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To determine the graph of the function \( f(x) = |x - h| + k \) when both \( h \) and \( k \) are positive, let's break down the steps and understand how the parameters \( h \) and \( k \) affect the graph of the absolute value function.
### Basic Graph of \( f(x) = |x| \)
First, consider the basic graph of the function \( f(x) = |x| \). This graph is a V-shaped graph with the vertex at the origin \((0, 0)\) and it opens upwards. The function is defined as:
- \( f(x) = x \) when \( x \geq 0 \)
- \( f(x) = -x \) when \( x < 0 \)
### Transformations Involving \( h \) and \( k \)
1. Horizontal Shift (shifting by \( h \)):
When we introduce \( h \) into the function and rewrite it as \( f(x) = |x - h| \), it shifts the graph horizontally. Specifically:
- The graph is shifted \( h \) units to the right if \( h \) is positive.
- The vertex of the graph moves from \((0, 0)\) to \((h, 0)\).
2. Vertical Shift (shifting by \( k \)):
Adding \( k \) to the function, making it \( f(x) = |x - h| + k \), moves the graph vertically. Specifically:
- The graph is shifted \( k \) units upwards if \( k \) is positive.
- The vertex of the graph moves from \((h, 0)\) to \((h, k)\).
### Combination of Both Shifts
When we combine both transformations, the resulting function \( f(x) = |x - h| + k \) has the following characteristics:
- The graph is shifted \( h \) units to the right.
- The graph is shifted \( k \) units upwards.
- The vertex of the graph is now at the point \((h, k)\).
### Summary of the Graph Properties
Given that both \( h \) and \( k \) are positive, the effect on the graph of \( f(x) = |x - h| + k \) is:
- The entire graph is translated \( h \) units to the right, meaning the vertex moves to the right of the y-axis, at \( h \).
- The entire graph is translated \( k \) units up, meaning the vertex moves up from the x-axis, at \( k \).
Thus, the vertex of the graph will be at the coordinate \((h, k)\). This transformation results in a graph that retains its V-shape, opening upwards and moving asymptotically towards infinity in both directions from the vertex.
So the graph of [tex]\( f(x) = |x - h| + k \)[/tex] when [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are both positive is shifted [tex]\( h \)[/tex] units to the right and [tex]\( k \)[/tex] units up, with a vertex at the point [tex]\((h, k)\)[/tex].
### Basic Graph of \( f(x) = |x| \)
First, consider the basic graph of the function \( f(x) = |x| \). This graph is a V-shaped graph with the vertex at the origin \((0, 0)\) and it opens upwards. The function is defined as:
- \( f(x) = x \) when \( x \geq 0 \)
- \( f(x) = -x \) when \( x < 0 \)
### Transformations Involving \( h \) and \( k \)
1. Horizontal Shift (shifting by \( h \)):
When we introduce \( h \) into the function and rewrite it as \( f(x) = |x - h| \), it shifts the graph horizontally. Specifically:
- The graph is shifted \( h \) units to the right if \( h \) is positive.
- The vertex of the graph moves from \((0, 0)\) to \((h, 0)\).
2. Vertical Shift (shifting by \( k \)):
Adding \( k \) to the function, making it \( f(x) = |x - h| + k \), moves the graph vertically. Specifically:
- The graph is shifted \( k \) units upwards if \( k \) is positive.
- The vertex of the graph moves from \((h, 0)\) to \((h, k)\).
### Combination of Both Shifts
When we combine both transformations, the resulting function \( f(x) = |x - h| + k \) has the following characteristics:
- The graph is shifted \( h \) units to the right.
- The graph is shifted \( k \) units upwards.
- The vertex of the graph is now at the point \((h, k)\).
### Summary of the Graph Properties
Given that both \( h \) and \( k \) are positive, the effect on the graph of \( f(x) = |x - h| + k \) is:
- The entire graph is translated \( h \) units to the right, meaning the vertex moves to the right of the y-axis, at \( h \).
- The entire graph is translated \( k \) units up, meaning the vertex moves up from the x-axis, at \( k \).
Thus, the vertex of the graph will be at the coordinate \((h, k)\). This transformation results in a graph that retains its V-shape, opening upwards and moving asymptotically towards infinity in both directions from the vertex.
So the graph of [tex]\( f(x) = |x - h| + k \)[/tex] when [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are both positive is shifted [tex]\( h \)[/tex] units to the right and [tex]\( k \)[/tex] units up, with a vertex at the point [tex]\((h, k)\)[/tex].
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