Describe how the graph of g(x) is related to the graph of ƒ (x) = x^3
.
g (x) = (3x)^3
can you explain in baby steps why the answer is :
horizontal compression of the
graph of f (x) by a factor of _
1/3
Thank you.
f(x) = x^3
g(x) = 3^3 x^3 = 27 x^3
sketch a graph
x f g
0 0 0
1 1 27
2 8 216
3 27 729 etc
at every value of x, the value of g is 27 times that of f
Sure! I'd be happy to explain it to you step-by-step.
Step 1: Start with the equation of the original function, ƒ(x) = x^3. The graph of this function is a simple cubic function.
Step 2: Now, let's consider the function g(x) = (3x)^3. In this equation, we have a horizontal compression factor of 1/3.
Step 3: To understand how this horizontal compression works, let's compare the two functions at a specific value of x. Take a value, say x = 2.
Step 4: For the original function ƒ(x) = x^3, the value of ƒ(2) would be (2)^3, which equals 8.
Step 5: Now, let's find the corresponding value for g(x) = (3x)^3. Plug in the same value of x = 2. The value of g(2) would be (3(2))^3, which simplifies to 6^3, and yields 216.
Step 6: If we compare the two outputs, we can see that g(2) = 216, while ƒ(2) = 8.
Step 7: This means that for each value of x, the corresponding y-value for g(x) will be 27 times larger than the corresponding y-value for ƒ(x).
Step 8: In other words, the function g(x) = (3x)^3 is steeper than the function ƒ(x) = x^3.
Step 9: Since this compression factor of 1/3 is applied horizontally, it means that the graph of g(x) will be compressed toward the y-axis, making it narrower compared to the graph of ƒ(x).
So, in summary, the equation g(x) = (3x)^3 describes a horizontal compression of the graph of the function ƒ(x) = x^3 by a factor of 1/3.
Sure! To understand the relationship between the graph of g(x) = (3x)^3 and the graph of ƒ(x) = x^3, let's break it down in baby steps.
Step 1: Start with the graph of ƒ(x) = x^3.
The graph of ƒ(x) = x^3 is a standard cubic function where the input value x is cubed. This means that as x increases, ƒ(x) also increases. The graph of ƒ(x) is symmetric about the y-axis and passes through the origin (0, 0).
Step 2: Introduce the function g(x) = (3x)^3.
The function g(x) = (3x)^3 is a transformation of ƒ(x). Notice that the only difference between the two functions is the 3x term inside the parentheses. This means that g(x) is obtained by multiplying the input value x by 3 before cubing it.
Step 3: Analyze the effect of the factor 3 on the graph.
Multiplying x by 3 before cubing it will stretch or compress the graph horizontally. When we multiply x by 3, it means that for any given x-value, the new x-value will be three times the original value. This has the effect of "compressing" or "shrinking" the graph horizontally.
Step 4: Calculate the compression factor.
To determine the exact amount of horizontal compression, we compare the original graph of ƒ(x) = x^3 to the transformed graph of g(x) = (3x)^3. Here, g(x) is compressed compared to ƒ(x) because the input values for g(x) are three times smaller than those of ƒ(x).
Therefore, the compression factor is the reciprocal of the multiplier, 1/3. This means that the graph of g(x) is horizontally compressed by a factor of 1/3, which implies that the graph of g(x) is "squished" horizontally, making it narrower compared to the graph of ƒ(x) = x^3.
I hope this explanation helps you understand the relationship between the graph of g(x) = (3x)^3 and the graph of ƒ(x) = x^3.