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:
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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 \( f(h) = -5 \).
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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.
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Determine \( k \) with the Vertex: Since the vertex is at \((h, -5)\), we can identify \( k \) as \(-5\). Thus, the function simplifies to \( f(x) = |x - h| - 5 \).
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Set Up Equations with Given Points:
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For the point \((-6, -2)\): \[ f(-6) = |(-6) - h| - 5 = -2 \ | -6 - h | - 5 = -2 \ | -6 - h | = 3 \]
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For the point \((0, -2)\): \[ f(0) = |0 - h| - 5 = -2 \ |0 - h| - 5 = -2 \ | -h | = 3 \]
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Solve the Absolute Value Equations:
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From the equation \( | -6 - h | = 3 \): \[ -6 - h = 3 \quad \Rightarrow \quad -h = 9 \quad \Rightarrow \quad h = -9 \ \text{or} \ -6 - h = -3 \quad \Rightarrow \quad -h = 3 \quad \Rightarrow \quad h = -3 \]
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From the equation \( | -h | = 3 \): \[ -h = 3 \quad \Rightarrow \quad h = -3 \ \text{or} \ -h = -3 \quad \Rightarrow \quad h = 3 \]
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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 \((-3, -5)\).
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