Sketch a graph of \(f(x)=-\frac{1}{2}|x+2|+1\)
Identify the base function
The base function is \(|x|\).
Transformations on the base function
The graph of \(|x|\) is shifted left by 2 units to get \(|x+2|\).
The graph of \(|x+2|\) is vertically compressed by a factor of \(\frac{1}{2}\) to get \(\frac{1}{2}|x+2|\).
The graph of \(\frac{1}{2}|x+2|\) is reflected across the x-axis to get \(-\frac{1}{2}|x+2|\).
The graph of \(-\frac{1}{2}|x+2|\) is shifted upwards by 1 unit to get \(-\frac{1}{2}|x+2|+1\).
Find the vertex
The vertex of \(f(x) = a|x-h|+k\) is at \((h, k)\).
Here, \(f(x)=-\frac{1}{2}|x+2|+1\), so \(h=-2\) and \(k=1\).
The vertex is at \((-2, 1)\).
Find some points on the graph
When \(x = -4\), \(f(-4) = -\frac{1}{2}|-4+2|+1 = -\frac{1}{2}|-2|+1 = -1+1 = 0\).
So, \((-4, 0)\) is a point on the graph.
When \(x = -3\), \(f(-3) = -\frac{1}{2}|-3+2|+1 = -\frac{1}{2}|-1|+1 = -\frac{1}{2}+1 = \frac{1}{2}\).
So, \((-3, \frac{1}{2})\) is a point on the graph.
When \(x = -2\), \(f(-2) = -\frac{1}{2}|-2+2|+1 = 1\). So, \((-2, 1)\) is the vertex.
When \(x = -1\), \(f(-1) = -\frac{1}{2}|-1+2|+1 = -\frac{1}{2}|1|+1 = -\frac{1}{2}+1 = \frac{1}{2}\).
So, \((-1, \frac{1}{2})\) is a point on the graph.
When \(x = 0\), \(f(0) = -\frac{1}{2}|0+2|+1 = -\frac{1}{2}|2|+1 = -1+1=0\).
So, \((0, 0)\) is a point on the graph.
Sketch the graph
Plot the vertex \((-2,1)\) and the points \((-4,0)\), \((-3, 0.5)\), \((-1, 0.5)\) and \((0,0)\) and connect them to make a V shape opening downwards.
\\(\boxed{\text{The graph is a V shape opening downwards, with vertex at }(-2, 1)\text{ and passing through points }(-4, 0), (-3, 0.5), (-1, 0.5), \text{ and }(0,0)}\\)
\\(\boxed{\text{The graph is a V shape opening downwards, with vertex at }(-2, 1)\text{ and passing through points }(-4, 0), (-3, 0.5), (-1, 0.5), \text{ and }(0,0)}\\)
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