I'm trying to describe how the graph is related to the parent function.
The parent function is x^3
and the problem is:
y=-(0.8x)^3
I know its a reflection over the x-axis, but what is the formula for how much it expands or shrinks?
Like on another problem it's
y=|x|
and y= |0.2x|
The answer is that it expands by 5, but I'm not exactly sure how to get it.
Thanks.
-(0.8x)^3 is stretching in the x-axis by a factor of (1/0.8) and as you noted, a reflection about the x-axis.
When ever the function f(x) is mapped to another by f(kx), it is stretching or contracting by a factor of (1/k). (1/k)>1 is a stretching, and (1/k)<1 is contracting.
The stretching factor (1/k) applies to the x-direction only.
To describe how the graph is related to the parent function y = x^3, we need to analyze the given function y = -(0.8x)^3.
First, let's consider the effect of the coefficient 0.8 on the x-values. The absolute value of this coefficient determines how the graph is horizontally expanded or compressed.
In this case, 0.8 is less than 1, indicating that the graph is horizontally compressed relative to the parent function y = x^3. To find the amount of compression, you can take the reciprocal of the coefficient, which gives you 1/0.8 = 1.25.
This means that the graph is horizontally compressed by a factor of 1.25, or it is squeezed towards the y-axis. In comparison to the parent function, the graph appears narrower.
Now, let's consider the negative sign in front of the function. It indicates that the graph is reflected over the x-axis. So, the entire graph is flipped upside down.
Combining the two transformations, the given function y = -(0.8x)^3 is a vertically flipped and horizontally compressed version of the parent function y = x^3.
Regarding your example with y = |x| and y = |0.2x|, to determine the expansion or shrinkage factor, you compare the absolute values of the coefficients.
In this case, the coefficient of the function y = |0.2x| is 0.2, which is smaller than 1. To find the amount of shrinkage, you can take the reciprocal of the coefficient, which gives you 1/0.2 = 5.
This indicates that the graph y = |0.2x| is horizontally expanded by a factor of 5 relative to the graph y = |x|. In other words, it appears five times wider than the original graph.
I hope this helps! Let me know if you have any further questions.
To describe how the graph of a function is related to its parent function, you need to analyze how changes to the equation affect the shape and position of the graph. Let's break down the given examples:
1. Parent function: y = x^3. The graph of the parent function is a cubic curve.
2. Given equation: y = -(0.8x)^3. Here, we have a similar cubic function as the parent function, but with an additional reflection and scaling.
To understand the effect of the reflection over the x-axis, note that the negative sign in front of (0.8x)^3 reflects the graph vertically. A positive sign would produce the same graph as the parent function, while a negative sign reflects it over the x-axis.
Regarding the scaling or expansion/shrinking, it depends on the coefficient in front of x. In this case, it is 0.8.
To determine the scaling factor, you need to compare the coefficients of x in the given equation and the parent function. In the parent function, the coefficient of x is 1, while in the given equation, it is 0.8. The scaling factor can be determined by comparing how many times the coefficient in the parent function needs to be multiplied by to yield the coefficient in the given equation.
In the example y = |x| versus y = |0.2x|, the coefficient in the parent function is 1, while in the given equation, it is 0.2.
To find the scaling factor, divide the coefficient in the given equation by the coefficient in the parent function:
0.2 / 1 = 0.2
So, y = |0.2x| is obtained by multiplying the parent function's coefficient by 0.2.
In this case, it shrinks the graph by a factor of 0.2. You can also look at scaling as expanding or shrinking by the reciprocal of the scaling factor. In the example, 1 / 0.2 is equal to 5, which means the graph is expanded by a factor of 5.
To summarize, in the given equation y = -(0.8x)^3:
- The graph is reflected over the x-axis due to the negative sign.
- The graph is scaled down or shrunk by a factor of 0.8 compared to the parent function y = x^3.