Transformations Of The Cubic Function: G(x)=(x+2)^3-7

by Alex Johnson 54 views

Let's dive into the fascinating world of function transformations and explore how the graph of $g(x)=(x+2)^3-7$ relates to its parent function, $f(x)=x^3$. Understanding these transformations is key to mastering function behavior and graphing. We'll break down how changes to the basic cubic function shift, stretch, or compress its graph, making it easier to visualize and analyze. The parent function $f(x)=x^3$ is our baseline, a simple yet powerful cubic equation that forms the foundation for more complex functions. Its graph is a smooth, S-shaped curve that passes through the origin (0,0), rises as x increases, and falls as x decreases. The point of inflection at the origin is a crucial characteristic. When we introduce changes to this basic form, we're essentially applying geometric transformations. These transformations can include horizontal shifts (left or right), vertical shifts (up or down), stretches, compressions, and reflections. In the case of $g(x)=(x+2)^3-7$, we are looking at a specific combination of these transformations. The key is to recognize how the structure of the equation directly translates to these graphical changes. By dissecting the equation term by term, we can pinpoint each transformation and understand its effect on the overall shape and position of the graph. This systematic approach allows us to accurately predict and sketch the graph of $g(x)$ without having to plot numerous points. It's like having a secret code that unlocks the mysteries of function behavior. So, let's get ready to decipher the transformations and gain a deeper appreciation for the elegance of mathematical functions.

Understanding Parent Functions and Transformations

The parent function is the simplest form of a particular type of function. For cubic functions, the parent function is $f(x)=x^3$. Its graph is a fundamental shape that we can use as a reference point. All other cubic functions can be seen as transformations of this parent function. Transformations are operations that move, resize, or reflect a graph. They are applied to the parent function to create new graphs. For $g(x)=(x+2)^3-7$, we can identify two main types of transformations: a horizontal shift and a vertical shift. The part of the equation that affects the horizontal position is inside the parentheses with the variable, in this case, $(x+2)$. The part of the equation that affects the vertical position is outside the parentheses, in this case, $-7$. Let's consider the horizontal shift first. When we have $(x-h)$ inside the function, it represents a horizontal shift. If $h$ is positive, the shift is to the right by $h$ units. If $h$ is negative, the shift is to the left by |$h$| units. In our function $g(x)=(x+2)^3-7$, we have $(x+2)$. This can be rewritten as $(x-(-2))$. Therefore, $h = -2$. This means there is a horizontal shift of 2 units to the left. It's important to remember this sign convention: a plus sign inside the parentheses indicates a shift to the left, and a minus sign indicates a shift to the right. Now, let's look at the vertical shift. When we have $+k$ outside the function, it represents a vertical shift. If $k$ is positive, the shift is up by $k$ units. If $k$ is negative, the shift is down by |$k$| units. In our function $g(x)=(x+2)^3-7$, we have $-7$ outside the parentheses. This means $k = -7$. Therefore, there is a vertical shift of 7 units down. Combining these two transformations, the graph of $g(x)=(x+2)^3-7$ is obtained by shifting the graph of $f(x)=x^3$ 2 units to the left and 7 units down. This understanding allows us to accurately sketch the graph of $g(x)$ relative to the familiar graph of $f(x)=x^3$. It’s a powerful tool for visualizing mathematical relationships.

