Given the following transformations 𝑦 = 3𝑓(−2𝑥 + 8) − 6 to the parent function 𝑓(𝑥) = 𝑙𝑜𝑔2(𝑥), describe (in words) how to determine the following key features of the transformed function without graphing:
Domain:
Range:
x-Int:
y-int:
Asymptotes:
I'm having a lot of trouble with the x-int for this one. Can't decide whether it's (2, 0) or (3.5, 0).
since log(1) = 0
you want -2x+8 = 1
x = 7/2
Thanks buddy! But I just want to make sure: how come you're not just subbing 0 into the equation to make the x-intercept 2?
Also if you graph 𝑦 = 3log2(−2𝑥 + 8) − 6 on demos, the x-int is 2. What's up with that?
Really sorry to bother everyone but is the y-intercept (0, 3)?
To determine the key features of the transformed function without graphing, we need to understand the effects of each transformation applied to the parent function.
Starting with the given function 𝑦 = 3𝑓(−2𝑥 + 8) − 6, we can break it down step by step:
1. Reflection in the x-axis:
The transformation -2𝑥 reflects the graph of 𝑓(𝑥) = 𝑙𝑜𝑔2(𝑥) in the x-axis. This means that any positive y-values become negative, and vice versa.
2. Horizontal compression/stretch:
The transformation -2𝑥 compresses or stretches the graph horizontally. In this case, since the coefficient is 2 in the expression -2𝑥, it indicates a compression by a factor of 2. This means that the horizontal distance between points on the graph is halved.
3. Horizontal shift:
The transformation -2𝑥 + 8 shifts the graph horizontally to the right by 8 units. This means that any x-values are shifted 8 units to the right.
4. Vertical stretch/compression:
The transformation 3𝑓 stretches or compresses the graph vertically. The coefficient 3 indicates a vertical stretch by a factor of 3. This means that the vertical distance between points on the graph is multiplied by 3.
5. Vertical shift:
Finally, the transformation -6 shifts the graph vertically downward by 6 units.
Now that we understand the effects of each transformation, let's determine the key features of the transformed function:
1. Domain:
The domain remains the same as the parent function 𝑓(𝑥) = 𝑙𝑜𝑔2(𝑥). Logarithmic functions have a domain that excludes non-positive values, so the domain would be all real numbers greater than 0.
2. Range:
The range of the transformed function is affected by the vertical stretch and the vertical shift. Since the vertical stretch multiplies the y-values by 3 and then the vertical shift subtracts 6, the range will be all real numbers less than or equal to -6.
3. x-Intercept:
To find the x-intercept of the transformed function, we need to solve 𝑦 = 0 for 𝑥. In this case, we have:
0 = 3𝑓(−2𝑥 + 8) − 6
Rearranging this equation, we get:
6 = 3𝑓(−2𝑥 + 8)
Solving for 𝑥, we divide both sides by 3 to get:
2 = 𝑓(−2𝑥 + 8)
For the parent function 𝑓(𝑥) = 𝑙𝑜𝑔2(𝑥), the x-intercept is at (1, 0), since the base of the logarithm is positive. However, due to the transformations, the x-intercept of the transformed function will be different. We substitute 𝑓(−2𝑥 + 8) = 𝑙𝑜𝑔2(−2𝑥 + 8) into the equation and solve for 𝑥.
4. y-Intercept:
The y-intercept of the transformed function is obtained by substituting 𝑥 = 0 into the equation 𝑦 = 3𝑓(−2𝑥 + 8) − 6. After simplifying, we can find the y-coordinate of the y-intercept.
5. Asymptotes:
The parent function 𝑓(𝑥) = 𝑙𝑜𝑔2(𝑥) has a vertical asymptote at 𝑥 = 0 and no horizontal asymptote. However, the transformations can affect the asymptotes. In this case, the vertical asymptote will shift to the right by 8 units due to the transformation -2𝑥 + 8. There will still be no horizontal asymptote.
As for your specific question about the x-intercept, it can be challenging to determine it without actually evaluating the equation. It is recommended to substitute the transformed function into a calculator or software to find the precise x-intercept. Remember to apply all the transformations in the correct order and be careful with the signs and coefficients.