To show a function on a graph, we first need to know what a function is.

A function is simply a way to match each input (which we call the $x$-value) to one and only one output (the $y$-value). This special connection is very important when we plot points on a graph.

Step 1: Choose a Function
Let’s look at the function $f(x) = x^2$. This type of function is called quadratic. It has some important parts that we need to know about, like its vertex, axis of symmetry, and intercepts.

Step 2: Identify Key Features

• Vertex: For our function $f(x) = x^2$, the vertex is located at the point $(0, 0)$.
• Axis of Symmetry: The line $x = 0$ is the axis of symmetry. This means that the graph looks the same on both sides of this line.
• Intercepts: The function meets the $y$-axis at $(0, 0)$ and doesn’t touch the $x$-axis anywhere else. It never goes below the $x$-axis.

Step 3: Plotting Points
Now, let’s make a table of values for our function:

| $x$ | $f(x)$ | |-----|---------| | -2 | 4 | | -1 | 1 | | 0 | 0 | | 1 | 1 | | 2 | 4 |

Step 4: Draw the Graph
Take some graph paper and plot the points from your table. Connect these points with a nice smooth curve to create a U-shape. This is what most quadratic functions look like! Remember, as the $x$-values move away from 0 in either direction, the $f(x)$ values will get bigger.

Step 5: Analyze the Graph
Finally, take a look at how the graph shows the important features of the function. You can see that the graph is symmetric around the line and it curves upwards, which is what we expect from the $x^2$ function. Drawing the graph not only helps to see the function, but also makes it easier to understand its main features!