Which could be the graph of [tex]f(x)=|x-h|+k[/tex] if [tex]h[/tex] and [tex]k[/tex] are both positive?

The function f ( x ) = ∣ x − h ∣ + k represents an absolute value function.
The graph of the function is V-shaped with the vertex at ( h , k ) .
Since 0"> h > 0 and 0"> k > 0 , the vertex is in the first quadrant.
The graph opens upwards, forming a V-shape in the first quadrant. Therefore, the graph is a V-shaped graph with its vertex in the first quadrant.

Explanation

Understanding the Function We are given the function f ( x ) = ∣ x − h ∣ + k , where 0"> h > 0 and 0"> k > 0 . We need to determine the general shape and location of the graph of this function.

Finding the Vertex The function f ( x ) = ∣ x − h ∣ + k is an absolute value function, which means its graph will be V-shaped. The vertex of the V-shaped graph is the point where the expression inside the absolute value is equal to zero, i.e., x − h = 0 , which gives x = h . At x = h , the value of the function is f ( h ) = ∣ h − h ∣ + k = ∣0∣ + k = k . Therefore, the vertex of the graph is at the point ( h , k ) .

Locating the Vertex Since we are given that 0"> h > 0 and 0"> k > 0 , this means that the coordinates of the vertex ( h , k ) are both positive. Therefore, the vertex of the graph is located in the first quadrant of the coordinate plane.

Describing the Graph The absolute value function opens upwards. Therefore, the graph of f ( x ) = ∣ x − h ∣ + k is a V-shaped graph that opens upwards, with its vertex located in the first quadrant.


Examples
Imagine you are designing a skateboarding ramp. The function f ( x ) = ∣ x − h ∣ + k can model the shape of the ramp, where h represents the horizontal shift and k represents the vertical height. If both h and k are positive, it means the lowest point of the ramp (the vertex) is located in front of you and above the ground. Understanding how h and k affect the ramp's position helps you design a ramp that is both fun and safe to use.

Answered by GinnyAnswer | 2025-07-08

The graph of the function f ( x ) = ∣ x − h ∣ + k is V-shaped, with its vertex at the point ( h , k ) located in the first quadrant because both h and k are positive. This means the graph opens upwards as it extends from the vertex. Understanding the placement of the vertex at ( h , k ) is essential to visualizing how the graph behaves.
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Answered by Anonymous | 2025-07-15