent function $ y=\sqrt[3]{x} $ at $ x = 0 $ is $ y = 0 $, at $ x = 8 $ is $ y = 2 $. The given graph at $ x = 0 $ is $ y = 1 $, at $ x = 8 $ is $ y = 3 $. So it is a vertical translation of 1 unit up and no horizontal translation? Wait, no, the slope of the cube root function: the derivative of $ y=\sqrt[3]{x}=x^{\frac{1}{3}} $ is $ y'=\frac{1}{3}x^{-\frac{2}{3}} $. The given graph has the same shape as the parent cube root function, so there is no reflection or stretch/compression. The key point (the point where the function changes its curvature) of the parent function is at $ (0,0) $, and for the given graph, it is at $ (0,1) $? No, looking at the graph, the curve passes through $ (0,1) $ and $ (2,2) $. So the transformation is: vertical translation 1 unit up and horizontal translation 2 units left? Wait, if we have $ y=\sqrt[3]{x + 2}+1 $. When $ x=-2 $, $ y=\sqrt[3]{0}+1 = 1 $, when $ x = 0 $, $ y=\sqrt[3]{2}+1\approx1.26 + 1=2.26 $, no. Wait, maybe the correct transformation is: vertical translation 1 unit up and no horizontal translation. Wait, the parent function $ y=\sqrt[3]{x} $ at $ x = 0 $ is $ y = 0 $, at $ x = 1 $ is $ y = 1 $, at $ x = 8 $ is $ y = 2 $. The given graph at $ x = 0 $ is $ y = 1 $, at $ x = 1 $ is $ y=\sqrt[3]{1}+1 = 2 $, at $ x = 8 $ is $ y=\sqrt[3]{8}+1 = 3 $. Ah! So the function is $ y=\sqrt[3]{x}+1 $. So the transformation is vertical translation 1 unit up.

# Brief Explanations:
The parent cube root function $ y=\sqrt[3]{x} $ undergoes a vertical translation 1 unit up. There is no reflection, stretch, or horizontal translation. The shape of the function remains the same as the parent cube root function, and it is shifted up by 1 unit.

# Answer:
The cube root parent function $ y=\sqrt[3]{x} $ is vertically translated 1 unit up to get the function $ y=\sqrt[3]{x}+1 $.

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QUESTION IMAGE

Question

Explanation:

Question 7

Step1: Recall the parent function

Step2: Apply vertical stretch

Step3: Reflect over the x - axis

Step4: Apply horizontal translation

Step5: Apply vertical translation

Step1: Recall the parent function

Step2: Reflect over the y - axis

Step3: Apply vertical compression

Step4: Apply horizontal translation

Answer:

Question 8