Question

Y = |x+1| +3 * |x-2|

Answer 1

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Answer 2

To understand the graph of the function $Y = |x+1| +3 * |x-2|$, we'll break it down into different intervals based on the absolute value expressions. The graph will have different behaviors depending on the sign of $x+1$ and $x-2$.

1. For $x < -1$:

   • Both $x+1$ and $x-2$ are negative, so their absolute values become positive.
   • The function becomes $Y = -(x+1) + 3(-(x-2)) = -x - 1 - 3x + 6 = -4x + 5$.

2. For $-1 \leq x < 2$:

   • $x+1$ is positive, but $x-2$ is negative.
   • The function becomes $Y = (x+1) + 3(-(x-2)) = x + 1 - 3x + 6 = -2x + 7$.

3. For $x \geq 2$:

   • Both $x+1$ and $x-2$ are positive, so their absolute values remain unchanged.
   • The function becomes $Y = (x+1) + 3(x-2) = x + 1 + 3x - 6 = 4x - 5$.

Now let's plot these three segments of the function:

1. For $x < -1$:

   • $Y = -4x + 5$

2. For $-1 \leq x < 2$:

   • $Y = -2x + 7$

3. For $x \geq 2$:

   • $Y = 4x - 5$

Since the function involves absolute values, it will have sharp turns (corner points) at $x = -1$ and $x = 2$. To find the behavior around these points, we'll evaluate the function at those points.

For $x = -1$: $Y = |(-1) + 1| + 3 * |(-1) - 2| = |0| + 3 * |-3| = 0 + 9 = 9$

For $x = 2$: $Y = |2 + 1| + 3 * |2 - 2| = |3| + 3 * |0| = 3 + 0 = 3$

Now, let's plot these points on the graph:

At $x = -1$: ( -1, 9 ) At $x = 2$: ( 2, 3 )

With this information, we can sketch the graph of the function $Y = |x+1| + 3 * |x-2|$ as follows:

perlCopy code
      |            /
      |           /
      |          /
  Y   |         /
  ^   |        / 
  |   |       /
  9   |______/______/______/______
      -1     2

The graph will look like a "V" shape with a sharp corner point at $x = -1$ and $x = 2$. The minimum point is at $x = -1$ with a value of $Y = 9$, and the graph approaches positive infinity as $x$ approaches negative infinity and approaches negative infinity as $x$ approaches positive infinity.