To determine the equation of the absolute value graph based on the given values, we can follow these steps:
a) Equation of the Absolute Value Graph
The absolute value graph can generally be represented by the function: \[ y = a |x - h| + k \] where \( (h, k) \) is the vertex of the V-shaped graph, and \( a \) determines the direction and stretch of the graph.
From the given values:
- \( x \) values: 10, 5, -10, -5, 0, 5, 10, -5
- Corresponding \( y \) values: 5, -10, -5, 0, 5, 10
It seems like you might have a symmetrical pattern centered around \( x = 0 \). The y-values seem to hint at a peak at \( y = 10 \) when \( x = 0 \) and decrease symmetrically as \( x \) moves away from 0.
However, to directly form an equation, we need to observe the outputs tied with given inputs:
- Points can be checked for deviations from the central point as follows:
- For \( x = 0 \), the maximum \( y \) is \( 10 \).
- The maximum \( y \) value decreases symmetrically from \( x = 0 \) as \( y = 10 - |x| \).
Therefore, the vertical translation indicates: \[ y = 10 - |x| \]
b) Explanation of How I Determined the Equation
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Identify the Vertex: The maximum \( y \)-value of \( 10 \) occurs at \( x = 0 \). Thus, \( (0, 10) \) is the vertex of the graph.
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Analyze Symmetry: The outputs for negative and positive inputs suggest symmetry. As \( |x| \) approaches 10, \( y \) decreases. When \( |x| = 10 \), \( y \) becomes zero, matching the maximum y-value pattern.
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Determine Function Behavior: The equation must reflect this decreasing nature from \( (0, 10) \) downward, showing that with \( a = -1 \) (the graph opens downwards) and translating down gives rise to the \( 10 - |x| \) form.
Therefore, the final equation of the absolute value graph is: \[ y = 10 - |x| \]
This reflects all observed outputs against inputs accurately, confirming the graph’s formation based on your provided data.