Given the following transformations 𝑦 = 3𝑓(− 2𝑥 + 8) − 6 to the parent function 𝑓(𝑥) = 𝑙𝑜𝑔(small2)(x), describe (in words) how to determine the following key features of the transformed function without

Question

Given the following transformations 𝑦 = 3𝑓(− 2𝑥 + 8) − 6 to the parent function 𝑓(𝑥) = 𝑙𝑜𝑔(small2)(x), describe (in words) how to determine the following key features of the transformed function without

graphing:

Domain:
Range:
x-Int:
y-int:
Asymptotes:

Please help me.

Answer 1

f(x) = log_2(x)

domain: x > 0
range: all reals
intercepts (1,0)
asymptotes: x=0

3f(-2x+8)=3f(-2(x-4)) is f(x)
shifted right by 4
reflected across the y-axis
compressed horizontally by 2
stretched vertically by 3
So that gives us
domain: (-2x+8) > 0 or x < 4
range: all reals
intercepts -2x+8=1, or (3.5,0) and (0,9)
asymptotes: x=4

see the graph at

https://www.wolframalpha.com/input/?i=3log_2%28-2x%2B8%29

Answer 2

oops. I forgot the downward shift of 6.

You can probably figure out what that affects., with the help of

https://www.wolframalpha.com/input/?i=3log_2%28-2x%2B8%29-6

Answer 3

To determine the key features of the transformed function without graphing, we will break down the given transformations step-by-step.

1. Domain:
Start with the domain of the parent function, which is all real numbers. In this case, there are no transformations applied directly to the variable x, so the domain of the transformed function remains the same as the parent function, which is also all real numbers.

2. Range:
To determine the range of the transformed function, we need to look at the range of the parent function and the applied transformations. The parent function, f(x) = log2(x), has a range of all real numbers. The transformation 3f(-2x + 8) - 6 does not affect the range because the logarithm function does not have any restrictions on the output values. Therefore, the range of the transformed function is also all real numbers.

3. x-Intercepts:
To find the x-intercepts, we need to set y = 0 and solve for x. In this case, we have the equation 3f(-2x + 8) - 6 = 0. To find the x-intercepts of the transformed function, we need to solve this equation.

4. y-Intercept:
To find the y-intercept, we set x = 0 and evaluate the function. Plugging in x = 0 into 𝑦 = 3𝑓(− 2𝑥 + 8) − 6, we obtain 𝑦 = 3𝑓(8) − 6. To find the y-intercept of the transformed function, we need to evaluate this expression.

5. Asymptotes:
The parent function, f(x) = log2(x), has a vertical asymptote at x = 0. The transformation 3f(-2x + 8) - 6 does not affect the vertical asymptote as it only translates and scales the function. Therefore, the vertical asymptote of the transformed function remains at x = 0.

Note: If there are any horizontal or slant asymptotes in the parent function, or if there are any transformations that change the shape of the function, they would affect the asymptotes of the transformed function. However, based on the given information, we do not have enough information to determine if there are any horizontal or slant asymptotes.

Answer 4

To determine the key features of the transformed function without graphing, we need to understand the effects of each transformation on the parent function.

The parent function is f(x) = log2(x). First, let's analyze the transformations applied to it:

1. Vertical Stretch/Shrink: The factor 3 in front of f(-2x + 8) tells us there is a vertical stretch of 3 in the y-direction. If the factor were less than 1, it would be a vertical shrink.

2. Horizontal Compression/Stretch: The expression -2x + 8 inside f tells us there is a horizontal compression or stretch. The horizontal shift is given by -2, but since it is inside the logarithmic function, it means the graph is horizontally stretched by a factor of 2 compared to the parent function. The "+ 8" indicates a horizontal shift to the right by 8 units.

3. Vertical Shift: The -6 at the end tells us there is a vertical shift downward by 6 units.

Let's now determine the key features of the transformed function:

1. Domain: The domain of the parent function f(x) = log2(x) is all positive real numbers (x > 0). Since the transformations applied do not alter the domain, the domain of the transformed function remains the same: all positive real numbers (x > 0).

2. Range: The range of the parent function f(x) = log2(x) is all real numbers. The vertical stretch or shrink and the vertical shift do not affect the range, so the range of the transformed function is still all real numbers.

3. x-Intercept: The x-intercept of the parent function f(x) = log2(x) occurs when y = 0. Solving log2(x) = 0, we find x = 1 as the x-intercept. However, the transformations applied do not alter the x-intercept, so the x-intercept of the transformed function is also x = 1.

4. y-Intercept: The y-intercept of the parent function f(x) = log2(x) occurs when x = 1. Substituting x = 1, we find y = 0 as the y-intercept. Again, the transformations applied do not change the y-intercept, so the y-intercept of the transformed function remains y = 0.

5. Asymptotes: The parent function f(x) = log2(x) has a vertical asymptote at x = 0 and no horizontal asymptote. The transformations do not affect the presence of asymptotes, so the transformed function also has a vertical asymptote at x = 0 and no horizontal asymptote.

By understanding the effects of each transformation on the parent function, we can determine the key features (domain, range, x-intercept, y-intercept, and asymptotes) of the transformed function without graphing.