Understanding how functions and their graphs relate is really important. It helps us see how functions behave and what they can do. Here are some key points to remember:

1. Types of Functions

• Linear Functions: These are written as $y = mx + b$. Here, $m$ is the slope, and $b$ is where the line crosses the y-axis. The graph of a linear function is a straight line. Linear functions are often used in real life, like when figuring out budgets or profits.

• Quadratic Functions: These are shown with the formula $y = ax^2 + bx + c$, where $a$ is not zero. The graph looks like a U-shape called a parabola. If $a$ is positive, the parabola opens up. If $a$ is negative, it opens down. Many parabolas have a symmetry around their highest or lowest point.

2. Looking at Graph Features

• Intercepts: You find the x-intercept(s) by making $y=0$, and the y-intercept by setting $x=0$. For example, with the function $y = x^2 - 4$, the x-intercepts are $x = -2$ and $x = 2$.

• Domain and Range: The domain includes all the possible x-values you can use, while the range includes all the possible y-values you can get. For the function $y = \sqrt{x}$, both the domain and the range start from 0 and go up to infinity.

3. How to Graph Functions

• Transformations: You can change how functions look by shifting them, stretching, or squishing them. For example, $y = f(x - 3) + 2$ means you move the graph 3 units to the right and 2 units up.

4. Function Behavior

• Increasing and Decreasing: Looking at where a function increases or decreases helps us understand what it’s doing. An increasing function goes up, and it has a positive slope. A decreasing function goes down and has a negative slope.

By using these ideas, students can better understand how functions connect to their graphs.