Answer :
To understand the graph of the function [tex]\( f(x) = |x - h| + k \)[/tex] where both [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are positive, let’s follow a systematic approach.
### Step-by-Step Analysis
1. Identify the Core Properties of the Graph:
- The function [tex]\( f(x) = |x - h| + k \)[/tex] represents a translated version of the basic absolute value function [tex]\( f(x) = |x| \)[/tex].
- The absolute value function [tex]\( f(x) = |x - h| \)[/tex] has a V-shaped graph with its vertex at [tex]\((h, 0)\)[/tex].
- Adding [tex]\( k \)[/tex] translates the entire graph vertically upwards by [tex]\( k \)[/tex] units.
2. Vertex of the Graph:
- The vertex (lowest point) of the graph [tex]\( f(x) = |x - h| \)[/tex] is at [tex]\((h, 0)\)[/tex].
- By adding [tex]\( k \)[/tex], the vertical translation moves the vertex up to the point [tex]\((h, k)\)[/tex].
3. Shape and Symmetry:
- The graph is symmetric about the vertical line [tex]\( x = h \)[/tex].
- For [tex]\( x \geq h \)[/tex], the graph increases linearly with a slope of 1.
- For [tex]\( x \leq h \)[/tex], the graph increases linearly with a slope of -1.
4. Defining Points and Values:
- Let’s generate some specific points for clarity.
- For [tex]\( h = 1 \)[/tex] and [tex]\( k = 2 \)[/tex] (as mentioned), evaluate the function at a few points:
- At [tex]\( x = h \)[/tex]: [tex]\( f(h) = |h - h| + k = 0 + k = k \)[/tex], the vertex is [tex]\((h, k) = (1, 2)\)[/tex].
- To the Left of [tex]\( h \)[/tex]:
- At [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = |0 - 1| + 2 = 1 + 2 = 3 \)[/tex].
- At [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = |-1 - 1| + 2 = 2 + 2 = 4 \)[/tex].
- To the Right of [tex]\( h \)[/tex]:
- At [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = |2 - 1| + 2 = 1 + 2 = 3 \)[/tex].
- At [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = |3 - 1| + 2 = 2 + 2 = 4 \)[/tex].
5. Plotting the Points:
- The points we evaluated are:
- [tex]\((-1, 4)\)[/tex], [tex]\( (0, 3)\)[/tex], [tex]\( (1, 2)\)[/tex], [tex]\( (2, 3)\)[/tex], [tex]\( (3, 4)\)[/tex]
- These points should be aligned along two linear segments forming a V-shape with the vertex at [tex]\( (1, 2) \)[/tex].
6. Connecting the Dots:
- For [tex]\( x < h \)[/tex] (left of the vertex), draw a straight line passing through the points [tex]\( (-1, 4) \)[/tex], [tex]\( (0, 3) \)[/tex], and approaching [tex]\( (1, 2) \)[/tex].
- For [tex]\( x > h \)[/tex] (right of the vertex), draw a straight line passing through the points [tex]\( (1, 2) \)[/tex], [tex]\( (2, 3) \)[/tex], and [tex]\( (3, 4) \)[/tex].
### Final Graph Description
Given [tex]\( h = 1 \)[/tex] and [tex]\( k = 2 \)[/tex], the graph of [tex]\( f(x) = |x - h| + k \)[/tex]:
- The vertex is at [tex]\((1, 2)\)[/tex].
- The graph is symmetric about [tex]\( x = 1 \)[/tex].
- The function value at the vertex is [tex]\( 2 \)[/tex].
- For [tex]\( x < 1 \)[/tex], the function increases linearly with a slope of [tex]\( -1 \)[/tex].
- For [tex]\( x > 1 \)[/tex], the function increases linearly with a slope of [tex]\( 1 \)[/tex].
Thus, the graph is a V-shaped curve opening upwards, with its vertex shifted to the point [tex]\((1, 2)\)[/tex].
Here is a typical sketch of such a graph:
[tex]\[ \begin{array}{c|ccccc} x & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & 4 & 3 & 2 & 3 & 4 \\ \end{array} \][/tex]
And plotted in a Cartesian coordinate system, it looks approximately like this:
```
4 + / \
| / \
3 +----/ \----
| / \
2 +--(1,2) \
|
1 +
|
+-------------------
-1 0 1 2 3
```
This provides a comprehensive understanding of the graph of the function [tex]\( f(x) = |x - h| + k \)[/tex] when [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are both positive.
### Step-by-Step Analysis
1. Identify the Core Properties of the Graph:
- The function [tex]\( f(x) = |x - h| + k \)[/tex] represents a translated version of the basic absolute value function [tex]\( f(x) = |x| \)[/tex].
