Sure! I can help you with these problems. Let's go through each problem step by step and explain how to get the answer.
1. f(x) = x; translated right 3 units:
To translate the function f(x) = x right 3 units, you need to subtract 3 from the x-coordinate of every point on the graph. This means that the value of x gets replaced with (x - 3). So the function g(x) with this transformation would be g(x) = (x - 3).
2. f(x) = x^2; vertical stretch by 5, translated down 3 units, and reflected across the x-axis:
To vertically stretch a function, you multiply the y-coordinate of every point on the graph by the vertical stretch factor. In this case, the vertical stretch factor is 5, so you multiply every y-coordinate by 5. To translate the function down 3 units, you subtract 3 from the y-coordinate of every point. Finally, to reflect the function across the x-axis, you negate the y-coordinates. So putting it all together, the function g(x) with these transformations would be g(x) = -5(x^2) - 3.
3. f(x) = |x|; vertical shrink by 1/4, right 1 unit, and up 2 units:
To vertically shrink a function, you multiply the y-coordinate of every point on the graph by the vertical shrink factor. In this case, the vertical shrink factor is 1/4, so you multiply every y-coordinate by 1/4. To translate the function right 1 unit, you add 1 to the x-coordinate of every point. And to translate the function up 2 units, you add 2 to the y-coordinate of every point. So the function g(x) with these transformations would be g(x) = (1/4)|x - 1| + 2.
4. f(x) = -(x - 4)^2; vertical stretch by a factor of 3, translated up 2, and right 8:
To vertically stretch a function, you multiply the y-coordinate of every point on the graph by the vertical stretch factor. In this case, the vertical stretch factor is 3, so you multiply every y-coordinate by 3. To translate the function up 2 units, you add 2 to the y-coordinate of every point. Finally, to translate the function right 8 units, you subtract 8 from the x-coordinate of every point. So the function g(x) with these transformations would be g(x) = 3(-(x - 8 - 4)^2) + 2.
Please note that in some steps, I simplified the expressions by performing necessary calculations. Make sure to carefully follow the sequence of transformations and simplify the expression as needed.