Part A: Given the function g(x) = |x − 5|, describe the graph of the function, including the vertex, domain, and range. (5 points)
Part B: If the parent function f(x) = |x| is transformed to h(x) = |x| + 3, what transformation occurs from f(x) to h(x)? How are the vertex and range of h(x) affected?
Part A:
The graph of the function g(x) = |x − 5| is a V-shaped graph with its vertex at (5,0). It opens upwards and has a y-intercept of (0,5). The domain of g(x) is all real numbers, because you can substitute any real value for x. The range of g(x) is all non-negative numbers, because the absolute value of a number is always non-negative.
Part B:
In the transformation of the parent function f(x) = |x| to h(x) = |x| + 3, a vertical shift occurs. The transformation is a vertical shift upwards by 3 units. The vertex of h(x) is shifted from (0,0) to (0,3). The range of h(x) is also affected by the vertical shift and becomes all non-negative numbers greater than or equal to 3.
Part A:
To describe the graph of the function g(x) = |x − 5|, let's start with the absolute value function f(x) = |x|.
The absolute value function f(x) = |x| is a V-shaped graph, with the vertex at the origin (0, 0). It consists of two straight lines meeting at the vertex and extending indefinitely in opposite directions. The graph is symmetric with respect to the y-axis.
Now, let's consider the function g(x) = |x − 5|. This function is a transformation of the absolute value function f(x) = |x|.
The transformation shifts the vertex horizontally to the right by 5 units, resulting in the vertex being at (5, 0). The graph of g(x) is still V-shaped, but it is centered around x = 5 instead of the origin.
For the domain of g(x), any real number can be substituted into the function, so the domain is (-∞, ∞).
For the range of g(x), the absolute value function always gives a non-negative output, so the range of g(x) is [0, ∞).
Therefore, the graph of g(x) = |x − 5| has a V-shape centered at (5, 0), with a domain of (-∞, ∞) and a range of [0, ∞).
Part B:
If the parent function f(x) = |x| is transformed to h(x) = |x| + 3, the transformation that occurs is a vertical shift upward by 3 units.
The "+3" in the function h(x) adds a constant value of 3 to the output of the absolute value function, which means that the entire graph of h(x) is shifted upward by 3 units.
The vertex of h(x) will have the same x-coordinate as the vertex of the parent function f(x), which is (0, 0). However, the y-coordinate of the vertex will change due to the vertical shift, resulting in the vertex of h(x) being at (0, 3).
In terms of the range, the range of f(x) is [0, ∞), as the absolute value function always gives non-negative values. Adding a constant value does not affect this range, so the range of h(x) will also be [0, ∞).
To summarize, the transformation from f(x) = |x| to h(x) = |x| + 3 is a vertical shift upward by 3 units. The vertex of h(x) is at (0, 3), and the range of h(x) is [0, ∞).