I need help with this question: write an equation g(x) for when the parent function f(x) = |x| is translated one unit to the left and two units up, opening downwards, and vertically stretched by a scale factor of 5. Thank you!!!
original is a V with vertex at origin
translate left:
y = |x+1| (that is 0 at -1)
translate up
y = 2 + |x+1|
stretch
y = 2 + 5|x+1|
turn upside down
y = 2 - 5|x+1|
To find the equation g(x) for the transformation described, you need to apply the following steps:
Step 1: Start with the parent function f(x) = |x|.
Step 2: Translate one unit to the left: To do this, you replace x with (x + 1).
Step 3: Translate two units up: To achieve this, you add 2 to the entire function.
Step 4: Reflect vertically (open downwards): Multiply the entire function by -1.
Step 5: Vertically stretch by a scale factor of 5: Multiply the function by 5.
Combining all these transformations, the equation g(x) is as follows:
g(x) = 5 * -(|x + 1| + 2)
To find the equation g(x) for the transformed function, we will go through the steps one by one.
Step 1: Start with the parent function f(x) = |x|.
Step 2: Translate one unit to the left.
- To translate f(x) one unit to the left, we need to subtract 1 from the x-coordinate.
- Therefore, we have f(x+1) which gives us the translation to the left.
Step 3: Translate two units up.
- To translate f(x+1) two units up, we need to add 2 to the y-coordinate.
- Therefore, we have f(x+1) + 2 which gives us the translation upward.
Step 4: Open it downwards.
- To open the graph downwards, we need to multiply the entire function by -1.
- Therefore, we have -[f(x+1) + 2] as the equation so far.
Step 5: Vertically stretch by a scale factor of 5.
- To vertically stretch the graph by a scale factor of 5, we need to multiply the function by 5.
- Therefore, we have -5[f(x+1) + 2].
Combining all the steps, the equation for g(x) is:
g(x) = -5[f(x+1) + 2]
And since the parent function is f(x) = |x|, we can substitute it into the equation to get the final answer:
g(x) = -5[|x+1| + 2]