You have provided the graph of the function $f(x)$, and you're looking to graph $h(x) = 2f(x) - 1$.

Transformation Breakdown:

1. Vertical Stretch by a factor of 2: The factor of 2 in $2f(x)$ stretches the graph of $f(x)$ vertically by multiplying all $y$-values by 2.
2. Vertical Shift down by 1: The term $-1$ shifts the graph down by 1 unit.

Applying Transformations:

• Start with $f(x)$. The given graph seems to be piecewise linear, with key points:
  • $(-3, 0)$
  • $(-2, 2)$
  • $(0, 2)$
  • $(2, 2)$
  • $(3, 0)$

For each key point:

• Stretch by 2: Multiply the $y$-coordinate by 2.
• Shift down by 1: Subtract 1 from the $y$-coordinate.

Let's calculate the new points for $h(x) = 2f(x) - 1$:

So, the points on the graph of $h(x)$ are:

• $(-3, -1)$
• $(-2, 3)$
• $(0, 3)$
• $(2, 3)$
• $(3, -1)$

These points outline the new shape of $h(x)$.

Would you like more details or an explanation of how to plot this? Here are a few related questions:

1. How do you identify key points on a piecewise graph?
2. How do different transformations (vertical stretch, shrink, translation) affect a graph?
3. Can horizontal shifts and stretches be combined with vertical ones?
4. How would the transformation change if the function was $h(x) = -2f(x) + 3$?
5. How can you graph transformations efficiently without recalculating each point?

Tip: When transforming graphs, always handle stretches first before applying translations.