Question

The graph of f(x) = |x – h| + k contains the points (–6, –2) and (0, –2). The graph has a vertex at (h, –5). Describe how to find the value of h. Then, explain how this value translates the graph of the parent function.

Answer 1

To find the value of \( h \) in the function \( f(x) = |x - h| + k \) given the points \((-6, -2)\) and \((0, -2)\), along with the vertex at \((h, -5)\), follow these steps:

1. **Identify the Key Characteristics:** From the information given, you know that the vertex of the graph is at \((h, -5)\). This means that the minimum (or maximum) value of the function occurs at \( x = h \). The value of \( f(h) = -5 \).

2. **Use the Given Points:** The points \((-6, -2)\) and \((0, -2)\) both have \( y \)-coordinates of \(-2\). Since the vertex at \((h, -5)\) is below these points, the graph must open upwards.

3. **Determine \( k \) with the Vertex:** Since the vertex is at \((h, -5)\), we can identify \( k \) as \(-5\). Thus, the function simplifies to \( f(x) = |x - h| - 5 \).

4. **Set Up Equations with Given Points:**
- For the point \((-6, -2)\):
\[
f(-6) = |(-6) - h| - 5 = -2 \\
| -6 - h | - 5 = -2 \\
| -6 - h | = 3
\]

- For the point \((0, -2)\):
\[
f(0) = |0 - h| - 5 = -2 \\
|0 - h| - 5 = -2 \\
| -h | = 3
\]

5. **Solve the Absolute Value Equations:**
- From the equation \( | -6 - h | = 3 \):
\[
-6 - h = 3 \quad \Rightarrow \quad -h = 9 \quad \Rightarrow \quad h = -9 \\
\text{or} \\
-6 - h = -3 \quad \Rightarrow \quad -h = 3 \quad \Rightarrow \quad h = -3
\]

- From the equation \( | -h | = 3 \):
\[
-h = 3 \quad \Rightarrow \quad h = -3 \\
\text{or} \\
-h = -3 \quad \Rightarrow \quad h = 3
\]

6. **Determine the Consistent Value:** We found the solutions for \( h \) are \( -9 \) or \( -3 \) from the first equation and either \( -3 \) or \( 3 \) from the second. The common solution that satisfies both conditions is \( h = -3 \).

So, the **value of \( h \) is \(-3\)**.

### Translating the Graph of the Parent Function:
The parent function of \( f(x) = |x| \) is the V-shaped graph of the absolute value function centered at the origin \((0, 0)\). By adjusting \( h \) in \( f(x) = |x - h| + k \):

- **Horizontal Translation:** The \( h \) value creates a **horizontal translation** of the graph. Specifically, replacing \( |x| \) with \( |x + 3| \) translates the graph \( 3 \) units to the left.
- **Vertical Translation:** The \( k \) value (which is \(-5\)) translates the graph **downward** by \( 5 \) units.

Thus, for the specified function \( f(x) = |x + 3| - 5 \), the graph is translated \( 3 \) units to the left and \( 5 \) units down from the position of the parent function. The vertex of the graph is at \((-3, -5)\).