The function $f(x) = px^2 - x + 3$ is a quadratic equation, and we are told that its graph does not touch or cross the x-axis. This means that the quadratic has no real solutions, and thus its discriminant must be negative.

For a quadratic equation of the form $ax^2 + bx + c = 0$, the discriminant is given by:

$\Delta = b^2 - 4ac$

In this case:

• $a = p$,
• $b = -1$,
• $c = 3$.

Substitute these values into the discriminant formula:

$\Delta = (-1)^2 - 4(p)(3) = 1 - 12p$

For the quadratic to have no real solutions, we need $\Delta < 0$:

$1 - 12p < 0$

Solve this inequality:

$1 < 12p$

$p > \frac{1}{12}$

Thus, the function $f(x) = px^2 - x + 3$ does not touch or cross the x-axis when $p > \frac{1}{12}$.

Would you like more details or have any questions?

Here are 5 related questions you might want to explore:

1. What happens to the function when $p = \frac{1}{12}$?
2. How do we find the vertex of a quadratic function?
3. What is the significance of the discriminant in quadratic equations?
4. How does changing the value of $p$ affect the shape of the parabola?
5. What are the x-intercepts of a quadratic equation?

Tip: The discriminant is a useful tool for quickly determining the nature of the roots of any quadratic equation.