Analyzing the Specific Transformations in $g(x)=(x+2)^3-7$

Now, let's meticulously analyze the specific transformations applied to the parent function $f(x)=x^3$ to obtain $g(x)=(x+2)^3-7$. The general form of a transformed cubic function that includes horizontal and vertical shifts is often represented as $g(x) = a(x-h)^3 + k$. In this form, $a$ controls vertical stretching or compressing and reflections across the x-axis, $h$ controls the horizontal shift, and $k$ controls the vertical shift. For our particular function, $g(x)=(x+2)^3-7$, we can identify the values of $h$ and $k$. First, let's focus on the horizontal shift. The term $(x+2)$ within the parentheses directly influences the x-values. To compare it to the standard $(x-h)$ form, we can rewrite $(x+2)$ as $(x - (-2))$. This reveals that $h = -2$. A negative value for $h$ indicates a shift to the left. Therefore, the graph of $g(x)$ is shifted 2 units to the left compared to the graph of $f(x)=x^3$. This means that a point that was at $(x, y)$ on the graph of $f(x)=x^3$ will now be at $(x-2, y)$ on the graph of $g(x)$ if we were only considering the horizontal shift. However, the entire function's behavior is shifted. The point $(0,0)$ on $f(x)=x^3$ (the origin and point of inflection) will be shifted to $(-2, 0)$ if only the horizontal shift was applied. Next, we examine the vertical shift. The term $-7$ outside the cubed expression is our $k$ value. Since $k = -7$, and negative values of $k$ indicate a downward shift, the graph of $g(x)$ is shifted 7 units down from the graph of $f(x)=x^3$. If we consider the point $(0,0)$ from the parent function, after the horizontal shift it moved to $(-2, 0)$. Now, applying the vertical shift of -7, this point moves to $(-2, -7)$. This point $(-2, -7)$ is the new point of inflection for the graph of $g(x)$. So, to summarize, the graph of $g(x)=(x+2)^3-7$ is the graph of $f(x)=x^3$ shifted 2 units to the left (due to the $+2$ inside the parentheses) and 7 units down (due to the $-7$ outside the parentheses). This precise understanding of how each component of the equation dictates a specific graphical movement is fundamental to mastering function analysis.

Comparing $g(x)$ to $f(x)=x^3$

When we compare the graph of $g(x)=(x+2)^3-7$ to the parent function $f(x)=x^3$, the core task is to identify the horizontal and vertical shifts. As established, the parent function $f(x)=x^3$ has its characteristic S-shape and its point of inflection at the origin $(0,0)$. Any transformation applied to this parent function will alter the position of this graph. The expression $(x+2)$ inside the cubic term $(x+2)^3$ dictates the horizontal movement. In function transformations, a term of the form $(x-h)$ inside the parentheses causes a horizontal shift of $h$ units. If $h$ is positive, the shift is to the right; if $h$ is negative, the shift is to the left. In our case, we have $(x+2)$, which can be written as $(x - (-2))$. This clearly indicates that $h = -2$. Therefore, there is a horizontal shift of 2 units to the left. This means that every x-coordinate on the graph of $f(x)=x^3$ is effectively decreased by 2 to get the corresponding x-coordinate on the graph of $g(x)$. The point that was at $(0,0)$ on $f(x)$ is now at $(-2,0)$ on $g(x)$ if we only consider the horizontal shift. Now, let's consider the vertical shift. The $-7$ term outside the cubic expression, $-7$, dictates the vertical movement. A term of the form $+k$ added to the function causes a vertical shift of $k$ units. If $k$ is positive, the shift is upwards; if $k$ is negative, the shift is downwards. Here, we have $-7$, so $k = -7$. This signifies a vertical shift of 7 units down. This means that every y-coordinate on the graph of $f(x)=x^3$ is decreased by 7 to get the corresponding y-coordinate on the graph of $g(x)$ (after considering the horizontal shift). Applying this vertical shift to the point that was originally at $(0,0)$ and then shifted horizontally to $(-2,0)$, it now moves to $(-2, -7)$. This point $(−2,−7)(-2, -7) is the new point of inflection for $g(x)$. Therefore, the graph of $g(x)=(x+2)^3-7$ is precisely the graph of $f(x)=x^3$ shifted 2 units to the left and 7 units down. This direct relationship between the equation's structure and the graph's position is fundamental in understanding how functions behave and can be manipulated.

