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Answer :
The equation of the absolute value function with the given vertex at (1, 3), passing from (-1, 5) and (3, 5) is f(x) = |x - 1| + 3.
Hence, the correct option 3rd.
Given is a graph of an absolute value function, having a vertex at (1, 3), passing from (-1, 5) and (3, 5).
We need to find the equation of the function.
To find the equation of the absolute value function, we need to consider its general form:
f(x) = a |x - h| + k
where (h, k) represents the vertex of the absolute value function.
In this case, the vertex is given as (1, 3), so h = 1 and k = 3.
Additionally, we are given two other points on the graph: descending at (-1, 5) and ascending at (3, 5).
Since absolute value functions are symmetrical around their vertex, the distances from the vertex to these two points are the same.
Therefore, the "a" value will be the slope of the line connecting the vertex to either of the given points.
Step 1: Find the slope "a."
The slope is given by (change in y) / (change in x). We can choose either of the given points to calculate the slope:
Slope =
[tex]\frac {(5 - 3)}{(3 - 1)}\\\\ = \frac{2}{2}\\\\ = 1[/tex]
Step 2: Write the equation using the vertex and slope:
Now, we have the values of a, h, and k:
a = 1
h = 1
k = 3
The equation of the absolute value function is:
f(x) = 1 × |x - 1| + 3
Simplifying further:
f(x) = |x - 1| + 3
This is the required equation.
Hence the correct option 3rd.
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