Let's go over the graphs and match each equation to its corresponding graph based on their characteristics.

Problem 3:

We have three equations and their graphs:

• $(a) \: y = x^2$
• $(b) \: y = x^5$
• $(c) \: y = x^8$

1. $y = x^2$: This is a basic parabola, symmetric with respect to the y-axis. Its shape is U-shaped and increases as $x$ moves away from 0 in either direction.

   • This graph is the light blue curve in the top graph.

2. $y = x^5$: This is an odd-degree polynomial, meaning it will resemble an "S" shape. For large positive $x$, $y$ increases steeply, and for large negative $x$, $y$ decreases steeply. Near the origin, it remains flatter than $x^2$.

   • This graph is the purple curve in the top graph.

3. $y = x^8$: This is an even-degree polynomial, similar to $x^2$ but steeper for larger values of $x$ and flatter near the origin. Its behavior is that of a sharply rising U-shape.

   • This graph is the pink curve in the top graph.

Problem 4:

Here, we have four equations to match with their corresponding graphs:

• $(a) \: y = 3x$
• $(b) \: y = 3^x$
• $(c) \: y = x^3$
• $(d) \: y = \sqrt{x}$

1. $y = 3x$: This is a linear equation with a slope of 3. The graph will be a straight line increasing at a constant rate.

   • This graph is the black line labeled $g$ in the bottom graph.

2. $y = 3^x$: This is an exponential function. It grows rapidly as $x$ increases and approaches 0 on the left as $x$ decreases.

   • This graph is the yellow curve labeled $F$ in the bottom graph.

3. $y = x^3$: This is an odd-degree polynomial similar to $y = x^5$, with an "S" shape. It increases steeply for positive values and decreases steeply for negative values of $x$, passing through the origin.

   • This graph is the purple curve labeled $f$ in the bottom graph.

4. $y = \sqrt{x}$: This is a square root function. It only exists for $x \geq 0$ and grows slowly for large $x$.

   • This graph is the blue curve labeled $G$ in the bottom graph.

Do you need more details or have any questions?

Related Questions:

1. What are the main differences between odd and even degree polynomials?
2. How does the behavior of exponential functions differ from polynomials?
3. Can you explain why the square root function only exists for non-negative values of $x$?
4. How does the slope of a linear function affect its graph?
5. What are the similarities between the graphs of $y = x^5$ and $y = x^3$?

Tip: When analyzing graphs, always check for symmetry, intercepts, and behavior as $x$ approaches large positive or negative values. These features help identify different functions.