Analyze the following characteristics for each graph.
- Is the graph an absolute value function? If yes, answer the following:
- What is the axis of symmetry \(x=h\)?
- What is the vertical shift \(k\)?
- What is the vertex \((h,k)\)?
- What is the dilation factor and orientation \(a\) from the vertex?
- What is the equation \(y=a|x-h|+k\)?
Graph 1 (Red): Is the graph an absolute value function? Yes
- The axis of symmetry is \(x=0\)
- The vertical shift is \(k=2\)
- The vertex is \((0,2)\)
- The dilation factor is 3 and the function opens up gives \(a = 3\)
- The equation is \(y = 3|x| + 2\)
Graph 2 (Blue): Is the graph an absolute value function? Yes
- The axis of symmetry is \(x=0\)
- The vertical shift is \(k=-2\)
- The vertex is \((0,-2)\)
- The dilation factor is 3 and the function opens down gives \(a = -3\)
- The equation is \(y = -3|x| – 2\)
Graph 3 (Green): Is the graph an absolute value function? Yes
- The axis of symmetry is \(x=-2\)
- The vertical shift is \(k=0\)
- The vertex is \((-2,0)\)
- The dilation factor is 3 and the function opens up gives \(a = 3\)
- The equation is \(y = 3|x+2| \)
Graph 4 (Orange): Is the graph an absolute value function? No, the graph is a piecewise function but not an absolute value function.