Questions: Sketch a graph of: f(x)=-1/2x+1+2

Transcript text: Sketch a graph of: $f(x)=-\frac{1}{2}|x+1|+2$

Solution

Solution Steps

Step 1: Identify the Vertex of the Absolute Value Function

The given function is $ f(x) = -\frac{1}{2} |x + 1| + 2 $. The vertex form of an absolute value function is $ f(x) = a|x - h| + k $, where $(h, k)$ is the vertex. Here, $ h = -1 $ and $ k = 2 $. Therefore, the vertex is $(-1, 2)$.

Step 2: Determine the Slope and Direction

The coefficient of the absolute value term is $-\frac{1}{2}$. This indicates that the graph opens downwards (since the coefficient is negative) and has a slope of $\frac{1}{2}$ on either side of the vertex.

Step 3: Plot Key Points

Using the vertex $(-1, 2)$ and the slope, we can find additional points:

For $ x = 0 $: $ f(0) = -\frac{1}{2} |0 + 1| + 2 = -\frac{1}{2} \cdot 1 + 2 = 1.5 $
For $ x = -2 $: $ f(-2) = -\frac{1}{2} |-2 + 1| + 2 = -\frac{1}{2} \cdot 1 + 2 = 1.5 $
For $ x = 1 $: $ f(1) = -\frac{1}{2} |1 + 1| + 2 = -\frac{1}{2} \cdot 2 + 2 = 1 $
For $ x = -3 $: $ f(-3) = -\frac{1}{2} |-3 + 1| + 2 = -\frac{1}{2} \cdot 2 + 2 = 1 $

Final Answer

Plot the points $(-1, 2)$, $(0, 1.5)$, $(-2, 1.5)$, $(1, 1)$, and $(-3, 1)$ on the graph. Connect these points to form a "V" shape that opens downwards, with the vertex at $(-1, 2)$. The graph should look like this:

  5 | 
  4 | 
  3 | 
  2 |         *
  1 |      *     *
  0 |   *           *
 -1 | 
 -2 | 
 -3 | 
 -4 | 
 -5 | 
    ----------------
    -5 -4 -3 -2 -1 0 1 2 3 4 5

The graph is symmetric about the vertex $(-1, 2)$ and decreases with a slope of $\frac{1}{2}$ on either side.

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