When we look at the graphs of different functions, we see how their shapes change based on the kind of function. Let’s break down some important types:

1. Linear Functions: These functions look like this: $y = mx + c$. Here, $m$ is the slope, and $c$ is where the line crosses the y-axis. The graph of a linear function is a straight line. The slope shows how steep the line is.

   For example:

   • The graph of $y = 2x + 1$ goes up steeply, which means it has a positive slope.
   • On the other hand, $y = -0.5x + 3$ goes down, which means it has a negative slope.

2. Quadratic Functions: These functions are written like $y = ax^2 + bx + c$. The graph of a quadratic function makes a U-shaped curve. This curve is called a parabola.

   Depending on the value of $a$:

   • If $a$ is positive (greater than 0), the U opens up.
   • If $a$ is negative (less than 0), the U opens down.

   For instance:

   • $y = x^2$ opens upwards.
   • $y = -x^2$ opens downwards.

3. Cubic Functions: These functions look like this: $y = ax^3 + bx^2 + cx + d$. The graphs of cubic functions can twist and turn, forming more complex shapes.

   An example is $y = x^3 - 3x$. This graph can change direction and might cross the x-axis at several points.

4. Exponential Functions: These functions are written as $y = a \cdot b^x$, where $b$ is greater than 0. The graph of an exponential function grows very fast for positive values of $x$ and gets closer to zero as $x$ gets smaller.

   A common example is $y = 2^x$. This graph climbs steeply.

By drawing these different types of functions, we can see how they create their unique shapes. This helps us understand math in a deeper way!