To plot five points on the graph of the function $f(x) = 3x^2 - 2$, we need to choose strategic values for $x$ and calculate the corresponding $y$-values using the given function. Let's select the point with $x = 0$, two points with negative $x$-values, and two points with positive $x$-values.

The function to be used is: $f(x) = 3x^2 - 2$

Here are some suggested $x$-values and their corresponding $y$-values:

1. $x = 0$
2. $x = -1$
3. $x = 1$
4. $x = -2$
5. $x = 2$

We will now calculate $f(x)$ for these values:

1. $f(0) = 3 \times 0^2 - 2 = -2$
2. $f(-1) = 3 \times (-1)^2 - 2 = 3 \times 1 - 2 = 1$
3. $f(1) = 3 \times 1^2 - 2 = 3 \times 1 - 2 = 1$
4. $f(-2) = 3 \times (-2)^2 - 2 = 3 \times 4 - 2 = 10$
5. $f(2) = 3 \times 2^2 - 2 = 3 \times 4 - 2 = 10$

The points to plot are:

• $(0, -2)$
• $(-1, 1)$
• $(1, 1)$
• $(-2, 10)$
• $(2, 10)$

Plot these on your graph to visualize the function.

Would you like more details on how to choose other points or need help with anything else? Here are some further questions you might consider:

1. How would the graph change if we altered the coefficients in the function $f(x) = 3x^2 - 2$?
2. What are the characteristics of the graph of a quadratic function like $f(x) = 3x^2 - 2$?
3. How do you find the vertex of the quadratic function?
4. Can you explain how the signs of the coefficients affect the orientation of the parabola?
5. What is the importance of plotting points manually in understanding the function's behavior?

Tip: To check the symmetry of the parabola, compare the $y$-values of points equidistant from $x = 0$ on both sides; they should be the same since the function is even.