Question

question 1-3 select values from an absolute value function are shown
| x | f(x)
| 0 | -1
| 2 | 0
| 4 | 1
| 6 | 0
| 8 | -1
| 10 | -2
| 12 | -3
what are the coordinates of the vertex?
○ (0, -1)
○ (2, 0)
○ (6, 0)
○ (4, 1)

Explanation:

Step1: Recall vertex properties

For an absolute value function (a V - shaped graph), the vertex is the point where the function changes direction (from decreasing to increasing or vice - versa). In a table of values, we look for the point where the function value is the maximum (if opening down) or minimum (if opening up) and also the point of symmetry.

Step2: Analyze the table values

Looking at the table:

• When \(x = 0\), \(f(x)=- 1\); \(x = 2\), \(f(x) = 0\); \(x = 4\), \(f(x)=1\); \(x = 6\), \(f(x)=0\); \(x = 8\), \(f(x)=-1\); \(x = 10\), \(f(x)=-2\); \(x = 12\), \(f(x)=-3\)
• The function value \(f(x)\) increases from \(x = 0\) to \(x = 4\) (since \(f(0)=-1\), \(f(2) = 0\), \(f(4)=1\)) and then decreases from \(x = 4\) to \(x = 12\) ( \(f(4)=1\), \(f(6)=0\), \(f(8)=-1\), etc.). Also, the function is symmetric about \(x = 4\) (the value at \(x = 0\) and \(x = 8\) is \(-1\), the value at \(x = 2\) and \(x = 6\) is \(0\)). The maximum value of \(f(x)\) in the given table occurs at \(x = 4\) where \(f(x)=1\). So the vertex (the point of maximum or minimum and the point of symmetry) has coordinates \((4,1)\)

Answer:

\((4,1)\)