Question text Consider the exponential function [math: f(x)=5x−3]\displaystyle f(x)={5^{x-3}}. a) If [math: y=f(x)] is dilated in the [math: y] direction (from the [math: x] - axis) by a factor of [math: 4]{4} then the new function [math: g] is Note: Type a*b for [math: a×b]a imes b. [math: g(x)=][input] [math: 45x−3] 4\,5^{x-3} b) If [math: y=f(x)] is dilated in the [math: x] direction (from the the [math: y]-axis) by a factor of [math: 16]\displaystyle\frac{1}{{6}} then the new function [math: h] is [math: h(x)=][input] Your last answer was interpreted as follows: 5*This answer is invalid. '*' is an invalid final character in 5* Check Question 5

Question

Question text Consider the exponential function [math: f(x)=5x−3]\displaystyle f(x)={5^{x-3}}. a) If [math: y=f(x)] is dilated in the [math: y] direction (from the [math: x] - axis) by a factor of [math: 4]{4} then the new function [math: g] is Note: Type a*b for [math: a×b]a 	imes b. [math: g(x)=][input] [math: 45x−3] 4\,5^{x-3} b) If [math: y=f(x)] is dilated in the [math: x] direction (from the the [math: y]-axis) by a factor of [math: 16]\displaystyle\frac{1}{{6}} then the new function [math: h] is [math: h(x)=][input] Your last answer was interpreted as follows: 5*This answer is invalid. '*' is an invalid final character in 5* Check Question 5

BlackTom AI · Accepted Answer

Let me restate the problem to ensure we’re examining the right parts. The original function is f(x) = 5^(x−3). There are two subparts:

a) If the function y = f(x) is dilated in the y-direction (vertical) by a factor of 4, what is the new function g(x)? The general rule is: vertical dilation by a factor a multiplies the output by a, so g(x) = a · f(x). Here a = 4, so g(x) = 4 · 5^(x−3).

b) If the function y = f(x) is dilated in the x-direction (horizontal) by a certain factor, what is the new function h(x)? The horizontal dilation rule depends on how the factor is applied. Replacing x with (x / k) in f gives a horizontal stretch by a factor k, while replacing x with (k x) gives a horizontal compression by a factor k. Concretely, h(x) = f(kx) corresponds to a horizontal compression by a factor of k, whereas h(x) = f(x / k) corresponds to a horizontal stretch by a factor k.

Now evaluate the provided candidate for part (b): 5^(6*x−3).

- If h(x) = f(6x), then h(x) = 5^(6x − 3). This corresponds to replacing x with 6x inside the exponent, which is the same as a horizontal compression by a factor of 6 (since the input is multiplied by 6). This is a valid form for a horizontal dilation, but you must be sure that the given factor in the prompt is indeed 6 (as part of the intended dilation). If the factor were 6, then h(x) = f(6x) = 5^(6x − 3) would be correct.

- If the factor were instead 16 or 1/6, the corresponding transformation would look different. For example, a horizontal dilation by a factor of 1/6 would be h(x) = f(6x) as well, producing 5^(6x − 3). A horizontal dilation by 16 would be f(x/16) leading to 5^(x/16 − 3). Generally, the exact form hinges on whether the problem intends x to be scaled by 6 (compression) or by 1/6 (stretch by 1/6) or by some other value.

Given the ambiguity in the prompt’s stated factor, the expression h(x) = 5^(6x − 3) would be the correct form specifically for the case where the horizontal dilation factor applied is 6 (i.e., replacing x with 6x). It would not be correct for a horizontal dilation by a factor of 16, nor for a dilation by a factor of 1/6 interpreted differently unless the convention used in the problem is stated.

Summarizing the reasoning for each option:
- For part (a), the correct form is g(x) = 4 · 5^(x − 3). The candidate "4*5^(x-3)" matches this exactly, since vertical dilation multiplies the output by 4.
- For part (b), the candidate h(x) = 5^(6x − 3) corresponds to a horizontal compression by a factor of 6 (i.e., replacing x with 6x). If the problem indeed meant a factor of 6 for the horizontal dilation, this is correct; if the problem intended a different factor (e.g., 16 or 1/6 interpreted as stretch), then this form would be incorrect. Without a clearly stated factor, we must acknowledge the dependence on the exact value given in the prompt.

In short:
- The first blank should be filled with 4*5^(x−3) for the vertical dilation by 4.
- The second blank would be 5^(6x−3) only if the horizontal dilation factor is 6 (i.e., a compression by 6). If the intended factor is something else, the form would change accordingly.

MCD2130 - T2 - 2025 MCD2130 Test 3

查看解析

登录即可查看完整答案

类似问题

The function passes through the point

The function passes through the point

The exponential function for the following data set is x y 0 1 1 6 2 36 3 216 4 1296

Question text Consider the exponential function [math: f(x)=25x−4+4]\displaystyle f(x)={2^{5\,x-4}+4}. Determine the equation of the horizontal asymptote of [math: y=f(x)]. Enter your answer as [math: y=a], where [math: a] is an integer. [input] Check Question 7

更多留学生实用工具

考试浏览器助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单