To solve this problem, let's examine the transformation that $y = 2h(x) - 1$ applies to the graph of $y = h(x)$.

1. Vertical Stretch by 2: The term $2h(x)$ means that all $y$-values of $h(x)$ will be doubled. This will stretch the graph vertically by a factor of 2. Points that were at $y = 1$, $y = 2$, and $y = -1$ will now be at $y = 2$, $y = 4$, and $y = -2$, respectively.

2. Vertical Shift Down by 1: The term $-1$ at the end means that all points will be shifted down by 1 unit.

Steps for Transforming Key Points:

1. Identify key points on $h(x)$ and apply the transformations.

   • For example, if a point on $h(x)$ is $(x, y)$, then on $y = 2h(x) - 1$, it will move to $(x, 2y - 1)$.

2. Apply these transformations to the graph:

   • Starting from each original point on $h(x)$, multiply the $y$-coordinate by 2 and subtract 1.

Example of Key Points Transformed:

• If there is a point on $h(x)$ at $(0, 0)$, after the transformations it will remain at $(0, 0)$ because $2 \cdot 0 - 1 = -1$.
• If there is a point at $(2, 2)$, it would transform to $(2, 2 \cdot 2 - 1) = (2, 3)$.
• Similarly, you would apply this to other points on the graph.

Would you like me to provide more specific points and a sketch of the graph after transformation?

Related Questions

1. How does a vertical stretch affect the slope of linear segments on a graph?
2. What happens if the transformation was $y = \frac{1}{2} h(x) - 1$ instead?
3. How does a vertical shift affect the x-intercepts of a graph?
4. What is the effect of multiplying $h(x)$ by a negative number in transformations?
5. How would we transform if we wanted to reflect $h(x)$ across the x-axis?

Tip

When transforming graphs, always start by identifying key points, and apply each transformation in sequence for accuracy.