- The absolute value function [tex]\( f(x) = |x - h| \)[/tex] has a V-shaped graph with its vertex at [tex]\((h, 0)\)[/tex].
- Adding [tex]\( k \)[/tex] translates the entire graph vertically upwards by [tex]\( k \)[/tex] units.
2. Vertex of the Graph:
- The vertex (lowest point) of the graph [tex]\( f(x) = |x - h| \)[/tex] is at [tex]\((h, 0)\)[/tex].
- By adding [tex]\( k \)[/tex], the vertical translation moves the vertex up to the point [tex]\((h, k)\)[/tex].
3. Shape and Symmetry:
- The graph is symmetric about the vertical line [tex]\( x = h \)[/tex].
- For [tex]\( x \geq h \)[/tex], the graph increases linearly with a slope of 1.
- For [tex]\( x \leq h \)[/tex], the graph increases linearly with a slope of -1.
4. Defining Points and Values:
- Let’s generate some specific points for clarity.
- For [tex]\( h = 1 \)[/tex] and [tex]\( k = 2 \)[/tex] (as mentioned), evaluate the function at a few points:
- At [tex]\( x = h \)[/tex]: [tex]\( f(h) = |h - h| + k = 0 + k = k \)[/tex], the vertex is [tex]\((h, k) = (1, 2)\)[/tex].
- To the Left of [tex]\( h \)[/tex]:
- At [tex]\( x = 0 \)[/tex]: [tex]\( f(0) = |0 - 1| + 2 = 1 + 2 = 3 \)[/tex].
- At [tex]\( x = -1 \)[/tex]: [tex]\( f(-1) = |-1 - 1| + 2 = 2 + 2 = 4 \)[/tex].
- To the Right of [tex]\( h \)[/tex]:
- At [tex]\( x = 2 \)[/tex]: [tex]\( f(2) = |2 - 1| + 2 = 1 + 2 = 3 \)[/tex].
- At [tex]\( x = 3 \)[/tex]: [tex]\( f(3) = |3 - 1| + 2 = 2 + 2 = 4 \)[/tex].
5. Plotting the Points:
- The points we evaluated are:
- [tex]\((-1, 4)\)[/tex], [tex]\( (0, 3)\)[/tex], [tex]\( (1, 2)\)[/tex], [tex]\( (2, 3)\)[/tex], [tex]\( (3, 4)\)[/tex]
- These points should be aligned along two linear segments forming a V-shape with the vertex at [tex]\( (1, 2) \)[/tex].
6. Connecting the Dots:
- For [tex]\( x < h \)[/tex] (left of the vertex), draw a straight line passing through the points [tex]\( (-1, 4) \)[/tex], [tex]\( (0, 3) \)[/tex], and approaching [tex]\( (1, 2) \)[/tex].
- For [tex]\( x > h \)[/tex] (right of the vertex), draw a straight line passing through the points [tex]\( (1, 2) \)[/tex], [tex]\( (2, 3) \)[/tex], and [tex]\( (3, 4) \)[/tex].
### Final Graph Description
Given [tex]\( h = 1 \)[/tex] and [tex]\( k = 2 \)[/tex], the graph of [tex]\( f(x) = |x - h| + k \)[/tex]:
- The vertex is at [tex]\((1, 2)\)[/tex].
- The graph is symmetric about [tex]\( x = 1 \)[/tex].
- The function value at the vertex is [tex]\( 2 \)[/tex].
- For [tex]\( x < 1 \)[/tex], the function increases linearly with a slope of [tex]\( -1 \)[/tex].
- For [tex]\( x > 1 \)[/tex], the function increases linearly with a slope of [tex]\( 1 \)[/tex].
Thus, the graph is a V-shaped curve opening upwards, with its vertex shifted to the point [tex]\((1, 2)\)[/tex].
Here is a typical sketch of such a graph:
[tex]\[ \begin{array}{c|ccccc} x & -1 & 0 & 1 & 2 & 3 \\ \hline f(x) & 4 & 3 & 2 & 3 & 4 \\ \end{array} \][/tex]
And plotted in a Cartesian coordinate system, it looks approximately like this:
```
4 + / \
| / \
3 +----/ \----
| / \
2 +--(1,2) \
|
1 +
|
+-------------------
-1 0 1 2 3
```
This provides a comprehensive understanding of the graph of the function [tex]\( f(x) = |x - h| + k \)[/tex] when [tex]\( h \)[/tex] and [tex]\( k \)[/tex] are both positive.