Determining the Correct Option

Based on our detailed analysis of the transformations applied to the parent function $f(x)=x^3$ to obtain $g(x)=(x+2)^3-7$, we can now confidently determine the correct option describing this relationship. We identified two primary transformations: a horizontal shift and a vertical shift. The term $(x+2)$ within the parentheses of $g(x)$ indicates a horizontal shift. Rewriting this as $(x - (-2))$, we determined that $h = -2$. According to the rules of function transformations, a negative value for $h$ signifies a shift to the left. Therefore, the graph of $g(x)$ is shifted 2 units to the left compared to $f(x)=x^3$. The $-7$ term outside the parentheses of $g(x)$ represents the vertical shift. A negative value for this term, $k = -7$, indicates a shift down. Thus, the graph of $g(x)$ is shifted 7 units down from the graph of $f(x)=x^3$. Combining these findings, the graph of $g(x)$ is shifted 2 units to the left and 7 units down. Now, let's evaluate the given options:

A. $g(x)$ is shifted 2 units to the right and 7 units down. This option incorrectly states a shift to the right. Our analysis clearly showed a shift to the left due to the $(x+2)$ term.

B. $g(x)$ is shifted 7 units to the right and 2 units up. This option has both the direction and magnitude of the horizontal and vertical shifts incorrect.

There seems to be a missing option in the provided choices that accurately reflects our findings. However, if we were to construct the correct option based on our analysis, it would state:

  • $g(x)$ is shifted 2 units to the left and 7 units down.

Let's re-examine the question and the provided options to ensure no misinterpretation. The question asks how $g(x)=(x+2)^3-7$ compares to $f(x)=x^3$. Our breakdown of $(x+2)$ meaning a shift of 2 units left and $-7$ meaning a shift of 7 units down is standard in function transformation analysis. If we assume there might be a typo in the options, or if the question intends to present options that test a common misunderstanding, we should stick to our derived transformations. Given the standard conventions: $(x+a)$ shifts left by $a$ units, and $(x-a)$ shifts right by $a$ units. $+b$ shifts up by $b$ units, and $-b$ shifts down by $b$ units.

Therefore, $(x+2)$ results in a shift of 2 units to the left. The $-7$ results in a shift of 7 units down.

None of the provided options A or B perfectly match this. However, if we must choose from the given options and assume a potential misunderstanding in how they are phrased or a typo, it's important to be precise. Let's consider the possibility that the options are structured to test the correct components of the shift, even if the phrasing isn't ideal.

Looking again at the options:

A. $g(x)$ is shifted 2 units to the right and 7 units down.

  • Incorrect horizontal shift direction.

B. $g(x)$ is shifted 7 units to the right and 2 units up.

  • Incorrect horizontal and vertical shift directions and magnitudes.

It appears there might be an error in the provided options. However, if we were to select the option that contains partially correct information or is closest to the truth, it would depend on how the question and options are intended to be interpreted. For the purpose of clear mathematical understanding, the correct description is a shift of 2 units left and 7 units down. Let's assume for a moment there was a typo and option A was intended to be '2 units to the left'. In that hypothetical case, it would be partially correct.

However, strictly adhering to the provided options and standard mathematical interpretation, neither option A nor option B is correct. The correct transformation is a shift of 2 units to the left and 7 units down.

Conclusion

In conclusion, by dissecting the equation $g(x)=(x+2)^3-7$, we've rigorously determined that its graph is a transformation of the parent cubic function $f(x)=x^3$. The key lies in understanding how specific terms within the function's equation correspond to geometric shifts on the coordinate plane. The $(x+2)$ component directly governs the horizontal positioning, indicating a shift of 2 units to the left. This is because the general form for a horizontal shift is $(x-h)$, and here $(x+2)$ is equivalent to $(x - (-2))$, making $h = -2$, which signifies a leftward movement. Simultaneously, the $-7$ term, situated outside the cubic expression, dictates the vertical movement, signifying a shift of 7 units down. This is because the general form for a vertical shift is $+k$, and here $k = -7$, denoting a downward displacement. Therefore, the complete transformation is a shift of 2 units left and 7 units down from the original graph of $f(x)=x^3$. It is crucial to correctly identify these components to accurately visualize and analyze the behavior of transformed functions. For further exploration into the fascinating realm of functions and their transformations, you might find the resources at Khan Academy to be incredibly helpful and comprehensive.