Let's go through the problems in this image:

Problems 1-4: Comparing Graphs of Two Functions

1. $f(x) = x$, $g(x) = -2x + 3$

   • $g(x)$ is a linear function as well, but it is a reflection of $f(x)$ across the x-axis (due to the negative sign) and a vertical stretch by a factor of 2. Additionally, it's shifted upward by 3 units.

2. $f(x) = |x|$, $g(x) = -\frac{1}{2}|x| + 3$

   • $g(x)$ involves a vertical compression of $f(x)$ by a factor of $\frac{1}{2}$ and reflection across the x-axis. Then, it's shifted up by 3 units.

3. $f(x) = x^2$, $g(x) = 3(x + 4)^2 - 1$

   • The graph of $g(x)$ is the graph of $f(x) = x^2$ shifted to the left by 4 units (due to $+4$), vertically stretched by a factor of 3, and shifted downward by 1 unit.

4. $f(x) = 3x$, $g(x) = \frac{3}{4}(x - 2)$

   • $g(x)$ involves a horizontal shift to the right by 2 units and a vertical compression by a factor of $\frac{3}{4}$.

Problems 5-8: Identifying Parent Function, Values of $a$, $h$, $k$ and Graphing

For each of the following functions, we'll identify the parent function and values of $a$, $h$, and $k$, and describe the transformations.

5. $y = x^2 - 3$

   • Parent function: $y = x^2$
   • $a = 1$ (no vertical stretch/compression)
   • $h = 0$ (no horizontal shift)
   • $k = -3$ (shifted down by 3 units)

6. $y = -3|x + 2| + 4$

   • Parent function: $y = |x|$
   • $a = -3$ (vertical stretch by 3 and reflection across x-axis)
   • $h = -2$ (shifted to the left by 2 units)
   • $k = 4$ (shifted up by 4 units)

7. $y = \frac{2}{3}(x - 1) + 2$

   • Parent function: $y = x$ (linear)
   • $a = \frac{2}{3}$ (vertical compression by $\frac{2}{3}$)
   • $h = 1$ (shifted to the right by 1 unit)
   • $k = 2$ (shifted up by 2 units)

8. $y = -(3 - x)^2 - 1$

   • Parent function: $y = x^2$
   • $a = -1$ (reflection across x-axis, no vertical stretch/compression)
   • $h = 3$ (shifted to the right by 3 units)
   • $k = -1$ (shifted down by 1 unit)

If you need detailed graphing or further explanation, feel free to ask!

Here are some related questions to explore:

1. How do vertical shifts affect the graph of a function?
2. What is the effect of multiplying a function by a negative number?
3. How can horizontal shifts be identified in function transformations?
4. How do you determine whether a function is stretched or compressed?
5. What is the geometric meaning of reflecting a function across the x-axis?

Tip:

Always pay close attention to the signs of $h$ and $k$ when determining shifts in a function's graph.