identify the translations of the parent function f(x)=|x| that give g(c)=2+|x+3|
Identify the transformations of the parent function f(x) = |x| that give
g(x) = −(3 + |x|).
To find the translations applied to the parent function f(x)=|x| that result in g(c)=2+|x+3|, we need to examine the two functions and identify the transformations applied.
1. Parent function f(x)=|x|:
- The absolute value |x| function represents a "V" shaped graph, which is symmetric with respect to the y-axis.
- It passes through the origin (0,0) and has a slope of 1 for x > 0, and a slope of -1 for x < 0.
2. Transformed function g(c)=2+|x+3|:
- The "+3" inside the absolute value |x+3| represents a horizontal translation of 3 units to the left.
- The "+2" outside the absolute value represents a vertical translation of 2 units upward.
Therefore, the translations applied to the parent function f(x)=|x| to obtain g(c)=2+|x+3| are:
- A horizontal translation of 3 units to the left.
- A vertical translation of 2 units upward.
To find the translations of the parent function f(x) = |x| that give g(c) = 2 + |x + 3|, we need to understand the effect of each translation on the function.
The parent function, f(x) = |x|, represents the absolute value of x. It is a piecewise function defined as f(x) = x for x ≥ 0 and f(x) = -x for x < 0.
Now let's look at the given function, g(c) = 2 + |x + 3|. This function adds 2 to the absolute value function, and it also translates the graph horizontally by shifting it 3 units to the left.
Here's how to break it down step by step:
1. Horizontal Translation:
The expression |x + 3| translates the graph horizontally by shifting it 3 units to the left. It means that the graph of g(c) = |x + 3| will have the points (0, 0) and (-3, 0) coincide.
2. Vertical Translation:
The addition of 2 to the function g(c) = |x + 3| shifts the graph vertically upwards by 2 units. This means that all y-values of the function will increase by 2. For example, the point (0, 0) on the graph of g(c) = |x + 3| will be shifted to (0, 2).
By combining the horizontal and vertical translations, we can determine the overall effect on the parent function f(x) = |x|. The final function g(c) = 2 + |x + 3| will be a graph that is shifted 3 units to the left and 2 units upwards compared to the parent function f(x) = |x|.
I hope this explanation helps you understand how to identify the translations in the given function! Let me know if you have any further questions.